Geometry of Algebraic Curves: Volume II

Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics Series editors M. Berge...

Author: Enrico Arbarello | Maurizio Cornalba | Phillip A. Griffiths

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Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics

Series editors M. Berger P. de la Harpe F. Hirzebruch N.J. Hitchin L. Hörmander A. Kupiainen G. Lebeau F.-H. Lin B.C. Ngô M. Ratner D. Serre Ya.G. Sinai N.J.A. Sloane A.M. Vershik M. Waldschmidt Editor-in-Chief A. Chenciner J. Coates S.R.S. Varadhan

268

For further volumes: http://www.springer.com/series/138

Enrico Arbarello Maurizio Cornalba Pillip A. Griffiths

Geometry of Algebraic Curves Volume II with a contribution by Joseph Daniel Harris

Enrico Arbarello Dipartimento di Matematica “Guido Castelnuovo” Università di Roma La Sapienza 00185 Roma Italy [email protected]

Phillip A. Griffiths Institute for Advanced Study Einstein Drive Princeton, NJ 08540 USA [email protected]

Maurizio Cornalba Dipartimento di Matematica “Felice Casorati” Università di Pavia Via Ferrata 1 27100 Pavia Italy [email protected] Contribution by: Joseph D. Harris Department of Mathematics Harvard University One Oxford Street Cambridge, MA 02138 USA

ISSN 0072-7830 ISBN 978-3-540-42688-2 e-ISBN 978-3-540-69392-5 DOI 10.1007/978-3-540-69392-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 84005373 Mathematics Subject Classification (2010): 14xx, 32xx, 30xx, 57xx, 05xx c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: VTeX UAB, Lithuania Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To the memory of Aldo Andreotti

Preface

This volume is devoted to the foundations of the theory of moduli of algebraic curves defined over the complex numbers. The first volume was almost exclusively concerned with the geometry on a fixed, smooth curve. At the time it was published, the local deformation theory of a smooth curve was well understood, but the study of the geometry of global moduli was in its early stages. This study has since undergone explosive development and continues to do so. There are two reasons for this; one predictable at the time of the first volume, the other not. The predictable one was the intrinsic algebro-geometric interest in the moduli of curves; this has certainly turned out to be the case. The other is the external influence from physics. Because of this confluence, the subject has developed in ways that are incredibly richer than could have been imagined at the time of writing of Volume I. When this volume, GAC II, was planned it was envisioned that the centerpiece would be the study of linear series on a general or variable curve, culminating in a proof of the Petri conjecture. This is still an important part of the present volume, but it is not the central aspect. Rather, the main purpose of the book is to provide comprehensive and detailed foundations for the theory of the moduli of algebraic curves. In addition, we feel that a very important, perhaps distinguishing, aspect of GAC II is the blending of the multiple perspectives—algebro-geometric, complex-analytic, topological, and combinatorial—that are used for the study of the moduli of curves. It is perhaps keeping this aspect in mind that one can understand our somewhat unusual choice of topics and of the order in which they are presented. For instance, some readers might be surprised to see a purely algebraic proof of the projectivity of moduli spaces immediately followed by a detailed introduction to Teichm¨ uller theory. And yet Teichm¨ uller theory is needed for our subsequent discussion of smooth Galois covers of moduli, which in turn is immediately put to use in our approach to the theory of cycles on moduli spaces. Besides, all the above are essential tools in Kontsevich’s proof of Witten’s conjecture, which is presented in later chapters. Concerning this, the main motivation of our choice of presenting Kontsevich’s original proof instead of one of the several more recent ones is—in addition to the great beauty of the proof itself—a desire to be as self-contained as possible. This same desire also motivates in part the presence, at the beginning of the book,

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Preface

of two introductory chapters on the Hilbert scheme and on deformation theory. In the Guide for the Reader we will briefly go through the material we included in this volume. Among the topics we did not cover are the theory of Gromov–Witten invariants, the birational geometry of moduli spaces, the theory of moduli of vector bundles on a fixed curve, the theory of syzygies for the canonical curve, the various incarnations of the Schottky problem together with the related theory of theta function, and the theory of stable rational cohomology of moduli spaces of smooth curves. Some of these topics are covered by excellent publications like [14] for syzygies and [532] for the birational geometry of moduli spaces. On other topics, like the intersection theory of cycles or the theory of the ample cone of moduli spaces of stable curves, we limited ourselves to the foundational material. Much of Volume I was devoted to the study of the relationship between an algebraic curve and its Jacobian variety. In this volume there is relatively little emphasis on the universal Jacobian or Picard variety and discussion of the moduli of abelian varieties. The latter is a vast and deep subject, especially in its arithmetic aspect, that goes well beyond the scope of this book. In some instances, important topics, such as the Kodaira dimension of moduli spaces of stable curves, the theory of limit linear series, or the irreducibility of the Severi variety, have appeared elsewhere, specifically in the book Moduli of Curves by Joe Harris and Ian Morrison [352]. This is in fact a good opportunity to thank Joe and Ian for their kind words in the introduction of their book. We believe that our respective books complement each other, and we encourage our readers to benefit from their work. In the bibliographical notes we try to point the reader to the most significant developments, not covered in this volume, of which we were aware at the time of writing. In fact, we view our bibliography and our bibliographical notes as, potentially, an ongoing project. There is virtually no area in the theory of moduli of curves where the contribution of David Mumford has not been crucial. Our first debt of gratitude is therefore owed to him. There is a long list of people to whom we would also like to express our gratitude. The first one is Joe Harris, whose generous contribution consists of approximately half of the exercises in this book. During the long years of preparation of this volume, the following people have greatly contributed with ideas, comments, remarks, and corrections: Gilberto Bini, Alberto Canonaco, Alessandro Chiodo, Herb Clemens, Eduardo Esteves, Domenico Fiorenza, Claudio Fontanari, Jeffrey Giansiracusa, John Harer, Eduard Looijenga, Marco Manetti, Elena Martinengo, Gabriele Mondello, Riccardo Murri, Filippo Natoli, Giuseppe Pareschi, Gian Pietro Pirola, Marzia Polito, Giulia Saccà, Edoardo Sernesi, Roy Smith, Lidia Stoppino, Angelo Vistoli. To all of them we extend our heartfelt sense of gratitude.

Preface

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We also wish to thank the students in the courses that we taught out of draft versions of parts of the book, who also offered a number of suggestions for improvements. The first two authors are also grateful to several institutions which hosted them during the preparation of this volume, in particular the Courant Institute of New York University, Columbia University, the Italian Academy in New York, IMPA in Rio de Janeiro, the Institut Henri Poincaré in Paris, the Accademia dei Lincei in Rome, and above all the Institute for Advanced Study. Special thanks go to Enrico Bombieri, who was instrumental in arranging the first two authors’ stays at the Institute. It was through his good offices that they were supported on one of these stays as “Sergio Serapioni, Honorary President, Società Trentina Lieviti – Trento (Italy) Members.” We gratefully acknowledge financial support provided by the PRIN projects “Spazi di moduli e teoria di Lie” funded by the Italian Ministry for Education and Research, and by the EAGER project funded by the European Union. Rome, Pavia, Princeton, 2010

Contents

Guide for the Reader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii Chapter IX. The Hilbert Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The idea of the Hilbert scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Construction of the Hilbert scheme . . . . . . . . . . . . . . . . . . . . . . . The characteristic system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mumford’s example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variants of the Hilbert scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . Tangent space computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C n families of projective manifolds . . . . . . . . . . . . . . . . . . . . . . . Bibliographical notes and further reading . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 12 19 27 40 43 49 56 64 65

Chapter X. Nodal curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elementary theory of nodal curves . . . . . . . . . . . . . . . . . . . . . . . . Stable curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stable reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isomorphisms of families of stable curves . . . . . . . . . . . . . . . . . . The stable model, contraction, and projection . . . . . . . . . . . . . . Clutching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vanishing cycles and the Picard–Lefschetz transformation . . . Bibliographical notes and further reading . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 83 99 104 113 117 126 127 143 161 161

Chapter XI. Elementary deformation theory and some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 1. 2. 3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Deformations of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Deformations of nodal curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

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4. 5. 6. 7. 8. 9. 10. 11.

The concept of Kuranishi family . . . . . . . . . . . . . . . . . . . . . . . . . . The Hilbert scheme of ν-canonical curves . . . . . . . . . . . . . . . . . . Construction of Kuranishi families . . . . . . . . . . . . . . . . . . . . . . . . The Kuranishi family and continuous deformations . . . . . . . . . The period map and the local Torelli theorem . . . . . . . . . . . . . . Curvature of the Hodge bundles . . . . . . . . . . . . . . . . . . . . . . . . . . Deformations of symmetric products . . . . . . . . . . . . . . . . . . . . . . Bibliographical notes and further reading . . . . . . . . . . . . . . . . . .

187 193 203 212 216 224 242 248

Chapter XII. The moduli space of stable curves . . . . . . . . . . . . . . 249 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Construction of moduli space as an analytic space . . . . . . . . . . Moduli spaces as algebraic spaces . . . . . . . . . . . . . . . . . . . . . . . . . The moduli space of curves as an orbifold . . . . . . . . . . . . . . . . . The moduli space of curves as a stack, I . . . . . . . . . . . . . . . . . . . The classical theory of descent for quasi-coherent sheaves . . . . The moduli space of curves as a stack, II . . . . . . . . . . . . . . . . . . Deligne–Mumford stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Back to algebraic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The universal curve, projections and clutchings . . . . . . . . . . . . . Bibliographical notes and further reading . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249 257 268 274 279 288 294 299 307 309 323 323

Chapter XIII. Line bundles on moduli . . . . . . . . . . . . . . . . . . . . . . . . 329 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Line bundles on the moduli stack of stable curves . . . . . . . . . . . The tangent bundle to moduli and related constructions . . . . . The determinant of the cohomology and some applications . . . The Deligne pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Picard group of moduli space . . . . . . . . . . . . . . . . . . . . . . . . Mumford’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Picard group of the hyperelliptic locus . . . . . . . . . . . . . . . . Bibliographical notes and further reading . . . . . . . . . . . . . . . . . .

329 332 344 347 366 379 382 387 396

Chapter XIV. Projectivity of the moduli space of stable curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 1. 2. 3. 4. 5. 6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A little invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The invariant-theoretic stability of linearly stable smooth curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical inequalities for families of stable curves . . . . . . . . . . Projectivity of moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical notes and further reading . . . . . . . . . . . . . . . . . .

399 400 406 414 425 437

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Chapter XV. The Teichm¨ uller point of view . . . . . . . . . . . . . . . . . . 441 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Teichm¨ uller space and the mapping class group . . . . . . . . . . . . . A little surface topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadratic differentials and Teichm¨ uller deformations . . . . . . . . The geometry associated to a quadratic differential . . . . . . . . . The proof of Teichm¨ uller’s uniqueness theorem . . . . . . . . . . . . . Simple connectedness of the moduli stack of stable curves . . . Going to the boundary of Teichm¨ uller space . . . . . . . . . . . . . . . Bibliographical notes and further reading . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

441 445 453 461 472 479 483 485 497 498

Chapter XVI. Smooth Galois covers of moduli spaces . . . . . . . . . 501 1. 2. 3. 4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Level structures on smooth curves . . . . . . . . . . . . . . . . . . . . . . . . Automorphisms of stable curves . . . . . . . . . . . . . . . . . . . . . . . . . . Compactifying moduli of curves with level structure; a first attempt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Admissible G-covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Automorphisms of admissible covers . . . . . . . . . . . . . . . . . . . . . . 7. Smooth covers of M g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Totally unimodular lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Smooth covers of M g,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Bibliographical notes and further reading . . . . . . . . . . . . . . . . . . 11. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

501 508 515 518 525 536 544 551 556 562 562

Chapter XVII. Cycles in the moduli spaces of stable curves . . 565 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic cycles on quotients by finite groups . . . . . . . . . . . . . . Tautological classes on moduli spaces of curves . . . . . . . . . . . . . Tautological relations and the tautological ring . . . . . . . . . . . . . Mumford’s relations for the Hodge classes . . . . . . . . . . . . . . . . . Further considerations on cycles on moduli spaces . . . . . . . . . . The Chow ring of M 0,P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical notes and further reading . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

565 566 570 573 585 596 599 604 605

Chapter XVIII. Cellular decomposition of moduli spaces . . . . . 609 1. 2. 3. 4. 5. 6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The arc system complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ribbon graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The idea behind the cellular decomposition of Mg,n . . . . . . . . . Uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hyperbolic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

609 613 616 623 624 627

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7. 8. 9. 10.

The hyperbolic spine and the definition of Ψ . . . . . . . . . . . . . . . The equivariant cellular decomposition of Teichm¨ uller space . Stable ribbon graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extending the cellular decomposition to a partial compactification of Teichm¨ uller space . . . . . . . . . . . . . . . . . . . . . 11. The continuity of Ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Odds and ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Bibliographical notes and further reading . . . . . . . . . . . . . . . . . .

636 643 648 652 655 661 665

Chapter XIX. First consequences of the cellular decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 1. 2. 3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The vanishing theorems for the rational homology of Mg,P . . Comparing the cohomology of M g,n to the one of its boundary strata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The second rational cohomology group of M g,n . . . . . . . . . . . . . 5. A quick overview of the stable rational cohomology of Mg,n and the computation of H 1 (Mg,n ) and H 2 (Mg,n ) . . . . . . . . . . . 6. A closer look at the orbicell decomposition of moduli spaces . 7. Combinatorial expression for the classes ψi . . . . . . . . . . . . . . . . 8. A volume computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Bibliographical notes and further reading . . . . . . . . . . . . . . . . . . 10. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

667 670 673 676 683 690 694 699 708 709

Chapter XX. Intersection theory of tautological classes . . . . . . . 717 1. 2. 3. 4. 5. 6. 7. 8.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Witten’s generating series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Virasoro operators and the KdV hierarchy . . . . . . . . . . . . . . . . . The combinatorial identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feynman diagrams and matrix models . . . . . . . . . . . . . . . . . . . . Kontsevich’s matrix model and the equation L2 Z = 0 . . . . . . . A nonvanishing theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A brief review of equivariant cohomology and the virtual Euler–Poincaré characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. The virtual Euler–Poincaré characteristic of Mg,n . . . . . . . . . . . 10. A very quick tour of Gromov–Witten invariants . . . . . . . . . . . . 11. Bibliographical notes and further reading . . . . . . . . . . . . . . . . . . 12. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

717 721 726 729 734 745 750 754 759 766 771 773

Chapter XXI. Brill–Noether theory on a moving curve . . . . . . . 779 1. 2. 3. 4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The relative Picard variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brill–Noether varieties on moving curves . . . . . . . . . . . . . . . . . . Looijenga’s vanishing theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .

779 781 788 796

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5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

The Zariski tangent spaces to the Brill–Noether varieties . . . . The μ1 homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lazarsfeld’s proof of Petri’s conjecture . . . . . . . . . . . . . . . . . . . . The normal bundle and Horikawa’s theory . . . . . . . . . . . . . . . . . Ramification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Hurwitz scheme and its irreducibility . . . . . . . . . . . . . . . . . Plane curves and gd1 ’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unirationality results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical notes and further reading . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

802 808 814 819 835 845 854 863 872 879 885

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945

Guide for the Reader

The first four chapters of this volume, that is, Chapters IX, X, XI, and XII, are devoted to the construction of the moduli space M g,n of stable npointed curves of genus g. The three main characters in these chapters are: nodal curves, deformation theory, and Kuranishi families. Chapter IX gives a self-contained introduction to the Hilbert scheme, explaining the various implications of the concept of flatness and highlighting the case of curves as, for instance, in Mumford’s example. Nodal curves are studied in Chapter X. There, we establish the Stable Reduction Theorem (4.11), the theorem on isomorphisms of families of stable curves (5.1), and, in Section 6, the basic constructions of clutching, projection, and stabilization. All these results are fundamental in the construction of the moduli space of stable curves and in the study of its boundary. The Kodaira–Spencer deformation theory is ubiquitous in this book. Its first appearance is in Section 5 of Chapter IX. It presents itself in its most classical guise as the study of the characteristic system which, in modern terms, translates into the study of the tangent space to the Hilbert scheme. The deformation theory of nodal curves, and in particular of stable ones, is the central theme of Chapter XI. There, (5.10) is the key exact sequence describing the tangent space to the local deformation space of a nodal curve. The concept of Kuranishi family is pivotal in the entire volume. The (bases of) Kuranishi families provide the analytic charts for the atlases of moduli stacks of curves. Kuranishi families are constructed by slicing the Hilbert scheme Hg,n,ν of ν-log-canonical embedded stable n-pointed curves of genus g, transversally with respect to the orbits of the natural projective group acting on Hg,n,ν , and then restricting to these slices the universal family of curves over Hg,n,ν (see Theorem (6.5) and the key Definitions (6.7) and (6.8) in Chapter XI). The moduli space M g,n is then constructed in Chapter XII. We exhibit M g,n first as an analytic space, then as an algebraic space, and finally as an orbifold and as a Deligne–Mumford stack. Actually, one of the purposes of this chapter, besides the construction of moduli spaces, and the study of the first properties of their boundary strata, is to give an utilitarian and essentially self-contained introduction to the theory of stacks. This is done in Sections 3–9.

xviii


Several topics treated in the first four chapters are not directly aimed at the construction of moduli spaces. Specifically: -

-

-

Section 9 of Chapter IX deals with the universal property of the Hilbert scheme with respect to continuous families of projective manifolds. Its natural continuation is Section 7 of Chapter XI, where it is shown that the universal property of the Kuranishi family holds also in this context of continuous families of Riemann surfaces. These results will be essential in our presentation of Teichm¨ uller theory in Chapter XV. Section 9 of Chapter X is devoted to the Picard–Lefschetz theory of vanishing cycles describing the topological picture of a family of smooth curves degenerating to a nodal one. Section 8 of Chapter XI deals with the classical theory of the period map for Riemann surfaces and its infinitesimal behavior. In Section 9 of the same chapter we study the positivity properties of the Hodge bundle from the viewpoint of its curvature. In the final section of Chapter XI we present Kempf’s study of deformations of the symmetric product of a curve leading to the proof of Green’s theorem about quadrics passing through the canonical curve (cf. Theorem (4.1) in Chapter VI).

In Chapter XIII we present the theory of line bundles on moduli stacks of curves, developing the necessary theory of descent. In the first two sections we introduce the Hodge bundle, the point-bundles Li , the tangent bundle to the stack Mg,n , the canonical bundle, the stack divisors corresponding to the codimension one components of its boundary, and the normal bundles to the various boundary strata. The following Section 4 is devoted to the theory of the determinant of the cohomology. This theory is well suited to producing line bundles on moduli stacks, and, at the end of this section, we treat the boundary of moduli as a determinant, leading to important formulae of “restriction to the boundary” as in Lemma (4.22), Proposition (3.10), and formula (4.31). In Section 5 we present the theory of the Deligne pairing, we introduce Mumford’s κ1 class, and we give a concrete version, “without denominators,” of the Riemann–Roch theorem for line bundles on families of nodal curves (cf. Theorem (5.31)). In Section 6 we compare the various notions of Picard group for moduli spaces of curves. Section 7 is devoted to Mumford’s remarkable idea that the Grothendieck–Riemann–Roch theorem can be effectively used to produce relations among classes in the moduli spaces of curves. There we prove the key formula κ1 = 12λ + ψ − δ for Mumford’s class and the formula KM g,n = 13λ+ψ−2δ−δ1,∅ for the canonical class. In the final Section 8 we study the Picard group of the closure H g ⊂ M g of the hyperelliptic locus. The fact that M g,n is a projective variety (and therefore a scheme) is established in Chapter XIV. To prove this we use a mixture of two techniques that are of independent interest. The first one is Mumford’s geometric


xix

invariant theory. In Sections 2 and 3, we prove the Hilbert–Mumford criterion of stability (Proposition (2.2)), and we use this criterion to prove the stability of the ν-log-canonically embedded smooth curves, viewed as points in the appropriate Hilbert scheme. We then take a sharp turn and use stability of smooth curves to find numerical inequalities among cycles in moduli spaces and, consequently, positivity results. Using the same techniques, we then prove the ampleness of Mumford’s class κ1 and hence the projectivity of M g,n . Chapter XV gives a self-contained treatment of Teichm¨ uller space and of the modular group. The Teichm¨ uller space TS is constructed in Section 2, as a complex manifold, by patching together bases of Kuranishi families. We then examine the natural map Φ : B → TS from the unit ball, in the space of quadratic differentials on the reference Riemann surface S, to the Teichm¨ uller space TS . The continuity of this map is an immediate consequence of the results proved in Section 7 of Chapter XI about the universal property of Kuranishi families with respect to continuous families of Riemann surfaces. To prove the injectivity of Φ we first study, in Section 5, the geometry associated to quadratic differentials and then prove, in the following section, Teichm¨ uller’s Uniqueness Theorem. As we explain at the end of Section 4, the fact that Φ is a diffeomorphism follows readily from Teichm¨ uller’s Uniqueness Theorem and from the elementary theory of the Beltrami equation. In the last section of this chapter we introduce a bordification of Teichm¨ uller space which is very close to the one defined in terms of Fenchel–Nielsen coordinates. Although this bordification is interesting in itself, its only use in our book is in Chapter XIX, where we present Kontsevich’s combinatorial expression for the point-bundle classes ψi . Teichm¨ uller space can be thought of as the space representing a rigidification of the moduli functor in which each Riemann surface C comes equipped with a marking (i.e., the homotopy class of a diffeomorphism onto a fixed reference surface). This marking eliminates the automorphism group of C, with the result that Teichm¨ uller space is smooth. The same process of rigidification of the moduli functor can be performed algebraically by considering, for example, pairs consisting of a Riemann surface and the group of points of order n in its Jacobian. More generally, one is looking for finite index normal subgroups Λ of the mapping class group Γg,n . Then Tg,n /Λ is a Galois cover of Mg,n with Galois group H = Γg,n /Λ. In many instances Tg,n /Λ is smooth, so that Mg,n can be represented as the quotient of a smooth variety by a finite group H. The main results in this circle of ideas are proved in the first two sections of Chapter XVI. When trying to naively push the same ideas to prove analogous results regarding M g,n , one encounters significant difficulties. These difficulties are addressed in Section 4, and the way to analyze them is to use the Picard–Lefschetz transformation. The problem of expressing M g,n as a quotient X/H where X is a smooth variety and H a finite group was solved by Looijenga. In the remaining part of the chapter we present a variation of Looijenga construction due to Abramovich, Corti, and

xx


Vistoli which exhibits X as a fine moduli space, in fact as a moduli space for admissible G-covers, where G is an appropriate finite group, and H is a quotient of the semidirect product Gn Aut(G). The fact that M g,n can be expressed as the quotient X/H, with X a smooth variety and H a finite group, makes it relatively easy to talk about its Chow rings. The theory of cycles in M g,n is the central subject of Chapter XVII. After presenting, in Section 2, the foundational material on the intersection theory of stacks of the form [X/H] with X smooth and H a finite group, in Section 3 we introduce the tautological classes. These are the Mumford–Morita–Miller classes (i.e., the κ-classes), the point-bundle classes (i.e., the ψ-classes), the Hodge classes (i.e, the λ-classes), and the boundary classes (i.e., the δ-classes). In Section 4 we describe the behavior of these classes under push-forward and pullback via the projection morphism π : M g,n+1 → M g,n and the clutching morphisms ξΓ : M Γ → M g,n from the various boundary strata. In Section 5, following Mumford, we use, on the one hand, Grothendieck’s Riemann–Roch theorem to find relations between the Hodge classes and the κ classes, and on the other hand, using the flatness of the Gauss–Manin connection, we exhibit a set of generators for the tautological ring R• (Mg ) (i.e., the ring generated by the tautological classes). At the end of the section we discuss Deligne’s canonical extension of the Gauss–Manin connection to the boundary of moduli. Section 6 offers a brief and informal discussion of the tautological ring, presenting two results, a nonvanishing theorem for the tautological class κg−2 due to Faber, and a vanishing theorem for polynomials of degree greater than g − 2 in the tautological classes, due to Looijenga. Both results are proved in subsequent parts of the book. In the last section we present Keel’s result on the Chow ring of M 0,n , and we give a direct computation of A1 (M 0,n ). The fact that Mg,n is a rational K(Γg,n , 1) for the mapping class group Γg,n hints to the possibility of studying Mg,n from a combinatorial point of view. This is done in Chapter XVIII, where we introduce a Γg,n -invariant triangulation of the Teichm¨ uller space Tg,n . Loosely speaking, the complex structure on a Riemann surface determines (and is determined by) a graph embedded in the Riemann surface itself. This makes it possible to give a Γg,n invariant cellular decomposition of Tg,n where the cells are labelled by these graphs. In the first two sections we introduce the arc system complex and, by duality, the ribbon graphs, which are the basic tools for the combinatorial description of Tg,n . To prove that (a subcomplex of) the arc system complex gives a combinatorial model of Tg,n , one may choose either the theory of Jenkins–Strebel differentials, or alternatively, via uniformization, the canonical hyperbolic metric on Riemann surfaces. We choose the latter since it more easily enables one to extend the cellular decomposition to the bordification of Tg,n . After explaining, in Section 4, how hyperbolic geometry is used to obtain the cellular decomposition of Tg,n and after recalling, in Sections 5 and 6, some basic facts about the uniformization theorem and the Poincaré metric, in Sections 7 and 8 we give the construction of the cellular decom-


xxi

position of Tg,n . In this book the cellular decomposition of moduli spaces is used in two ways. First of all to give a simple and direct proof of the vanishing of the rational homology of Mg,n in high degree. These applications are given in Chapter XIX. The second enters when computing the intersection number of tautological classes in Kontsevich’s proof of Witten conjecture, which is given in Chapter XX. In fact, this last application requires that the cellular decomposition of Mg,n be extended to a suitable compactification of moduli space. This task, which is technically more demanding, is carried out in Sections 9–12. Chapter XIX discusses the first consequences of the cellular decomposition constructed in Chapter XVIII. We begin by computing the rational cohomology of M g,n in degrees one and two. This computation can be performed by elementary methods by virtue of the vanishing of the high homology of Mg,n which, in turn, is a direct consequence of the cellular decomposition. This is carried out in Sections 2, 3, and 4. In Section 5, after a very brief discussion of Harer’s stability theorem and of the Madsen–Weiss and Tillmann theorems on the stable rational cohomology of Mg,n , we prove Harer’s theorem on the second homology of Mg,n . This we do by using the knowledge of H 2 (M g,n ; Q) and Deligne’s spectral sequence for the complement of a divisor with normal crossings. Further uses of the cellular decomposition are presented in Section 7, where we give Kontsevich’s combinatorial expression for the point-bundle classes ψ, and in Section 8, where we give Kontsevich’s combinatorial expression for an orientation form on Mg,n . Chapter XX is almost entirely devoted to Kontsevich’s proof of Witten’s conjecture on the intersection numbers of the ψ-classes. The proof is self-contained, with the exception of an algebraic result by Itzykson for which there is a very clear and well-written reference. In the first two sections we review Witten’s generating series for the intersection numbers of the ψ-classes, introduce the Virasoro operators, and describe their link with the KdV hierarchy. In Section 4 we prove Kontsevich’s combinatorial formula expressing Witten’s generating series as a sum over ribbon graphs. We then give a selfcontained treatment of the Feynman diagram expansion of matrix integrals, and finally, in Section 6, we express Kontsevich’s combinatorial sum as a matrix integral and, using this, conclude the proof of Witten’s conjecture. As we show in Section 7, the knowledge of the intersection numbers of the ψ-classes can be used to prove the nonvanishing of the class κg−2 . This result by Faber gives the threshold for the non-vanishing of the tautological ring of Mg . In fact, in Section 4 of Chapter XXI we prove a theorem by Looijenga stating that the ring of tautological classes on Mg vanishes in degree strictly larger than g − 2. After recalling some basic facts about equivariant cohomology, in the last two sections of Chapter XX we present Harer and Zagier’s computation of the virtual Euler–Poincaré characteristic of Mg,n . The Brill–Noether theory is one the central themes of the first volume of this book. There we study the static aspect of this theory, namely the theory of special linear series on a fixed curve. In our final Chapter XXI we

xxii


study the Brill–Noether theory for smooth curves moving with moduli. In the first few sections, aside for an intermission in which we prove the vanishing theorem of Looijenga we mentioned above, we construct the basic varieties of the Brill–Noether theory for smooth moving curves, and we describe their tangent spaces in terms of the fundamental homomorphisms μ0 : H 0 (C; L) ⊗ 2 ), where C is a smooth H 0 (C; ωC L−1 ) → H 0 (ωC ) and μ1 : ker μ0 → H 0 (ωC curve and L a line bundle on it. We also connect these maps to the normal sheaf relative to the morphism φL : C → Pr , where r = h0 (C, L) − 1. In Section 7 we present Lazarsfeld’s elegant proof of Petri’s conjecture. In the remaining part of the chapter we concentrate mostly on the study of gd1 ’s and gd2 ’s on smooth curves. We revisit a number of classical results and present some nonclassical ones, related to, among others, the Hurwitz scheme, the Severi variety of plane curves of given degree and genus and the unirationality of Mg for small values of g.

Notational conventions and blanket assumptions -

-

Unless otherwise stated, all schemes are implicitly assumed to be of finite type over C. If V is a vector space or a vector bundle, PV is the projective space, or projective bundle, of lines in V , or in the fibers of V . If ϕ : X → S is a morphism of schemes or of analytic spaces and T is a locally closed subscheme or subspace of S, we write XT to denote the fiber product X ×S T . Likewise, if s is a point of S, we write Xs to denote the fiber ϕ−1 (s). We usually write Symq V to indicate the q-th symmetric product of the module or coherent sheaf V . Occasionally, we instead use the notation S q V , especially when V is a vector space.

List of Symbols

AX Ap,q X Aq (E) AqX/Y G(k, n) G(k, V ) IX KX PV SmV Symm V TX Tx (X) V∨ < xν1 xν2 > < τd1 · · · τdn > < xij xkl > t→0

11X χvirt (Γ) ΔΓ ΔP Δa,A

Sheaf of differentiable functions on X Sheaf of smooth (p, q)-forms on the complex manifold X Sheaf of smooth q-forms with values in the vector bundle E Sheaf of relative smooth q-forms of a smooth fibration X → Y Grassmannian of k-planes in Cn Grassmannian of k-planes in the vector space V Ideal sheaf of subscheme X Class of canonical bundle of X in the Picard group of X Projective space of lines in the vector space V , or in the fibers of the vector bundle V mth symmetric power of module or coherent sheaf V mth symmetric power of module or coherent sheaf V Tangent bundle to X Tangent space to X at x Dual of module or coherent sheaf V Propagator Intersection number of point bundle classes Expectation value Propagator in the matrix model Asymptotic expansion Unit graded line bundle Virtual Euler–Poincaré characteristic of Γ Boundary stratum of moduli space of stable curves attached to the graph Γ Locus in moduli space parameterizing curves with at least one separating node of type P Locus in moduli space parameterizing curves with at least one separating node of type (a, A)

735 721 734 741 736 349 758 312 261 262

xxiv Δirr δa,A δirr ηi |Γ| Γ(L) ΓS Γg Γg ˜g Γ Γg,n Γg [ψ] Γg [m] Γ(S;q1 ,...,qn ) ιv (ψ) κ1 κa κ a Λψ Λ[m] λ λ(ν) λi λi (ν) μ0 μL μW μψ μ0,W μ1,W Ω1X/Y ωC ∂Mg,n ∂Mg,P

List of Symbols

Locus in moduli space parameterizing curves with at least one nonseparating node Class of boundary divisor of curves with a separating node of type (a, A) Class of boundary divisor of curves with a nonseparating node Class of the divisor Ei in the Picard group of the stack of stable hyperelliptic curves Geometric realization of graph Γ Subgroup of ΓS,P generated by a system of curves L Mapping class group of the surface S Mapping class group of a reference genus g surface Mapping class group for a genus g Riemann surface Torelli group Mapping class group of a reference n-pointed genus g surface

Mapping class group of the n-pointed surface (S; q1 , . . . , qn ) Interior product of vector v and q-covector ψ Codimension one Mumford class Mumford class Modified Mumford class

Hodge class Generalized Hodge class Hodge class Generalized Hodge class Petri homomorphism

Petri homomorphism mu-one map Sheaf of relative Kähler differentials of X → Y Dualizing sheaf of C Boundary of moduli space of stable n-pointed curves of genus g Boundary of moduli space of stable P -pointed curves of genus g

261 339 339 390 93 491 144 145 451 460 144 510 512 144 220 377 572 572 510 512 334 334 572 573 794 807 807 511 807 808 95 90 261 261

xxv

List of Symbols

Φ : B(KS2 ( → Tg,n ψ ψi ΣL

pi ))

Σ[ pq ] ξΓ ξa,A ξirr A(S, P ) |A(S, P )| A (S, P ) |A (S, P )| A0 (S, P ) A0 (S, P ) |A0 (S, P )| A∞ (S, P ) A∞ (S, P ) Admg (G) Admg (G) Aut(X →Y ) Aut(X →Y )G Aut(X/Y ) Aut(X/Y )G AutB (X ) k = (−1)k B2k B B(KS2 ( pi )) B(a(e), l, m) BlD (M ) BlD (X) Cd Cd Cdr CX/Y C g,P C g,P

Teichm¨ uller homeomorphism Sum of all point classes Point class Sheaf of differential operators of order less than or equal to 1 acting on sections of the line bundle L Clutching morphism associated to the graph Γ Clutching morphism Clutching morphism Arc complex Geometric realization of the arc complex

Complex of proper simplices

Subcomplex of improper simplices Moduli stack of admissible G-covers Moduli space of admissible G-covers

Hilbert scheme parameterizing automorphisms of fibers of X → B Bernoulli numbers Unit ball in the space H 0 (S, KS2 ( pi )) Building block for the hyperbolic decomposition of a Riemann surface Real oriented blow-up of M along D Real oriented blow-up of X along D d-fold symmetric product of curve C Relative d-fold symmetric product Brill–Noether subvariety of the relative d-fold symmetric product Conormal sheaf of X in Y Universal curve over the moduli space of stable P -pointed curves of genus g Universal family over the moduli stack of P -pointed genus g curves

465 335 335

804 542 312 313 313 613 613 661 661 614 661 661 614 661 535 535 536 536 536 536 209 586 462 642 487 149 242 784 788 31 310 138

xxvi

List of Symbols

C g,P

Universal curve over the moduli stack of stable P -pointed curves of genus g cl(X/G) Fundamental class of X/G CollF Collapsing map to stable model pth contraction functor Contrp DΓ Boundary stratum of moduli stack of stable curves attached to the graph Γ Boundary stratum of moduli stack of stable Da,A curves parameterizing curves with at least one separating node of type (a, A) Boundary stratum of moduli stack of stable Dirr curves parameterizing curves with at least one nonseparating node dπ (F ) Determinant of the cohomology Volume element associated to a quadratic dAω differential ω det Determinant functor Diff + (Σ, p1 , . . . , pn ) Group of orientation preserving diffeomorphisms of a pointed topological oriented surface Diff 0 (Σ, p1 , . . . , pn ) Group of orientation preserving diffeomorphisms of a pointed topological surface which are homotopic to the identity E(Γ) Set of edges of graph Γ E Hodge bundle Component of the divisor cut out by Dirr on Ei the boundary of the stack of stable hyperelliptic curves e Half edge of a ribbon graph thought of as an oriented edge [e]0 Orbit of the oriented edge e under the action of σ0 Orbit of the oriented edge e under the action [e]1 of σ1 [e]2 Orbit of the oriented edge e under the action of σ2 evC,L Evaluation map H 0 (C, L) ⊗ OC → L Teichm¨ uller deformation fω Ga Ribbon graph corresponding to simplex a Dual of the ribbon graph Ga Ga (2),m G G{p,q} G[ pq ] Gdr Relative Brill–Noether variety parameterizing gdr ’s for a Kuranishi family

310 567 124 125 312

313

313 356 462 348

454

454 93 334

390 617 618 618 618 813 463 620 620 541 544 542 794

xxvii

List of Symbols

Gdr (p) r Gg,d

Graph(C) Graph(C; D) GraphS (C) (G, x) (G, x, m) (G, x, m, [f ]) H(d, w) H (2),m Hg Hg G Hg,n Hν,g,n

Hg Hg hX (t) hF (n) Hg HilbX/S p(t) HilbX/S

Hilbp(t) r Homext (G, H) HomS (X, Y ) Isoext (G, H) IsomS (X, Y ) Jg J(C) Kφ

Relative Brill–Noether variety parameterizing gdr ’s Relative Brill–Noether variety parameterizing gdr ’s for a Kuranishi family Dual graph of C Dual graph of the curve with marked points (C; D) Dual graph of C with respect to the set S of nodes T -marked ribbon graph P -marked ribbon graph with unital metric P -marked ribbon graph with unital metric and with an isotopy class of embedding Hurwitz space Moduli space of smooth hyperelliptic curves of genus g Moduli space of stable hyperelliptic curves of genus g Subscheme of Hilbert scheme parameterizing stable n-pointed curves of genus g embedded by the ν-fold log-canonical sheaf Moduli stack of smooth hyperelliptic curves of genus g Moduli stack of stable hyperelliptic curves of genus g Hilbert polynomial of the scheme X Hilbert polynomial of the coherent sheaf F Siegel upper half-space of genus g Hilbert scheme of subschemes of fibers of X→S Hilbert scheme of subschemes of fibers of X → S with Hilbert polynomial p(t) Hilbert scheme of closed subschemes of Pr with Hilbert polynomial p(t) Group of exterior homomorphisms from a group G to a group H Hilbert scheme of S-morphisms from X to Y Group of exterior isomorphisms from G to H Hilbert scheme of S-isomorphisms from X to Y Jacobian locus Jacobian of C Torsion subsheaf of the normal sheaf

790 794 88 93 88 619 619 620 857 511 388 388 559

196 388 387 5 5 217 46 43 7 454 47 455 48 461 89 836

xxviii L(Γ) Ln L, M L, M π lω (γ) Ld (p) Li G Mg Mg Mg [ψ] M g [ψ] Mg [m] Mg,n M g,n Mg,P M g,P

M g,P r Mg,d comb Mg,P

List of Symbols

Set of half-edges of graph Γ Virasoro operator Deligne pairing Deligne pairing ω-length of γ Relative Poincaré line bundle of degree d Point bundle Moduli space of genus g curves with Teichm¨ uller structure of level G Moduli space of stable curves of genus g Moduli space of curves with a ψ-structure Compactification of the moduli space of curves with ψ structure Moduli space of genus g curves with level m structure Moduli space of smooth n-pointed curves of genus g Moduli space of stable n-pointed curves of genus g Moduli space of smooth P -pointed curves of genus g Moduli space of stable P -pointed curves of genus g Locus in Mg parameterizing curves with a gdr

comb

M g,P comb Mg,P (r)

Mg [ψ] Mg [ψ] Mg,n M(T ) MΓ Mg,n Mg,P [M/G] Nf

508 104 510 518 512 261 257 261 257 663 794 664 664 664

comb

M g,P (r) G Mg

93 717 367 369 473 781 334

664 Moduli stack of genus g curves with Teichm¨ uller structure of level G Moduli stack of curves with a ψ-structure Moduli stack of smooth n-pointed curves of genus g Category of sections of groupoid M over T Product of moduli spaces of curves attached to vertices of graph Γ Moduli stack of stable n-pointed curves of genus g Moduli stack of P -pointed genus g curves Orbifold quotient of manifold M by the finite group G Normal sheaf to the morphism f

509 511 510 281 281 311 281 138 277 345

xxix

List of Symbols

Nφ NX/Y N ormD/S O(x) Out(G) PicdC/S Picd Picd (p) Picdg PL PrL Prp QCoh QCoh(H, G) Resp (ϕ) ∨

SL Sch Sch/S Sp2g (Z) Stab StabG (p) StMd tψ (Tg ) Tg Tg,n Tg,P TΣ,P T(Σ,p1 ,...,pn )

Normal sheaf modulo torsion 836 Normal sheaf of X in Y 31 Norm map 367, 375 Orbit of x 401 Group of outer automorphisms of a group G 454 Relative degree d Picard functor 782 Relative Picard variety of a Kuranishi family 794 Relative Picard variety 781 Relative Picard variety of a Kuranishi family 794 Picard–Lefschetz representation 145 Projection morphism 311 pth projection functor 125 Category of quasi-coherent sheaves 294 Category of G-equivariant quasi-coherent sheaves on H 340 Residue of ϕ at p 240 Category of schemes Category of schemes over S Symplectic group Stabilization functor Stabilizer of p in G Stable model functor Moduli space of curves with a ψ-structure Teichm¨ uller space of genus g Riemann surfaces Teichm¨ uller space of n-pointed Riemann surfaces of genus g Teichm¨ uller space of P -pointed Riemann surfaces of genus g Teichm¨ uller space based on a topological pointed surface (Σ, P ) Teichm¨ uller space of genus g Riemann surfaces pointed by {p1 , . . . , pn }

comb

T g,P TS,P V (Γ) Wdr

Wdr (p) r Wg,d

X0 (G)

Bordification of Teichm¨ uller space Set of vertices of graph Γ Brill–Noether subvariety of the relative Picard scheme of a Kuranishi family Brill–Noether subvariety of the relative Picard scheme Brill–Noether subvariety of the relative Picard scheme of a Kuranishi family Set of vertices of a ribbon graph

490 283 279 460 138 527 124 509 446 446 446 446 446 663 485 93 794 788 794 616

xxx X1 (G) [X/G] (X(G), σ0 , σ1 ) |X| [X/G]

List of Symbols

Set of edges of a ribbon graph Quotient groupoid of the scheme X modulo the group scheme G Data defining a ribbon graph Orbit space of orbifold groupoid |X| Orbifold quotient of orbifold groupoid X by the finite group G

616 286 616 276 278

Chapter IX. The Hilbert Scheme

1. Introduction. In this chapter we introduce the Hilbert scheme which, roughly speaking, parameterizes subschemes of a fixed projective space with a prescribed Hilbert polynomial. Our immediate motivation for discussing Hilbert schemes at this point is to be able to construct the moduli space of stable curves, first as an analytic space and then as an algebraic space and as a stack. The Hilbert scheme will continue to play a major role throughout the rest of the book. In the first section we describe the basic results on families

(1.1)

X ⊂ Pr × S f u S

of closed subschemes of Pr parameterized by a scheme S. We give an informal introduction to Hilbert schemes, and we discuss a few elementary examples, including the Grassmannian which is, at the same time, the basic example of a Hilbert scheme and the ambient space where any Hilbert scheme is naturally embedded. The theorems announced in this section are proved in the following two. One of these results anticipates the meaning of flatness, which is the key notion in the deformation theory of schemes: if the family (1.1) is flat, then all of its fibers Xs = f −1 (s) have the same Hilbert polynomial. In fact, on a reduced base S, this property can be taken as a characterization of flatness. Alternatively, the flatness of f is equivalent to the requirement that f∗ OX (n) is locally free for large n. These results can be thought of as global consequences of flatness. In the second section we prove what was announced in the first one. Specifically, using the theory of base change in cohomology, we show that, when the family (1.1) is flat, one gets a rather good control on how the cohomology of the fibers Xs varies, as the point s travels in S. At the end of this section we prove yet another property of flatness, which is an analogue of Sard’s theorem, saying that, given a morphism ϕ : Z → T , with T reduced, there is a Zariski dense open subset of T over which ϕ is flat. E. Arbarello et al., Geometry of Algebraic Curves, Grundlehren der mathematischen Wissenschaften 268, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-540-69392-5 1,

2

9. The Hilbert Scheme

The third section is devoted to the actual construction of the Hilbert scheme and its universal family and to the realization of the Hilbert scheme as a subscheme of a Grassmannian. Fixing a positive integer represents r and a rational polynomial p(t), the Hilbert scheme Hilbp(t) r the functor, from the category of schemes to the category of sets, which associates to a scheme S the set of flat families of projective subschemes of Pr parameterized by S and with Hilbert polynomial equal to p(t). The closed points of Hilbp(t) are just the projective subschemes of Pr r with Hilbert polynomial equal to p(t). Apart from the use of flatness, which is ubiquitous in the construction of the Hilbert scheme, another tool is needed, namely the following strengthening of a classical results by Serre on projective varieties. Let r be a nonnegative integer, and let p(t) be a rational polynomial. Then there exists an integer n0 , depending only on r and p(t), such that, for any subscheme X ⊂ Pr with Hilbert polynomial p(t) and for any n ≥ n0 , the homomorphism ϕn

X H 0 (Pr , OPr (n)) −−→ H 0 (X, OX (n))

is onto, and the scheme X is completely determined by its kernel. Moreover, the dimension of ker ϕnX depends only on r and p(t) and is given by n+r n q(n) = dim ker ϕX = − p(n). r Now fix n ≥ n0 . The assignment X → [ker ϕnX ] gives then an injective to the Grassmannian G of q(n)-planes set-theoretical map from Hilbp(t) r in H 0 (Pr , OPr (n))). It is in the Grassmannian G that one looks for equations defining the appropriate scheme structure of Hilbp(t) r , and the p(t) universal property of Hilbr will be a reflection of the universal property of the Grassmannian. As we will show in Section 6, a Hilbert scheme can be quite nasty. Of course, one of its advantages is that it represents a functor. Another one is that it is projective. We prove this result at the end of Section 4. In Section 5, in order to compute the tangent space to the Hilbert scheme at a given point, we discuss another aspect of flatness. The basic picture that we analyze is now the local picture of a flat family

(1.2)

X ⊂ Y ×S f u S = Spec C[ε]

of closed affine subschemes of an affine scheme Y = Spec(R) parameterized by Spec C[ε] or, as one says, the picture of a first-order embedded deformation of the closed subscheme X = Xs0 ⊂ Y × {s0 }. Here s0 is

§1 Introduction

3

the closed point of Spec C[ε]. Set X = Spec(R/I) and X = Spec(R[ε]/J), where I is an ideal in R, and J is an ideal in R[ε]. It turns out that saying that the family (1.2) is flat is equivalent to saying that every presentation by generators and relations of the ideal I of the central fiber X = X mod ε extends to a presentation of J. From this property it is not hard to see that the first-order embedded deformations of X are classified by HomR/I (I/I 2 , R/I). All of this easily globalizes to show that the first-order embedded deformations of a subscheme X ⊂ Y are in a one-to-one correspondence with the sections of the normal sheaf NX/Y = HomOX (I/I 2 , OX ), where I is the ideal sheaf of X in Y . This is the first instance of the Kodaira–Spencer theory, which in fact goes back to the classical notion of the characteristic system. In classical algebraic geometry, given, for example, an algebraic family of curves on a surface Y and a curve C in this family, the first-order approximation of this algebraic family was studied via the linear series cut out on C by the infinitely near curves belonging to the algebraic family. This linear series was called the characteristic linear series of the given algebraic family. The divisors belonging to the characteristic series are nothing but zero-divisors of sections of the normal bundle NC/Y . Having acquired the normal bundle description of the tangent spaces to the Hilbert scheme, in Section 6 we present Mumford’s example which illustrates how bad a Hilbert scheme can be, even in innocent-looking cases. The case at hand is the one of a smooth connected space curve such that the corresponding point in the appropriate Hilbert scheme H lies in a component along which H is everywhere nonreduced. Section 7 is rather technical but of central importance. There we study a number of variants of the Hilbert schemes. Let us very briefly present a few of these. Given a scheme S and a closed subscheme X of Pr × S, the first variant comes from the desire of parameterizing couples (s, Y ), where s is a point of S, and Y is a closed subscheme of the fiber of X → S over s with given Hilbert polynomial. Another useful space that can be constructed is a scheme parameterizing nested pairs of subschemes in the fibers of a family of projective schemes. This construction will be routinely used when dealing with pointed curves. Another important variant is the scheme (not of finite type) IsomS (X, Y ), where X ⊂ Pr × S and Y ⊂ Pt × S are closed subschemes, and both X and Y are flat over S. This is a scheme representing the functor which associates to each scheme T over S the set of all isomorphisms, as schemes over T , from X ×S T to Y ×S T . This scheme and its properties are essential in studying how the automorphism group of a curve varies as one varies the curve. Finally the Hilbert scheme of isomorphisms is one of the central objects in the theory of Deligne–Mumford stacks. Section 8 contains tangent space computations for several of the Hilbert space variants introduced in the previous section. The final section has a partially nonalgebraic flavor. The repre-

4


sentability properties of the Hilbert functor can be effectively used also when dealing with differentiable, or even continuous, families of projective manifolds. From our perspective, this section introduces the necessary tools for proving analogous properties for Kuranishi families, which, in turn, will play an important role in our treatment of Teichm¨ uller theory. 2. The idea of the Hilbert scheme. We recall our general conventions. Unless otherwise stated, all schemes are implicitly assumed to be of finite type over C. Fix a projective space Pr . In a very imprecise sense, we would like to parameterize closed subschemes of Pr with “fixed numerical characters.” Naively, this means putting a scheme structure on the set H of all such subschemes which is, in some sense, natural. One way of making the adjective “natural” slightly more precise is to ask that the scheme H be the parameter space for a family {Xh }h∈H , where Xh is the closed subscheme of Pr corresponding to h. Our first goal, then, is to formalize the notion of family of closed subschemes of Pr with fixed numerical characters. Clearly, given an ambient scheme Y , a family of subschemes of Y parameterized by a scheme S is nothing but a subscheme X ⊂ Y ×S, viewed as fibered over S via the natural projection f

→S. Y ×S ⊃X −

(2.1)

We shall write Xs to denote the fiber f −1 (s). Now we fix our attention on the case where Y = Pr and f is proper, or, which is the same, X is closed in Pr × S. As most of the numerical invariants of a closed subscheme of Pr , like the dimension, the degree, and so forth, are encoded in its Hilbert polynomial, we may implement the vague requirement that the numerical characters of the fibers Xs be fixed by asking that the Hilbert polynomial hXs (t) = (−1)i dim H i (Xs , OXs (t)) i

be independent of s. Now it is a basic result that, at least when the base S is reduced, the local constancy of the Hilbert polynomial is equivalent to the flatness of f . To explain this, we begin by recalling a few basic definitions. A module M over a commutative ring R is said to be flat if tensoring with M is an exact functor. As is well known, each of the following conditions is equivalent to the flatness of M : - TorR q (M, H) = 0 - TorR q (M, H) = 0 q > 0; - TorR 1 (M, H) = 0 - TorR 1 (M, H) = 0

for any R-module H and any q > 0; for any finitely generated R-module H and any for any R-module H; for any finitely generated R-module H.

§2 The idea of the Hilbert scheme

5

A morphism of schemes ϕ:X→S is flat if for any x ∈ X, OX,x is a flat OS,ϕ(x) -module. More generally, a coherent OX -module F is said to be flat over S if Fx is a flat OS,ϕ(x) module for every x ∈ X. This is equivalent to saying that, for any pair of affine open subsets U ⊂ X and V ⊂ S such that ϕ(U ) ⊂ V , Γ(U, F ) is a flat Γ(V, OS )-module. It is useful to notice that to check that F is flat over S, it suffices to verify that Fx is OS,ϕ(x) -flat for all closed points x ∈ X. We shall say that a family as in (2.1) is flat if f is. The notion of flatness carries over, with the same definition, to the context of analytic spaces and coherent analytic sheaves over them. Moreover, the analytic notion of flatness and the algebraic one agree in the following sense. Let ϕ : X → S be a morphism of schemes, and let F be a coherent OX -module. Denote by ϕan : X an → S an and F an the corresponding analytic objects. Then F is flat over S if and only if F an is flat over S an . The proof is based on a fundamental result of Serre [626], asserting that, if Z is a scheme of finite type over C and z ∈ Z is a closed point, then OZ an ,z is flat over OZ,z . The rest of the argument is a simple application of the basic properties of tensor products. One may, for instance, apply Exercise 1 for this chapter, with A = OS,ϕ(x) , A = OS an ,ϕan (x) , B = OX,x , and B = OX an ,x . These observations will allow us, when talking about flatness, to go back and forth without risks between the algebraic and the analytic category. If F is a coherent sheaf on Pr , its Hilbert polynomial is hF (n) =

(−1)i hi (Pr , F (n)) .

i

The Hilbert polynomial hX (t) of a projective scheme X is nothing but the Hilbert polynomial of OX . The link between flatness and Hilbert polynomials is formally expressed by the following result. Proposition (2.2). Let f

Pr × S ⊃ X − →S be a family of closed subschemes of Pr . Then i) If f is flat, then the Hilbert polynomial hXs (t) is locally constant as a function of s ∈ S, ii) If S is reduced, then the converse of i) holds. In fact, it suffices to check the local constancy of hXs (t) as s varies among the closed points of S.

6


The proposition captures one of the main geometric aspects of flatness. We shall study some of the technical aspects of flatness in the next section, where the proposition will be proved. In view of (2.2), it seems reasonable to take flatness as a good formalization of the notion of having “fixed numerical characters” for families of closed subschemes of projective space. Let us then fix a rational polynomial p(t). According to our initial program, it is natural to consider the set H of all closed subschemes of Pr with Hilbert polynomial equal to p(t) and to try and give the set H a scheme structure. Actually whose closed points are in one-to-one we shall construct a scheme Hilbp(t) r correspondence with the points of H and a flat family π

⊃X − → Hilbp(t) Pr × Hilbp(t) r r of subschemes of Pr such that the identification between closed points of Hilbp(t) and points of H is given by r Hilbp(t) h → Xh = π−1 (h) . r is universal Even more will turn out to be true. The family over Hilbp(t) r in the sense that any flat family f

Pr × S ⊃ X − →S of closed subschemes of Pr with Hilbert polynomial p(t) is the pullback X ×Hilbp(t) S → S r

via a unique morphism α : S → Hilbp(t) r . commutative diagram of cartesian squares Pr × S C A AA A J A X A A f A D u A α S

In other words, there is a

id ×α wX π

w Pr × Hilbp(t) r A A A A A D A

) ' ' '' 0

u w Hilbp(t) r

Another way of saying this is that Hilbp(t) represents the Hilbert functor, r that is, the functor h from schemes to sets defined by h(S) = {families of closed subschemes of Pr with Hilbert polynomial p(t) parameterized by S} .


7

This means that there is an isomorphism of functors between h and the functor S → Hom(S, Hilbp(t) r ). The isomorphism is given as follows:

Hom(S, Hilbp(t) r ) −→ h(S) α : S → Hilbp(t) → X × S → S . p(t) r Hilb r

is called the Hilbert scheme of closed subschemes The scheme Hilbp(t) r of Pr with Hilbert polynomial p(t). If Y is a subscheme of Pr with Hilbert polynomial p(t), the corresponding point in Hilbp(t) will usually r be denoted by [Y ]. Let us immediately give a few elementary examples. Example (2.3). Maybe the simplest example is the one of hypersurfaces in Pr of a fixed degree d. Observe first that, given such a hypersurface X, its Hilbert polynomial is p(n) =

n−d+r n+r , − r r

as can be seen by taking cohomology of the exact sequence 0 → OPr (n − d) → OPr (n) → OX (n) → 0 . Notice that p(t) does not depend on the particular X we are considering, but only on its degree, and that its leading term is d

tr−1 . (r − 1)!

Moreover, hypersurfaces of degree d are characterized, among all subschemes of Pr , by having p(t) as Hilbert polynomial. In fact, suppose that the Hilbert polynomial of Y ⊂ Pr is p(t). Let Y1 , . . . , Yh be the be the irreducible components of Y of dimension r − 1, and let μ1 , . . . , μh multiplicities with which they occur in Y . The hypersurface X = μ i Yi is a subscheme of Y , and the quotient Q = IX /IY is supported on a subscheme of Pr of dimension at most r − 2. Since, as is well known, the degree of the Hilbert polynomial of a coherent sheaf equals the dimension of its support, it follows that the Hilbert polynomial of Q has degree at most r − 2. On the other hand, p(t) = hY (t) = hX (t) + hQ (t) ,

8


so it follows that the leading term of hX (t) is the same as that of p(t), hence that X has degree d and that hX (t) = p(t). But then hQ (t) = 0, so Q = 0, that is, X = Y . Now set d+r − 1. N= r r Denote by x0 , . . . , xr homogeneous coordinates in P and by aI , where I ik = d, homogeneous runs through all multiindices (i0 , . . . , ir ) such that coordinates in PN . Consider the hypersurface X in Pr × PN defined by the equation a I xI = 0 . (2.4) I

Then X , together with its projection π onto PN , is a family of hypersurfaces of degree d in Pr . As the Hilbert polynomial of the fiber π−1 (a) is independent of the point a ∈ PN , by part ii) of Proposition (2.2) this family is flat. It is essentially evident that PN is the Hilbert scheme of hypersurfaces of degree d in Pr and that π : X → PN is its universal family. To prove this, let f

Pr × S ⊃ X − →S be a flat family of hypersurfaces of degree d in Pr . We wish to show that this family comes from a morphism α : S → PN . We shall rely on an elementary characterization of flatness which we now state and whose proof will be found in the next section. Proposition (2.5). Let F be a coherent sheaf on Pr × S, and denote by ξ the projection of Pr × S onto S. Then F is flat over S if and only if ξ∗ (F (n)) is locally free for any sufficiently large n. Now let I be the ideal sheaf of X in Pr × S. Consider the exact sequence 0 → I(n) → OPr ×S (n) → OX (n) → 0 . For large n, R1 ξ∗ I(n) vanishes, so we obtain the exact sequence 0 → ξ∗ I(n) → ξ∗ OPr ×S (n) → ξ∗ OX (n) → 0 . Clearly, ξ∗ OPr ×S (n) = H 0 (Pr , OPr (n)) ⊗ OS is free, while, by the proposition, ξ∗ OX (n) is locally free, so that ξ∗ I(n) is locally free as well. Again by the proposition, I is flat over S. Now consider ξ∗ (I(d)). Since, for any hypersurface Y of degree d in Pr , H 0 (Pr , IY (d)) is onedimensional, the theory of base change (which we will very briefly review at the beginning of Section 3) tells us that ξ∗ I(d) is a line bundle on S. Thus we can find a finite cover {Ui } of S and a generator σi of ξ∗ I(d)


9

on each Ui . In concrete terms, we may think of σi as a homogeneous polynomial of degree d bi,I xI I

whose coefficents are regular functions on Ui . In addition, for any s ∈ Ui , the equation of f −1 (s) as a subscheme of Pr is precisely

bi,I (s)xI = 0.

I

We may then define a map αi : Ui → PN which, from an intuitive point of view, is given by s → [· · · : bi,I (s) : · · · ] . More exactly, the morphism αi corresponds to the homomorphism from the homogeneous coordinate ring of PN to the coordinate ring of Ui defined by aI → bi,I . The fact that σi and σj are local generators of the line bundle ξ∗ I(d) implies that there is a unit u on Ui ∩ Uj such that σj = uσi . In other words, bj,I = ubi,I for any multiindex I. This shows that the morphisms αi patch together to define a global morphism α : S → PN , and it is obvious from the definitions that the family f : X → S is the pullback, via α, of the universal family π : X → PN . The uniqueness of α comes from the fact that to say that there is a commutative diagram of cartesian squares Pr × S 6 4 4 A 4 < A X Aξ A f A u A D α S

id ×α

w Pr × PN 6 4 4 A 4 < A wX A A π A D u A N wP

means that X, as a subscheme of Pr × S, is defined by the pullback, via α, of the universal equation (2.4). Since this equation has degree 1 in the aI variables and degree d in the x variables, the pullback in question, in intrinsic terms, is nothing but a nowhere vanishing section of ξ∗ IX (d) ⊗ α∗ OPN (1) . Any local trivialization of α∗ OPN (1) gives rise to a local generator of ξ∗ IX (d). It is evident that applying to this section the construction performed in the existence part, one gets back α. This proves the uniqueness.

10


Example (2.6). The second elementary example is the Grassmannian G = G(k + 1, r + 1) of (k + 1)-dimensional subspaces of Cr+1 . Let us show that G is the Hilbert scheme of k-dimensional linear subspaces of Pr ; of k+t course, the relevant Hilbert polynomial is p(t) = k . Let S be the universal subbundle on G; its projectivization PS yields a flat family of k-planes in Pr parameterized by G π

Pr × G ⊃ PS − →G This turns out to be the universal family on G = Hilbp(t) r . In fact, given any flat family f Pr × S ⊃ X − →S of k-planes in Pr , we can attach to it the rank k + 1 vector bundle on S defined by f∗ OX (1). We also have an inclusion (2.7)

(f∗ OX (1))∨ → H 0 (Pr , OPr (1))∨ ⊗ OS = Cr+1 ⊗ OS .

It is immediate to check that the projectivization of this inclusion can be identified with the inclusion of X in Pr × S. At this point we may use the standard universal property of the Grassmannian to get a unique morphism α : S → G such that the inclusion (2.7) is the pullback, via α, of the inclusion of the universal subbundle S in the trivial bundle of rank r + 1 on G. This proves that G coincides with the Hilbert scheme of k-planes in Pr . The preceding example can be viewed as a Hilbert-scheme theoretic interpretation of the universal property of the Grassmannian. As we shall see momentarily, this universal property is one of the key ingredients in the construction of general Hilbert schemes. Example (2.8). For a zero-dimensional subscheme Z of Pr , the Hilbert polynomial is the constant polynomial d = deg Z = dim H 0 (Z, OZ ) . Let H be the Hilbert scheme of degree d, zero-dimensional subschemes of Pr . It is easy to give an explicit model for an open subset of H. Consider the d-fold symmetric product of Pr minus the big diagonal. This is a smooth variety S which is the parameter space of the (flat) family of all zero-dimensional subschemes of Pr consisting of d distinct points. By universality, S maps to H. In Section 5 we shall see that (as is intuitively evident) S actually embeds in H as an open subset; this will be just a tangent space computation. We shall see in Section 4 that Hilbert schemes are complete, as was clearly the case in the two preceding examples. Thus, in the case at hand, H is a compactification of S. It is worth pointing out that this compactification is not the d-fold symmetric product of Pr , except, of course, for d = 1, when S is already complete. Even worse, for r > 2 and d ≥ 2, H is not even irreducible, contrary to what one might intuitively expect (cf. [380])


11

Example (2.9). Even innocent-looking subschemes of projective space may degenerate, inside the Hilbert scheme, to very ugly ones. As a simple example, consider curves of degree two in P3 . The smooth ones come in two kinds, the smooth conics and the unions of two skew lines. These can be told apart by looking at their Hilbert polynomial; the one of a smooth conic is Q(n) = 2n + 1, the one of a union of skew lines is P (n) = 2n + 2. We shall look at the Hilbert scheme HilbP 3. Let X0 , . . . , X3 be homogeneous coordinates in P3 . For t = 0, let Ct be the union of the lines {X1 = X3 = 0} and {X2 = X3 − tX0 = 0}. The family {Ct } extends across {t = 0} to a flat family of subschemes Let C ⊂ P3 × C be the subscheme defined by of P3 as follows. the equations X1 X2 = X1 (X3 − tX0 ) = X2 X3 = X3 (X3 − tX0 ) = 0. The fiber of C → C over t = 0 is Ct , while the fiber C0 over t = 0 is the nonreduced subscheme of P3 defined by the equations X1 X2 = X1 X3 = X2 X3 = X32 = 0. Informally, C0 can be described as the reducible conic Γ defined by the equations X1 X2 = X3 = 0 plus the embedded component X1 = X2 = X32 = 0. To see that C → C is flat, it suffices to notice that the Hilbert polynomial of C0 is P (n) = 2n + 2. In fact, the kernel of OC0 → OΓ is a one-dimensional complex vector space concentrated at X1 = X2 = X3 = 0, while, as we observed, the Hilbert polynomial of Γ is Q(n) = 2n + 1. Before closing this section, let us give a quick preview of the construction of the Hilbert scheme. Fix a rational polynomial p(t). Let X be a subscheme of Pr with p(t) as Hilbert polynomial. Look at the map ϕn

H 0 (Pr , OPr (n)) −−→ H 0 (X, OX (n)) . By Serre’s theorems there is an integer n0 such that, for all n ≥ n0 , the following properties hold: i) ϕn is surjective; ii) h0 (X, OX (n)) = p(n); iii) X is completely determined by the kernel of ϕn , in the sense that the degree n hypersurfaces passing through X generate H 0 (Pr , IX (m)) for any m ≥ n. As we shall see in a crucial lemma, one can find an n0 that depends only on r and p(t) but not on the particular X under consideration. Set n+r − p(n) . q(n) = h0 (Pr , OPr (n)) − p(n) = r For any fixed n ≥ n0 , we may then associate to X the point H 0 (Pr , IX (n)) = ker ϕn ∈ G = G(q(n), H 0 (Pr , OPr (n))) .

12


If H is the set of all subschemes of Pr with Hilbert polynomial equal to p(t), the above construction gives an injection H → G X→ [H (P , IX (n)) ⊂ H 0 (Pr , OPr (n))] . 0

r

Suppose that f

→S Pr × S ⊃ Y − is a flat family of subschemes of Pr with Hilbert polynomial p(t). Then S is equipped with the vector subbundle f∗ (I(n)) ⊂ H 0 (Pr , OPr (n)) ⊗ OS , where I is the ideal sheaf of Y in Pr × S. Of course, we have f∗ (I(n)) ⊗ k(s) = H 0 (Ys , If −1 (s) (n)) for any s ∈ S. Then the universal property of the Grassmannian provides a morphism α:S→G which, set theoretically, lands in H. Now that a certain measure of the required universal property is built into H, the remaining technical challenge will be to equip H with a scheme structure such that two conditions are satisfied. First of all, any α as above must land into H as a morphism of schemes. As for the second condition, look at the universal subbundle on G. It generates a sheaf of ideals on Pr × G and hence determines a family of closed subschemes η

→ G. Pr × G ⊃ Y − The fibers of η are all possible subschemes of Pr defined by q(n) linearly independent equations of degree n. What we will have to show is that π → H of this family to H is flat. At this point H will the restriction X − π be the Hilbert scheme Hilbp(t) and X − → H its universal family. r 3. Flatness. In this section we have collected a few technical results about flatness that either have been announced in the previous section or will be needed in the construction of the Hilbert scheme. One of the most important applications of flatness is to the theory of base change in cohomology (cf., for instance, Mumford’s book [552], Section 5). Since this will be the main tool in this section, we shall briefly digress on it. A typical setup is the following. We have a scheme

§3 Flatness

13

S, a coherent sheaf F on Pr × S, and a morphism of schemes f : T → S. Look at the diagram Pr × T (3.1)

g

η u T

f

w Pr × S ξ u wS

where ξ and η are the projections, and g = id ×f . Given a point s ∈ S, we write Fs to denote F ⊗ Oξ−1 (s) , viewed as a sheaf on Pr = ξ −1 (s). The problem of base change is to compare Rq η∗ g ∗ F with f ∗ Rq ξ∗ F for any q ≥ 0. In general, these two sheaves are far from being equal. When F is flat over S, we have better control on the situation. The theory of base change asserts that, under this assumption, any point of S has an open neighborhood U ⊂ S over which there is a bounded complex (3.2)

K0 → K1 → K2 → · · ·

of locally free coherent sheaves which calculates the direct images of F , functorially under base change. Thus, for any base change f : T → S such that f (T ) ⊂ U , the cohomology of the complex f ∗K 0 → f ∗K 1 → f ∗K 2 → · · · is R• η∗ g ∗ F . As a corollary, one gets the following well-known result. Proposition (3.3). Let S be a scheme, and F a coherent sheaf on Pr × S, flat over S. Let ξ : Pr × S → S be the projection. Then the following statements are equivalent: i) ii) iii) iv)

Rq ξ∗ F = 0 for any q > 0, Rq η∗ g ∗ F = 0 for any base change (3.1) and any q > 0, H q (Pr , Fs ) = 0 for any s ∈ S and any q > 0, H q (Pr , Fs ) = 0 for any closed point s of S and any q > 0.

Moreover, if one of the above holds, ξ∗ F is locally free, and the natural homomorphisms f ∗ ξ ∗ F → η∗ g ∗ F , ξ∗ F ⊗ k(s) → H 0 (Pr , Fs ) are isomorphisms for any base change (3.1) and any s ∈ S. Proof. The statement is local on S, so we may as well assume that a complex (3.2) exists on all of S. Condition iii) is a special case of ii), and iv) a special case of iii). Likewise, that ξ∗ F ⊗ k(s) = H 0 (Pr , Fs ) for any s ∈ S is a special case of the statement that f ∗ ξ∗ F = η∗ g ∗ F for any base change (3.1).

14


Let us show that i) implies ii). Condition i) means that K • is a locally free resolution of ξ∗ F . Therefore, ξ∗ F is locally free as well. Put otherwise, 0 → ξ∗ F → K 0 → K 1 → · · · is an exact sequence of locally free, and hence flat, OS -modules. This implies that the pulled-back sequence 0 → f ∗ ξ∗ F → f ∗ K 0 → f ∗ K 1 → · · · is also exact. Thus Rq η∗ g ∗ F = 0 for q > 0, and η∗ g ∗ F = f ∗ ξ∗ F . It remains to show that iv) implies i). It follows from iv) that the complex K 0 ⊗ k(s) → K 1 ⊗ k(s) → K 2 ⊗ k(s) → · · · is exact for any closed s, and what must be shown is that (3.2) is exact as well. More generally, we shall show, by induction on n, that any complex L 0 → L1 → · · · → L n → 0 → · · · of locally free coherent sheaves on U is exact if L• ⊗ k(s) is exact for any s. This is obviously true when n = 0. For n > 0, the assumption says in particular that Ln−1 ⊗k(s) → Ln ⊗k(s) is onto for any s, and Nakayama’s lemma then implies that Ln−1 → Ln is also onto. It follows that the kernel of this homomorphism, which we denote by Z n−1 , is locally free and that Z n−1 ⊗ k(s) equals the kernel of Ln−1 ⊗ k(s) → Ln ⊗ k(s). In particular, tensoring (3.4)

L0 → · · · → Z n−1 → 0 → · · ·

with k(s) yields an exact complex for any closed point s. By induction hypothesis, this implies that (3.4) is exact, concluding the proof. We now begin to prove the results we have announced in the previous section, starting with Proposition (2.5). We briefly recall its statement. We are given a coherent sheaf F on Pr × S, where S is a scheme, and denote by ξ the projection of Pr × S onto S. The proposition says that F is flat over S if and only if ξ∗ (F (n)) is locally free for any sufficiently large n. Suppose first that F is flat over S. If n is large enough, the higher direct images of F (n) vanish, and hence (3.3) tells us that ξ∗ (F (n)) is locally free. Conversely, assume that ξ∗ (F (n)) is locally free for all large n. To show that F is flat over S, we must show that, for any injection G1 → G2 of coherent OS -modules, ξ ∗ G1 ⊗ F injects into ξ ∗ G2 ⊗ F . Since the question is local on S, we may assume that S is affine; let A be its coordinate ring. Let R = ⊕n≥0 Rn be the homogeneous coordinate ring

i , i = 1, 2, and F = F , of Pr × S, so that R0 = A. We may write Gi = G

§3 Flatness

15

where the Gi are finitely generated A-modules, and F = ⊕n≥n0 Fn is a finitely generated graded R-module. To say that ξ∗ (F (n)) is locally free for large n is the same as saying that Fn is a projective A-module for large n. Since changing a finite number of summands of F does not change F , we may then assume that Fn is projective, and hence in particular flat over A, for all n ≥ n0 . That ξ ∗ G1 ⊗ F injects into ξ ∗ G2 ⊗ F then follows from the remark that ξ ∗ Gi ⊗ F = (Gi ⊗ A R) ⊗R F = Gi ⊗A F and from the flatness of F over A. Proposition (2.5) is now fully proved. We next turn to Proposition (2.2). It pays to prove the following, slightly more general, result. Proposition (3.5). Let F be a coherent sheaf on Pr × S, where S is a scheme. Denote by ξ the projection of Pr × S onto S. Then i) if F is flat over S, then the Hilbert polynomial hFs (t) is locally constant as a function of s ∈ S; ii) if S is reduced, then the converse of i) holds. In fact, it suffices to check the local constancy of hFs (t) as s varies among the closed points of S. Notice that ii) does not necessarily hold when S is not reduced. The simplest example is provided by the case S = Spec C[ε], r = 0, and F = C[ε]/(ε) C. That F is not flat can be seen by noticing that tensoring the injection of C[ε]-modules (ε) → C[ε] with F C yields the zero homomorphism C → C. Proposition (3.5) is an almost immediate consequence of the following lemma. Lemma (3.6). Let F be a coherent sheaf on Pr × S, and let ξ be the projection from Pr × S to S. Then there is an integer n0 such that, for any point s ∈ S and any n ≥ n0 , i) the natural map ξ∗ F (n) ⊗ k(s) → H 0 (Pr , Fs (n)) is an isomorphism, ii) H q (Pr , Fs (n)) = 0 for any q > 0. Assuming for the moment that the lemma has been proved, we deduce Proposition (3.5) from it. Lemma (3.6) implies that, for large n, hFs (n) equals the dimension of ξ∗ F (n) ⊗ k(s). If F is flat, Proposition (2.5) implies that this dimension is locally constant. Conversely, if S is reduced, it is well known that a coherent sheaf G such that the dimension of G⊗k(s) is locally constant as s varies among the closed points of S, is locally free; this is the content of Lemma (3.7) below. Thus the local constancy of the dimension of ξ∗ F (n) ⊗ k(s) implies that ξ∗ F (n) is locally free. That F is flat then follows from Proposition (2.5).

16


Recall that a commutative ring is said to be a Jacobson ring if every prime ideal is the intersection of maximal ones. As is well known, the general form of the Nullstellensatz asserts that a finitely generated algebra over a Jacobson ring is also Jacobson (cf., for instance, [194], Theorem 4.19), so that in particular finitely generated algebras over a field are Jacobson rings. Lemma (3.7). Let A be a reduced commutative ring, and let M be a finitely generated A-module. For any prime ideal P in A, let k(P ) be the quotient of AP modulo its maximal ideal. The following are equivalent: i) M is projective; ii) the dimension of M ⊗k(P ) over k(P ) is locally constant on Spec(A); When A is a Jacobson ring, i) and ii) are equivalent to: iii) the dimension of M ⊗ k(P ) over k(P ) is locally constant on the maximal spectrum of A. That i) implies ii) and iii) is clear. Conversely, suppose ii) holds. Let Q be a prime ideal. We must show that Mf is a free Af -module for a suitable f ∈ A Q. Let m1 , . . . , mn be elements of M whose classes in M ⊗ k(Q) form a basis over k(Q). Nakayama’s lemma then implies that the classes of m1 , . . . , mn generate MQ . Replacing A with Af for a suitable f ∈ A, we may in fact assume that the classes of m1 , . . . , mn generate MP for all P ∈ Spec(A) and hence that the homomorphism An → M sending the elements of the canonical basis to m1 , . . . , mn is onto; we may also assume that the dimension of M ⊗ k(P ) is constant on Spec(A). We claim that M is a free A-module. To see this, let K be the kernel of An → M , and tensor the exact sequence 0 → K → An → M → 0 with k(P ) to get the exact sequence K ⊗ k(P ) → k(P )n → M ⊗ k(P ) → 0. The middle homomorphism is an isomorphism by assumption. This means that K ⊂ P An = P ⊕n . Since this is true for all prime ideals P , it follows that K ⊂ N ⊕n , where N is the nilradical of A. Since A is assumed to be reduced, K = {0}, and our claim is proved. To prove that iii) also implies i) when A is a Jacobson ring, it suffices to observe that the argument above shows that K is contained in I ⊕n , where I stands for the intersection of the maximal ideals of A, and that I equals the nilradical since A is Jacobson. Notice also that replacing A with Af is legitimate since Af = A[1/f ] is also a Jacobson ring. This concludes the proof of (3.7). It remains to prove Lemma (3.6). The crux of the matter is to find an n0 that works uniformly for all points s ∈ S. In fact, if the uniformity requirement were dropped, (3.6) would be a special case of the following variant of Serre’s correspondence for coherent sheaves on projective space.

§3 Flatness

17

Lemma (3.8). Let F be a coherent sheaf on Pr × S, and let ξ be the projection of Pr × S onto S. Let f : T → S be any morphism of schemes, and set g = id ×f : Pr × T → Pr × S. Then there is an n0 such that, for any n ≥ n0 , i) the natural map f ∗ ξ∗ F (n) → η∗ g ∗ F (n) is an isomorphism, ii) Rq η∗ g ∗ F (n) = 0 for any q > 0. Let us quickly prove this lemma. Part ii) is just Serre’s vanishing theorem. Statement i) is local on the base, so we may assume that S = Spec A, T = Spec B. We set RA = A[x0 , . . . , xr ], where x0 , . . . , xr are indeterminates, and similarly for RB . Notice that RB = RA ⊗A B. Serre’s theorem tells us that there is a graded RA -module F = ⊕Fh such that F = F , and that moreover H 0 (Pr × S, F (h)) = Fh for large h. The sheaf version of this statement is that ξ∗ (F (h)) = F h , always for large h. The pullback g ∗ F is nothing but F ⊗ RA RB . On the other hand, F ⊗RA RB = F ⊗RA (RA ⊗A B) = F ⊗A B . In particular, for large h, ⊗A B . η∗ g ∗ F (h) = Fh But, again for large h, one has that f ∗ ξ∗ F (h) = f ∗ F h = Fh ⊗A B , proving the lemma. Returning to the problem of finding a uniform n0 in Lemma (3.8), that this is possible would be an immediate consequence of (3.3) if the sheaf F were flat over S. It would in fact suffice to take as n0 an integer m such that Rq ξ∗ F (n) = 0 for any n ≥ m and q > 0. Unfortunately, we cannot afford to limit ourselves to the flat case. To meet the uniformity requirement on n0 , we will have to use flatness in a more indirect way, combining it with (3.8). The key ingredient to do so is the following result, which is, in some sense, reminiscent of Sard’s lemma from differential geometry.

18


Proposition (3.9) (Sard’s lemma for flatness). Let α : X → Y be a morphism of schemes, and let G be a coherent sheaf on X. Assume that Y is reduced. Then there exists a Zariski dense open subset U ⊂ Y such that G|α−1 (U ) is flat over U . In proving (3.9) we may as well assume that Y is irreducible. Moreover we may assume that X = Spec B and Y = Spec A. Thus G corresponds to a finitely generated B-module M . We will be done if we can show that there is a ∈ A such that the localization Ma is a free Aa -module. Notice that, if M sits in an exact sequence 0 → L → M → N → 0, and there are a, a ∈ A such that La is free over Aa and Na is free over Aa , then Maa is free over Aaa . Thus it suffices to deal with the quotients Mi /Mi−1 of a composition series 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mn = M . Since we may find a composition series whose successive quotients are isomorphic to modules of the form B/P , where P is a prime ideal, we are reduced to the case where B is a domain and M = B. Let K be the quotient field of A; then BK = B ⊗A K is a Kalgebra of finite type. We argue by induction on the dimension d of Spec(BK ). If d = −1, that is, if BK = {0}, then there is a nonzero element a in the kernel of A → B. Thus Ba = {0} is free over Aa . If d ≥ 0, by Noether’s normalization lemma we may find algebraically independent elements b1 , . . . , bd of BK such that BK is integral over K[b1 , . . . , bd ]. Thus there exists c ∈ A such that b1 , . . . , bd ∈ Bc and Bc is integral over Ac [b1 , . . . , bd ]. It follows, in particular, that Bc is a finitely generated Ac [b1 , . . . , bd ]-module, so there is an exact sequence of Ac [b1 , . . . , bd ]-modules 0 → Ac [b1 , . . . , bd ]m → Bc → C → 0 such that C is a torsion module. Since Ac [b1 , . . . , bd ]m is free over Ac , it suffices to show that there is a ∈ A such that Ca is free over Aca . As before, using a composition series for C, we may reduce to the case where C is an integral Ac -algebra. But in this case, since C is a torsion Ac [b1 , . . . , bd ]-module, the dimension of Spec(CK ) is strictly less than d, and we are done by induction. We are now in a position to prove lemma (3.6). We first construct a sequence of subschemes of S. The first one is S0 = Sred . By Sard’s lemma for flatness the pullback of F to Pr × S0 is flat over an open subset of S0 , whose complement we call S1 . Now we restrict F to Pr ×S1

§4 Construction of the Hilbert scheme

19

and proceed in the same way to obtain S2 , and so on. The process terminates in a finite number of steps by noetherianity. We thus get a finite sequence S0 ⊃ S1 ⊃ · · · ⊃ SN ⊃ SN +1 = ∅ of reduced closed subschemes of S with the property that, setting Ti = Si Si+1 , the pullback of F to Pr × Ti is flat over Ti . Let s be a point of S. Clearly, s belongs to a unique Ti . Let us apply Lemma (3.8) when T = Ti and f is the inclusion of Ti in S. We get f ∗ ξ∗ F (n) = η∗ g ∗ F (n) for n ≥ νi . On the other hand, (g ∗ F )s = Fs , ξ∗ F (n) ⊗ k(s) = f ∗ ξ∗ F (n) ⊗ k(s) . Finally, since g ∗ F is flat over Ti , by the remark immediately preceding the statement of Sard’s lemma for flatness there is an integer μi , not depending on s ∈ Ti , such that H 0 (Pr , (g ∗ F )s (n)) = η∗ g ∗ F (n) ⊗ k(s) , H q (Pr , (g ∗ F )s (n)) = 0 , q > 0, for n ≥ μi . Putting everything together, we conclude that H 0 (Pr , Fs (n)) = ξ∗ F (n) ⊗ k(s) , H q (Pr , Fs (n)) = 0 ,

q > 0,

whenever n ≥ νi , n ≥ μi . Thus it suffices to take as n0 the maximum among all the νi and μi . This concludes the proof of Lemma (3.6). 4. Construction of the Hilbert scheme. Fix a projective space Pr and a rational polynomial p(t). We are ready to construct the Hilbert scheme Hilbp(t) parameterizing subschemes r of Pr with Hilbert polynomial p(t), following the strategy outlined at as a the end of Section 2. According to it, we wish to realize Hilbp(t) r subscheme of the Grassmannian G = G(q(n), H 0 (Pr , O(n))), for large n, where n+r − p(n) . q(n) = h0 (Pr , O(n)) − p(n) = r More exactly, given a subscheme X ⊂ Pr with Hilbert polynomial p(t), the idea is to associate to X the point of G corresponding to the vector subspace H 0 (Pr , IX (n)) ⊂ H 0 (Pr , O(n)), where n is so large that h0 (Pr , IX (n)) = q(n) and H 0 (Pr , IX (n)) generates the homogeneous ideal of X in degree greater than n. The first problem is to find an n that does not depend on X, but only on r and p(t). That this is possible is a consequence of the following result.

20


Lemma (4.1). Let r be a nonnegative integer, and let q(t) be a rational polynomial. Then there exists an integer n0 such that, for any ideal sheaf I ⊂ OPr with Hilbert polynomial q(t) and for any n ≥ n0 , i) H i (Pr , I(n)) = 0 for every i ≥ 1, ii) the natural map H 0 (Pr , I(n)) ⊗ H 0 (Pr , O(1)) −→ H 0 (Pr , I(n + 1)) is onto. It is well known that an integer n0 , possibly depending on I, such that i) and ii) hold for n ≥ n0 , exists. We shall see that one of the standard proofs of this fact actually yields (4.1), provided that a little more attention is paid to details. We argue by induction on r. For r = 0, there is nothing to prove. If r > 0, denote by X the projective scheme defined by I, choose a hyperplane H not containing any of the components of X, including the embedded ones, and set J = I ⊗ OH . Notice that J injects into OH and hence can be viewed as a sheaf of ideals in OH ; in fact, tensoring 0 −→ OPr (−1) −→ OPr −→ OH −→ 0

(4.2)

with OX , one gets the exact sequence α

0 = T or1 (OPr , OX ) −→ T or1 (OH , OX ) −→ OX (−1) −→ OX −→ · · · By the choice of H, the homomorphism α is injective, so T or1 (OH , OX ) vanishes. Tensoring 0 −→ I −→ OPr −→ OX −→ 0 with OH , we find that J = I ⊗ OH injects in OH . Tensoring (4.2) with I(m + 1), one gets the exact sequence (4.3)

0 −→ I(m) −→ I(m + 1) −→ J (m + 1) −→ 0 .

Thus the Hilbert polynomial of J satisfies the identity hJ (t) = hI (t) − hI (t − 1) = q(t) − q(t − 1) and therefore depends only on q(t) and not on I and H. By induction there exists n1 such that i) and ii) are satisfied for J whenever n ≥ n1 . In particular, it follows from (4.3) that, for such an n, the vector space H i (Pr , I(n)) is isomorphic to H i (Pr , I(n + 1)) whenever i ≥ 2; as H i (Pr , I(m)) vanishes for very large m, it follows that H i (Pr , I(n)) vanishes for i ≥ 2. It remains to deal with H 1 (Pr , I(n)). For n ≥ n1 , there is an exact sequence α

n H 0 (Pr , J (n+1)) → H 1 (Pr , I(n)) → H 1 (Pr , I(n+1)) → 0 . H 0 (Pr , I(n+1)) −−→


21

Thus, either αn is surjective, or h1 (Pr , I(n + 1)) is strictly less than h1 (Pr , I(n)). Observe that if αn is surjective, the same is true for αn+1 . In fact, the map H 0 (Pr , I(n + 1)) ⊗ H 0 (Pr , O(1)) −→ H 0 (H, J (n + 1)) ⊗ H 0 (H, O(1)) is surjective because αn is, and the map H 0 (H, J (n + 1)) ⊗ H 0 (H, O(1)) −→ H 0 (H, J (n + 2)) is surjective by induction, so the image in H 0 (Pr , I(n + 2)) of H 0 (Pr , I(n + 1)) ⊗ H 0 (Pr , O(1)) already surjects onto H 0 (H, J (n + 2)). In conclusion, as n increases, h1 (Pr , I(n)) decreases strictly for a while and then stabilizes; since h1 (Pr , I(m)) is certainly zero for very large m, it stabilizes at zero. It follows, in particular, that H 1 (Pr , I(n)) = 0 if n ≥ n1 + h1 (Pr , I(n1 )). On the other hand, there is an upper bound for h1 (Pr , I(n1 )) which is independent of I; in fact, h1 (Pr , I(n1 )) = h0 (Pr , I(n1 )) − q(n1 ) ≤ h0 (Pr , O(n1 )) − q(n1 ) . We claim that n0 = n1 + h0 (Pr , O(n1 )) − q(n1 ) + 1 will do. By what has just been said, i) certainly holds for any n ≥ n0 −1. Now we turn to ii). Look at the diagram H 0 (Pr , I(n))

η

w H 0 (Pr , I(n + 1)) u

αn

β H 0 (Pr , I(n)) ⊗ H 0 (Pr , O(1))

w H 0 (Pr , J (n + 1)) u δ

γ

w H 0 (Pr , J (n)) ⊗ H 0 (Pr , OH (1))

The top row is exact, and γ is surjective for n ≥ n0 , since i) holds for any n ≥ n0 − 1. As n0 > n1 , δ is also surjective for n ≥ n0 , by induction hypothesis. On the other hand, the image of η is contained in the image of β, and a diagram chase shows that β is onto. This ends the proof of the lemma. Remark (4.4). It may be of some use to observe that the result we just finished proving continues to hold, with the same proof, if in the statement I is replaced with a coherent subsheaf of a fixed coherent sheaf on Pr . This is a crucial step in generalizing the construction of the Hilbert scheme to the one of the Quot scheme parameterizing coherent quotients of a fixed coherent sheaf. In practice, we shall often use the following, seemingly more general, version of Lemma (4.1).

22


Corollary (4.5). Let r be a nonnegative integer, and let p(t) be a rational polynomial. Then there exists an integer n0 with the following property. Let Pr × S ⊃ X → S be any flat family of subschemes of Pr with Hilbert polynomial p(t), let ψ : Pr × S → S be the projection, and denote by IX the ideal sheaf of X in Pr × S. Then, for any n ≥ n0 , the following hold: i) ψ∗ IX (n) is locally free of rank q(n); ii) Ri ψ∗ IX (n) = 0 , i ≥ 1 ; iii) the multiplication map ψ∗ IX (n) ⊗ ψ∗ OPr ×S (1) → ψ∗ IX (n + 1) is onto; iv) for any morphism α : T → S, the natural homomorphism α∗ ψ∗ IX (n) → ϕ∗ IY (n) is an isomorphism, where Y = X ×S T ⊂ Pr × T , and ϕ : Pr × T → T is the projection. We first observe that IX is flat over S. This can be seen, for instance, by noticing that, for large m, the sequence 0 → ψ∗ IX (m) → ψ∗ OPr ×S (m) → ψ∗ OX (m) → 0 is exact, and then that, by (2.5), ψ∗ OX (m) is locally free for large m; it then follows that ψ∗ IX (m) is also locally free for large m, and another application of (2.5) proves our claim. We next let n0 be the integer provided by Lemma (4.1), where q(m) = m+r −p(m). Combining part i) of Lemma (4.1) with Proposition r (3.3) proves i), ii), and also iv), since (id ×α)∗ IX (n) = IY (n). Finally, the fact that ψ∗ IX (n) ⊗ k(s) = H 0 (Xs , IXs (n)) reduces iii) to part ii) of (4.1). We may now complete the construction of the Hilbert scheme Hilbp(t) r . Let X be a closed subscheme of Pr with Hilbert polynomial p(t). Let n+r − p(n) q(n) = r be the Hilbert polynomial of the ideal sheaf IX of X. Fix an integer n greater or equal to the integer n0 provided by Corollary (4.5). We may then associate to X ⊂ Pr its nth Hilbert point, that is, the surjective homomorphism ϕn : H 0 (Pr , OPr (n)) → H 0 (X, OX (n)) , or, dually, the kernel of ϕn , that is, the subspace H 0 (Pr , IX (n)) ⊂ H 0 (Pr , OPr (n)) .


23

The nth Hilbert point of X ⊂ Pr is a point of the Grassmannian G = G(q(n), H 0 (Pr , OPr (n))) parameterizing q(n)-dimensional subspaces of H 0 (Pr , OPr (n)). The nth Hilbert point determines X ⊂ Pr completely; in fact, part ii) of (4.1) says that the homogeneous ideal of X is generated, in degree n or more, by H 0 (Pr , IX (n)). Moreover, by part i) of (4.1), the dimension of H 0 (Pr , IX (m)) equals q(m) for any m ≥ n (in fact, for any m ≥ n0 ). In view of part ii) of (4.1), this means that the nth Hilbert point of X belongs to the subset H of the Grassmannian G whose points are the vector subspaces V ⊂ H 0 (Pr , OPr (n)) such that the image of the multiplication map ρm,V : V ⊗ H 0 (Pr , OPr (m − n)) → H 0 (Pr , OPr (m)) has dimension q(m). Conversely, suppose that V belongs to H. Then, by definition, V generates a homogeneous ideal whose Hilbert polynomial is q(t) and therefore comes from a closed subscheme of Pr with Hilbert polynomial p(t). Thus attaching to any X ⊂ Pr its nth Hilbert point establishes a one-to-one correspondence between closed subschemes of Pr with Hilbert polynomial p(t) and points of H. The Hilbert scheme Hilbp(t) will be exhibited as a subscheme of G whose set of closed points r is H. To construct it, we begin by globalizing the multiplication maps ρh,V . We let ξ : Pr × G → G be the projection, and F → ξ∗ OPr ×G (n) the universal subsheaf on G. We then let ρh : F ⊗ ξ∗ OPr ×G (h − n) → ξ∗ OPr ×G (h) be the multiplication map. We denote by Σh the determinantal locally closed subscheme of G defined by the condition rank ρh = q(h) . To be more precise, Σh is the intersection of the open subset of G where at least one q(h) × q(h) minor of ρh does not vanish with the subscheme of G whose ideal sheaf is generated by the (q(h) + 1) × (q(h) + 1) minors of ρh . Clearly, H= (Σh )red . h≥n

We would like to define Hilbp(t) to be the scheme-theoretic intersection r of all the Σh . The problem is that it is not clear that this makes sense,

24


as we are dealing with an infinite intersection of locally closed, and not necessarily closed, subschemes of G. The way out is to show that the finite intersections Σh Ξk = n≤h≤k

stabilize for large k. To do this, it suffices to show that the Ξk stabilize as sets. In fact, if this happens, then there is a k0 such that, for k larger than k0 , {Ξk } is a decreasing sequence of closed subschemes of Ξk0 and hence stabilizes by noetherianity. Denote by |Ξk | the set of closed points of Ξk . It is enough to show that the sequence {|Ξk |} is constant for large k. To prove this, it suffices to find, for every closed point s in G, a closed subscheme Ys of Pr and an integer N ≥ n, independent of s, such that (4.6)

rank ρh,s = hIYs (h)

for h ≥ N ,

where IYs is the ideal sheaf of Ys . Suppose in fact that this has been done. By definition, given a closed point s ∈ ΞN +r , we have rank ρh,s = q(h)

for n ≤ h ≤ N + r .

As both q(h) and hIYs (h) are polynomials in h of degree not exceeding r, they must coincide. This shows that rank ρh,s = q(h) for h ≥ N , or equivalently that s ∈ Ξh for all h ≥ N , proving the claim. It is clear that a candidate for Ys is the subscheme of Pr corresponding to the ideal generated by the subspace V ⊂ H 0 (Pr , OPr (n)), where [V ] = s ∈ G. To prove the existence of an integer N such that (4.6) holds, it is best to exhibit Ys as the fiber over s of a closed subscheme (4.7)

Y ⊂ Pr × G.

We will later realize that this subscheme, when restricted to Hilbp(t) r , is nothing but the universal family over Hilbp(t) . We set r Fm = ρm (F ⊗ ξ∗ OPr ×G (m − n)) . Then ⊕j Fj is a graded sheaf of ideals in ⊕j ξ∗ OPr ×G (j). Hence there are a sheaf of ideals J ⊂ OPr ×G and an integer N ≥ n such that Fm = ξ∗ J (m)

for m ≥ N .

We denote by Y the subscheme of Pr × G corresponding to J . We may assume that R1 ξ∗ J (m) = 0


25

for m ≥ N and also, by Lemma (3.6), that (4.8)

H 0 (Ys , OYs (m)) = ξ∗ OY (m) ⊗ k(s) , H i (Ys , OYs (m)) = 0 , i > 0 ,

for m ≥ N and for any s ∈ G, where Ys stands for the fiber of Y → G over s. Then the sequence ρm

F ⊗ ξ∗ OPr ×G (m − n) −−→ ξ∗ OPr ×G (m) → ξ∗ OY (m) → 0 is exact for m ≥ N , and tensoring it with k(s) we get another exact sequence ρm,s

(F ⊗k(s))⊗H 0 (Pr , OPr (m−n)) −−−→ H 0 (Pr , OPr (m)) → H 0 (Ys , OYs (m)) → 0. It follows that the image of ρm,s is H 0 (Ys , IYs (m)). On the other hand, this same exact sequence and (4.8) imply that H i (Ys , IYs (m)) vanishes for i > 0 and m ≥ N . In conclusion, rank ρm,s = hIYs (m) when m ≥ N , as we had to show. This concludes the construction of the scheme structure of Hilbp(t) r . represents the Hilbert functor. There It remains to prove that Hilbp(t) r is already a natural candidate for the universal family on Hilbp(t) r , namely of Y → G, which we denote by the restriction to Hilbp(t) r π : X → Hilbp(t) . r This family is flat. To show this, we use the criterion (2.5). Let j : Hilbp(t) → G be the inclusion. Lemma (3.8) implies that, for large h, r π∗ OX (h) = j ∗ ξ∗ OY (h) = j ∗ (coker(ρh )) . On the other hand, since the maps ∗ Hilbp(t) r , the sheaves j (coker(ρh )) are h ≥ n. Hence X is flat over Hilbp(t) r . induces by pullback a α : S → Hilbp(t) r

ρh have constant rank q(h) on locally free of rank p(h) for any As a consequence, any morphism flat family f

Pr × S ⊃ X = X ×Hilbp(t) S − →S r

of subschemes of Pr with Hilbert polynomial p(t). Denote by ϕ and ψ the projections from Pr × S and Pr × Hilbp(t) to their second factors. r Then, by dimension reasons, it follows from part i) of Corollary (4.5) of the universal that ψ∗ IX (n) ⊂ ψ∗ O(n) is just the restriction to Hilbp(t) r subbundle on G. It also follows from part iv) of (4.5) that this pulls back, via α, to ϕ∗ IX (n) ⊂ ϕ∗ OPr ×S (n).

26


Conversely, let Pr × S ⊃ X → S be any flat family of subschemes of P with Hilbert polynomial p(t), and let ϕ : Pr ×S → S be the projection. Again by Corollary (4.5), the sheaf ϕ∗ IX (m) is locally free of rank q(m) for any m ≥ n, and moreover the multiplication map r

ϕ∗ IX (n) ⊗ ϕ∗ OPr ×S (m − n) → ϕ∗ IX (m) is onto. The universal property of the Grassmannian tells us that there is a unique morphism α : S → G such that the universal subbundle on G pulls back to ϕ∗ IX (n). By what we have just observed, α factors through Hilbp(t) r , and, by construction X = X ×Hilbp(t) S . r

This ends the proof that Hilbp(t) represents the Hilbert functor. The r construction of the Hilbert scheme is complete. We end this section with a few remarks. The first is that the universal property of the Hilbert scheme holds also with respect to analytic families of proper subschemes of Pr , that is, for analytic subspaces X ⊂ Pr × S, where S is an analytic space, and the projection from X to S is proper and flat. In other words, any such family is obtained, in a unique way, by pullback via an analytic map from S to an appropriate Hilbert scheme. The proof of this fact is essentially the same as in the algebraic case. Here is an immediale and useful consequence. Lemma (4.9). Every flat analytic family of projective schemes is locally the pullback of an algebraic one. The second remark is the following. is projective. Proposition (4.10). The Hilbert scheme Hilbp(t) r By construction, the Hilbert scheme is a subscheme of a Grassmannian G, which is projective; thus to prove projectivity, it suffices to prove properness. We will use the valuative criterion, in the following form. We will show that, given a smooth curve S and a closed point s ∈ S, any morphism ψ : S ∗ = S {s} → Hilbp(t) extends to all of r S. First of all, ψ extends to a morphism ϕ : S → G. Recall, from the construction of the Hilbert scheme, that the universal family X → Hilbp(t) r is the restriction of the (nonflat) family Y → G introduced in (4.7). By base change via ϕ, the latter pulls back to a family q

Pr × S ⊃ Z − → S. Clearly, the restriction Z ∗ → S ∗ to S ∗ of this family is just the pullback via ψ, and hence is flat. Let m be the ideal sheaf of s of X → Hilbp(t) r

§5 The characteristic system

27

in OS , and denote by T the ideal sheaf in OZ consisting of the sections which are annihilated by m. It is evident that T is concentrated on q−1 (s). Now let J be the inverse image of T via OPr ×S → OZ , and denote by W the subscheme of Pr × S it defines. We claim that W is flat over S. Since W agrees with Z ∗ over S ∗ , this will prove the proposition, by virtue of the universal property of the Hilbert scheme. To prove flatness, we use the criterion (2.5). We must show that the pushforward q∗ OW (n) is locally free for large n. Away from s, this follows from the flatness of Z ∗ over S ∗ . At s, we argue as follows. Let t be a local parameter at s, that is, a local generator for m. By construction, t is not a zero divisor in OW . Thus, q∗ OW (n) is torsion-free at s for all n. Since S is a smooth curve, this implies that q∗ OW (n) is locally free at s for all n. It may be worth noticing that, as shown by Hartshorne [354], the is also connected. Hilbert scheme Hilbp(t) r 5. The characteristic system. In this section we will study the tangent spaces to the Hilbert scheme. We denote by C[ε] the ring of dual numbers, by Σ the scheme Spec C[ε], and by s0 its closed point. The tangent space to H = Hilbp(t) r at a point h corresponding to a subscheme X0 ⊂ Pr is in one-to-one correspondence with the set of morphisms of pointed schemes Hom((Σ, s0 ), (H, h)) . Since H represents the Hilbert functor, this can be identified with the set of all flat families f

→ Σ = Spec C[ε] Pr × Σ ⊃ X − whose fiber over s0 is X0 ⊂ Pr . More generally, given a scheme Y and a closed subscheme X0 ⊂ Y , we shall study the first-order embedded deformations of X0 in Y , meaning those flat families f

→ Σ = Spec C[ε] Y ×Σ⊃X − whose fiber over s0 is X0 ⊂ Y We first wish to explain the case where Y is affine. Let then R be the coordinate ring of Y so that R[ε] = R ⊗C C[ε] is the coordinate ring of Y × Σ. A subscheme X of Y × Σ corresponds to an ideal J ⊂ R[ε], and the flatness of the natural projection from X to Σ means that R[ε]/J is a flat C[ε]-module. We wish to show that the flatness of this family of subschemes of Y is equivalent to saying that every presentation by generators and relations of the ideal I of the central fiber X0 = X mod (ε) extends to a presentation of J. Formally, we shall prove the following more general result.

28


Lemma (5.1). Let ϕ : A → B be a homomorphism of noetherian commutative rings, with B flat over A. Assume either that A is a local artinian ring or that A and B are both local rings and ϕ is a local homomorphism. Let J be an ideal in B, and set C = B/J. Let k be the quotient of A modulo its maximal ideal mA , and set Bk = B ⊗A k, Ck = C ⊗A k. Then the following statements are equivalent: i) C is flat over A; ii) every exact sequence (5.2)

Bkl → Bkh → Bk → Ck → 0 is the reduction modulo mA of an exact sequence: Bl → Bh → B → C → 0 ;

iii) there are generators F1 , . . . , Fh of J such that, denoting by fi the image of Fi in Bk , i = 1, . . . , h, every relation among the fi extends to a relation among the Fi . The case of first-order deformations of embedded affine subschemes introduced above is the special one in which A = C[ε] and B = R[ε]. The reader may find it helpful to follow the proof with this specific example in mind. In proving Lemma (5.1) we shall rely on the following well-known results. Lemma (5.3). Let R be a commutative ring, and let 0 → E → F → G → 0 be an exact sequence of R-modules. If G is flat, then E is flat if and only if F is flat. Tensoring the exact sequence with an R-module H, we get an exact sequence of Tor’s R R R · · · → TorR 2 (G, H) → Tor1 (E, H) → Tor1 (F, H) → Tor1 (G, H) → · · · .

The terms on the right and on the left vanish since G is flat, so the middle terms are isomorphic. Thus E is flat, that is, TorR 1 (E, H) vanishes for any H, if and only if the same is true for F . Lemma (5.4) (Local criterion of flatness). Let ϕ : A → B be a homomorphism of commutative rings. Assume either that A is a local artinian ring or that A and B are both local noetherian rings and ϕ is a local homomorphism. Then a finitely generated B-module M (or any B-module if A is artinian) is flat over A if and only if TorA 1 (M, k) = 0, where k stands for the quotient of A modulo its maximal ideal mA .


29

The only if part is obvious. For a proof of the converse in the case where ϕ is a local homomorphism of local noetherian rings, we refer to [194], Theorem 6.8. If A is artinian, we need to prove that TorA 1 (M, N ) vanishes for any finitely generated A-module N . There is an integer v such that mvA = 0, and it suffices to show that TorA 1 (M, Ni ) vanishes for N of the composition series each quotient Ni = miA N/mi+1 A N = m0A N ⊃ m1A N ⊃ · · · ⊃ mvA N = 0 . That TorA 1 (M, Ni ) is zero follows immediately from the assumptions since Ni is a finite-dimensional vector space over k. We now return to the proof of Lemma (5.1). We shall show that i) implies ii) and that iii) implies i); that iii) follows from ii) is obvious. We may write Ck = Bk /I, where I is an ideal. Since Ck = B/(J + mA B), the ideal I is isomorphic to J/(J ∩ mA B). Suppose now that C is flat over A, and let an exact sequence of the form (5.2) be given. Lemma (5.3), applied to the exact sequence 0 → J → B → C → 0, implies that J is flat over A. Moreover, tensoring this sequence with k, we obtain the exact sequence · · · → TorA 1 (C, k) → J ⊗A k → Bk → Ck → 0 . The term on the left vanishes by the flatness of C, and hence J ⊗A k = I, or, equivalently, mA J = J ∩mA B, since J ⊗A k = J/mA J. We can construct the commutative diagram Bkh u

β

w Iu

Bh

α

wJ

w0

We claim that α is onto. In fact if Q is the cokernel of α, we have that Q ⊗A k can be identified with the cokernel of β and hence is zero; this means that Q = mA Q. If B is a local ring, this implies that Q = mB Q, so Q = 0 by Nakayama’s lemma. If A is artinian, there is an integer v such that mvA = 0, so Q = mA Q = m2A Q = · · · = mvA Q = 0. In any case the conclusion is that Q is zero. At this stage we have extended the generators of I to generators of J. It remains to extend relations. Let N be the kernel of β, and M the kernel of α. We then have the commutative diagram 0

w Nu

w Bkh u

β

w Iu

w0

0

wM

w Bh

α

wJ

w0

30


Tensoring the lower sequence with k and using the flatness of J, we find that M ⊗A k equals N . We can construct a commutative diagram w Nu

Bkl u γ

Bl

w0

wM

and arguing exactly as before, we see that γ is onto, so that (5.5)

γ

α

Bl − → Bh − →B→C→0

is an exact sequence extending (5.2). This proves that i) implies ii). Now assume that iii) holds. The choice of the generators F1 , . . . , Fh corresponds to a surjective homomorphism B h → J; likewise, f1 , . . . , fh give a surjective homomorphism Bkh → I. Denoting by M and N the kernels of these homomorphisms, condition iii) means that M maps onto N . Thus, if B l → M is onto, then Bkl maps surjectively onto N . In other words, tensoring the exact sequence (5.6)

Bl → Bh → B → C → 0

with k yields an exact sequence Bkl → Bkh → Bk → Ck → 0 . We can complete (5.6) to a free resolution of C; tensoring with k, we obtain a complex whose homology calculates the groups TorA i (C, k). On the other hand, we just noticed that this complex is exact in degree one, so TorA 1 (C, k) vanishes. By Lemma (5.4), this is sufficient to infer that C is flat over A. This concludes the proof of the lemma. A way of rephrasing the equivalence of i)–iii) in the preceding lemma is the following: Corollary (5.7). Let A and B be as in (5.1). Let f1 , . . . , fh be elements of B/mA B, and, for each i = 1, . . . , h, let Fi be an element in B which reduces to fi modulo mA B. Then B/(F1 , . . . , Fh ) is flat over A if and only if every relation among the fi extends to a relation among the Fi . Lemma (5.1) makes it possible to completely classify embedded firstorder deformations of a subscheme X0 ⊂ Y . We begin with the affine case. Lemma (5.8). Let R be a commutative noetherian C-algebra, and let I be an ideal in R. The first-order embedded deformations of X0 = Spec(R/I) within Y = Spec(R) are in one-to-one correspondence with HomR/I (I/I 2 , R/I) = HomR (I, R/I).


31

Proof. Given an element g of R, we write [g] to indicate its class modulo I. We must classify the ideals J ⊂ R[ε] such that R[ε]/J is flat over C[ε] and J/((ε) ∩ J) = I. Let one such J be given. Given i ∈ I, pick j ∈ J whose reduction modulo (ε) is i. We can then write j = i − εh, where h is an element of R which depends R-linearly on j and which is uniquely determined by i modulo I; in fact, if i = 0, then εh belongs both to J and to (ε), and, during the proof of Lemma (5.1), we observed that the flatness of R[ε]/J implies, in particular, that J ∩ (ε) = εJ = εI, so that h ∈ I. Conversely, suppose we are given a homomorphism α : I → R/I. Choose generators f1 , . . . , fn for I, write α(fi ) = [gi ], where gi ∈ R, and set Fi = fi − εgi and J = (F1 , . . . , Fn ). Clearly, J/(J ∩ (ε)) = I. We wish to show that R[ε]/J is flat over C[ε]. By Corollary (5.7), this follows if we can show that any relation among the fi is the reduction modulo (ε) of a relation among the Fi . Let then ai fi = 0 be a relation and notice that ai f i = 0 , ai [gi ] = α ai gi = bi fi for some meaning that ai gi ∈ I, so that we can write elements bi in R. Thus,

(ai + εbi )Fi =

ai fi + ε

b i fi −

ai g i = 0

ai fi = 0. Summing up, we is a relation among the Fi which extends have associated to each ideal J in R[ε] extending I and such that R[ε]/J is flat over C[ε], a homomorphism of R-modules from I to R/I, and conversely. It is then a trivial exercise to check that these two maps are inverse to each other, thus proving the lemma. Let X be a closed subscheme of a fixed scheme Y , and let I be the ideal of X in Y . The sheaf I/I 2 , or rather its restriction to X, is called the conormal sheaf of X in Y and denoted CX/Y . Its dual HomOX (CX/Y , OX ) = HomOY (I, OX ) is the normal sheaf of X in Y and is denoted NX/Y . From Lemma (5.8) one immediately obtains a description of all first-order embedded deformations of X in Y . Proposition (5.9). Let X be a closed subscheme of Y , and let I be the ideal sheaf of X in Y . Then the first-order embedded deformations of X in Y are in a ono-to-one correspondence with H 0 (X, NX/Y ) = HomOX (CX/Y , OX ) = HomOY (I, OX ). This is a natural place to introduce, in the case at hand, a general machinery which will occur several times, in different contexts, throughout this volume. Let Y ×B ⊃X →B

32


be a flat family of subschemes of Y parameterized by a scheme B. Let X = Xb0 for some closed point b0 ∈ B. Let v be a tangent vector to B at b0 . Such a vector can be interpreted as a morphism of pointed schemes v : Spec C[ε] −→ (B, b0 ) . Pulling back the family X via v yields a first-order embedded deformation of X in Y and hence, by Proposition (5.9), an element of H 0 (X, NX/Y ). This assignment defines a map (5.10)

Tb0 (B) −→ H 0 (X, NX/Y ) .

Following a classical terminology, this map is called by Kodaira and Spencer the characteristic map. The map (5.10) is linear; a proof is outlined in exercises A-1 to A-8. It is perhaps of some interest to relate Proposition (5.9) with the classical notion of characteristic system. The setting is the one of a family X = {Xs }s∈S of divisors in a smooth projective variety Y , parameterized by a smooth variety S. Let X = Xs0 be a reduced and irreducible member of this family. Then the characteristic linear system of the family at X is the linear system cut out on X by “infinitely near” members of the family. More precisely, suppose that X is defined locally by the equation f (y1 , . . . , yn , t1 , . . . , tk ) = 0 , where y1 , . . . , yn are local coordinates on Y , and t1 , . . . , tk are local coordinates on S centered at s0 , so that the equation of X in Y is f (y1 , . . . , yn , 0, . . . , 0) = 0 . For each i, the equation ∂f (y1 , . . . , yn , t1 , . . . , tk ) =0 ∂ti t1 =···=tk =0 defines, locally, a divisor on X. Since f is uniquely determined up to multiplication by a unit g, this divisor does not depend on the choice of f . In fact, ∂f ∂gf ≡ g ∂ti t1 =···=tk =0 ∂ti t1 =···=tk =0

mod

f |t1 =···=tk =0 .

Thus (∂f /∂ti )|t1 =···=tk =0 determines a global divisor Di on X. The linear system generated by the Di is the classical characteristic system. In our notation, {∂/∂ti } is a basis for the tangent space to S at s0 ,


33

and (∂f /∂ti )|t1 =···=tk =0 is the image in H 0 (X, NX/Y ) of {∂/∂ti } via the characteristic map (5.10). Of course, in this particular case, I = OY (−X), so that NX/Y = OY (X) ⊗ OX OX (Di ). Let us go back to the Hilbert scheme Hilbp(t) r . Let h be a (closed) , and let X be the corresponding subscheme of Pr . point of Hilbp(t) r Proposition (5.9) has the following important consequence. Corollary (5.11). The tangent space to Hilbp(t) at h is given by r 0 Th (Hilbp(t) r ) = H (X, NX/Pr ) .

In particular, this result shows that h0 (X, NX/Pr ) is an upper bound at h; a lower bound for this same dimension for the dimension of Hilbp(t) r is given by the following important proposition. Proposition (5.12). Let X be a closed local complete intersection subscheme of Pr , and let h be the corresponding point of Hilbp(t) r . Then the dimension of every irreducible component of Hilbp(t) at h is at r least h0 (X, NX/Pr ) − h1 (X, NX/Pr ) . We shall not prove this fundamental existence result but refer to [440] or [625] for a proof. Now let us give some applications. Example (5.13). We revisit Example (2.8). Let H be the Hilbert scheme of degree d zero-dimensional subschemes of Pr and consider a point of H corresponding to a subscheme Z ⊂ Pr . If Z consists of d distinct points p1 , . . . , pd , then H 0 (Z, NZ/Pr ) = ⊕di=1 Tpi (Pr ) = Tp1 +···+pd (Symd (Pr )) , where Symd ( ) stands for d-fold symmetric product. This shows that the open subset of Symd (Pr ) consisting of d-uples of distinct points embeds in H as an open subset. Example (5.14). Consider hypersurfaces of degree d in Pr . We know that the Hilbert scheme of these is isomorphic to a projective space of dimension d+r − 1. r In fact, if X is a degree d hypersurface in Pr , then d+r − 1. h0 (X, NX/Pr ) = h0 (X, OX (d)) = r Example (5.15). Consider a smooth complete nondegenerate degree d curve C in Pr , and let g be its genus. The curve C corresponds to a

34


point [C] in the Hilbert scheme Hilbp(t) r , where p(t) = dt + 1 − g. We shall write H to denote this Hilbert scheme. We might naively expect that dim[C] H = h0 (C, NC/Pr ) = χ(NC/Pr ), as is certainly the case for r = 2, by the preceding example. Actually, all we can say is that dim[C] H ≤ h0 (C, NC/Pr ) . On the other hand, in the case under consideration the lower bound on the dimension of H given by (5.12) equals the Euler characteristic of NX/Pr , whence the inequality (5.16)

dim[C] H ≥ χ(NX/Pr ) .

The Euler characteristic of NX/Pr is readily calculated. From the sequence 0 → TC → TPr ⊗ OC → NC/Pr → 0 we see that the degree of the normal bundle is deg NC/Pr = (r + 1)d + 2g − 2, and then, by Riemann–Roch, we have (5.17)

χ(NC/Pr ) = deg(NC/Pr ) − (r − 1)(g − 1) = (r + 1)d − (r − 3)(g − 1) .

We now sketch an alternative proof of (5.16), referring to Section 8 of Chapter XXI, and specifically to Remark (8.20) therein, for more details. As we shall see in that chapter, in any family of line bundles of degree d on curves of genus g, the locus of those line bundles having r + 1 or more sections has codimension at most (r + 1)(g − d + r) in the neighborhood of a line bundle with exactly r + 1 sections. Applying this to the “family of all line bundles of degree d on all curves of genus g,” one concludes that the family of all linear series of degree d and dimension r on curves of genus g has local dimension at least 4g − 3 − (r + 1)(g − d + r) everywhere. Since such a linear series, when without base points, determines a map of a curve to Pr up to the (r2 + 2r)-dimensional family P GL(r + 1, C) of automorphisms of Pr , we may conclude that the dimension of H at [C] is at least dim[C] H ≥ 4g − 3 − (r + 1)(g − d + r) + (r2 + 2r) = (r + 1)d − (r − 3)(g − 1) , as desired. Note one curious feature of the behavior of χ(NC/Pr ) as a function of g. When r = 2, it increases with g; when r = 3, it equals 4d and hence is independent of the genus, while for r ≥ 4, it decreases with g.


35

There is one important case where the dimension of H is exactly equal to χ(NC/Pr ). Suppose that C is as above and that, in addition, OC (1) is nonspecial. We claim that H is smooth at [C] of dimension χ(NC/Pr ). To see this, it suffices to show that h0 (C, NC/Pr ) = χ(NC/Pr ) or, equivalently, that h1 (C, NC/Pr ) = 0. Look at the cohomology of the diagram 0 u TC 0

w OC

w

r+1 OC (1)

w TPr

u ⊗ OC

w0

u NC/Pr u 0 where the horizontal sequence is the restriction to C of the Euler sequence for the tangent bundle of projective space. We get, in particular, H 1 (C, OC (1))r+1

w H 1 (C, TPr ⊗ OC )

w0

u H 1 (C, NC/Pr ) u 0 From the assumption that OC (1) be nonspecial, it then follows that H 1 (C, TPr ⊗ OC ), and hence H 1 (C, NC/Pr ), vanishes. We can summarize what has been proved in the following statement. Proposition (5.18). Let C be a smooth, irreducible, nondegenerate curve of degree d and genus g in Pr such that OC (1) is nonspecial. Set p(t) = dt + 1 − g. Then: i) dim[C] Hilbp(t) = dim T[C] (Hilbp(t) r r ) = χ(NC/Pr ) = (r+1)d−(r−3)(g− 1), ii) H 1 (C, NC/Pr ) = 0. We close this section by discussing some other consequences of Lemma (5.1), centered around the permanence, under small deformations, of the property of being a local complete intersection. Recall that a regular

36


sequence in a commutative ring R is a finite sequence f1 , . . . , fn of elements of R such that the ideal (f1 , . . . , fn ) is proper and fi is not a zero divisor in R/(f1 , . . . , fi−1 ) for all i with 1 ≤ i ≤ n. An embedding of schemes Y → X is said to be a regular embedding if every point of Y has an affine neighborhood U = Spec R in X such that the ideal of U ∩ Y in R is generated by a regular sequence; the length of the sequence, which is independent of the particular sequence chosen (cf., for instance, Lemma (5.22) below), is called the codimension of the embedding. When X is smooth, the notion of regularly embedded subscheme of X coincides with the one of local complete intersection subscheme of X (cf. [503], pp. 105 and 121). The relations among the elements of a regular sequence are particularly easy to describe. The following elementary lemma describes the first step of what usually goes under the name of Koszul resolution. be a regular Lemma (5.19). Let R be a commutative ring, let f1 , . . . , fh sequence in R, and let a1 , . . . , ah be elements of R. Then ai fi = 0 if ) with entries in and only if there is an h × h skew-symmetric matrix (c ij R such that ai = j cij fj for all i. The “if” part f1 , . . . , fh be a using induction a1 = 0, and we (5.20)

is obvious and does not require the assumption that regular sequence. The converse is also easy to prove, on h. Suppose in fact that ai fi = 0. If h = 1, then are done. If h > 1, we have instead that bi fi ah = i0 mk = 0, it follows that a = 0. di,j bj . Substituting We now prove ii). Write bm = ci fi and fi = the latter relations into the first yields a relation among the bi . If all the coefficients ci did belong to m, the coefficient of bm in this relation would be invertible, contradicting the minimality of b1 , . . . , bm . Thus, possibly rearranging the fi , we may assume that cn is a unit. It is then immediate to check that f1 , . . . , fn−1 , bm is a regular sequence and that it generates I. Now write bm−1 as a linear combination of f1 , . . . , fn−1 , bm . If the coefficients of f1 , . . . , fn−1 all belonged to m, we could conclude, arguing as before, that b1 , . . . , bm is not a minimal system of generators of I. Possibly rearranging the fi , we may thus suppose that the coefficient of fn−1 is a unit. Hence f1 , . . . , fn−2 , bm−1 , bm is a regular sequence generating I. This procedure can be repeated until either all the bi have been used, or there are no more fi to be replaced. The first alternative must occur, since otherwise b1 , . . . , bm would not be a minimal system of generators of I. On the other hand, the final regular sequence one obtains cannot contain elements of R other than b1 , . . . , bm , since these already generate I. In conclusion, m = n, and b1 , . . . , bm is a regular sequence. Q.E.D. Here is another consequence of (5.19). Lemma (5.23). Let ϕ : A → B be a local homomorphism of noetherian local rings, with B flat over A. Let J = (F1 , . . . , Fh ) be an ideal in B, and let k be the quotient of A modulo its maximal ideal mA . Denote by f1 , . . . , fh the images of F1 , . . . , Fh in Bk = B ⊗A k. Then, if f1 . . . , fh is a regular sequence, C = B/J is flat over A. Moreover, F1 , . . . , Fh is a regular sequence.

38


We first prove flatness. Let ai fi = 0 be a relation among the fi . We know, by Lemma (5.19), that there is a skew-symmetric matrix (cij ) with entries in Bk such that ai = j cij fj for every i. For each choice of i and j with i > j, pick an element dij of B mapping tocij , and set dji = −dij ; also set dii = 0. Then bi Fi = 0, where bi = j dij Fj , is a ai fi = 0. Summing up, any relation relation among the Fi extending among the fi extends to a relation among the Fi , and the conclusion follows from (5.1), or better from (5.7). It remains to prove that F1 , . . . , Fh is a regular sequence. Set Ji = (F1 , . . . , Fi ). Since B is A-flat by assumption, and B/Ji is A-flat for any i, as we just proved, applying Lemma (5.3) successively to the exact sequences of B-modules 0 → Ji → B → B/Ji → 0, 0 → Ji /Ji−1 → B/Ji−1 → B/Ji → 0, ×F

i 0 → Ki → B/Ji−1 −−−→ Ji /Ji−1 → 0

shows that Ji , Ji /Ji−1 and Ki are A-flat, where Ki is the kernel of the multiplication map ×Fi : B/Ji−1 → B/Ji−1 . It follows that Ki ⊗A k can be identified with the kernel of ×fi : Bk /(f1 , . . . , fi−1 ) → Bk /(f1 , . . . , fi−1 ) and hence vanishes. This means that Ki = mA Ki . A fortiori, Ki = mB Ki , so Ki vanishes by Nakayama’s lemma. This concludes the proof of (5.23). The following property of regular sequences is well known. Lemma (5.24). Let I = (f1 , . . . , fh ) be a proper ideal in a local ring A. If f1 , . . . , fh is a regular sequence, then I/I 2 is a free A/I-module of rank h. The proof of this result is immediate. It is clear that the classes of f1 , . . . , fh generate I/I 2 ; all we have to do is2 show that they are independent. Suppose that ai fi belongs to I , i.e., that we may write a i fi = bi fi , where the bi belong to I. By Lemma (5.19), the coefficients of the relation (ai − bi )fi = 0 must be of the form ai − bi = j cij fj for some skew-symmetic matrix (cij ). This implies that ai ∈ I for each i, finishing the proof. A consequence of Lemma (5.24) is that, if Y → X is a regular embedding, then the conormal sheaf CY /X to Y in X, and hence also the normal sheaf NY /X , are locally free OY -modules. Lemma (5.25). Let π : X → S be a flat morphism of schemes, and let Y be a subscheme of X, flat and proper over S. Suppose that Ys0 → Xs0 is a regular embedding for some s0 ∈ S. Then Ys → Xs is a regular embedding for all s in a suitable Zariski neighborhood of s0 .


39

Let x be a point of Ys0 . There is an affine neighborhood of x in Xs0 over which the ideal of Ys0 in Xs0 is generated by a regular sequence f1 , . . . , fn . By (5.1), the fi extend to generators F1 , . . . , Fn of the ideal of Y on an affine neighborhood of x in X; by (5.23), we may assume that F1 , . . . , Fn is a regular sequence. To prove (5.25), we are thus reduced to proving the following result. Lemma (5.26). Let ϕ : A → B be a homomorphism of noetherian rings, J an ideal in B, and P a prime ideal in B containing J. Set Q = ϕ−1 (P ). Suppose that B and B/J are flat over A and that J is generated by a regular sequence F1 , . . . , Fn . Then the images of F1 , . . . , Fn in BP ⊗AQ AQ /mAQ form a regular sequence. Set Jh = (F1 , . . . , Fh ), 1 ≤ h ≤ n, and denote by gi the image of Fi in BP ⊗AQ AQ /mAQ . We know that ×F

h 0 → B/Jh−1 −−−→ B/Jh−1 → B/Jh → 0

is exact. Hence, 0 → (B/Jh−1 )P → (B/Jh−1 )P → (B/Jh )P → 0 is also exact. If (B/Jh )P were flat over AQ , ×Fh ⊗ 1 : (B/Jh−1 )P ⊗AQ AQ /mAQ → (B/Jh−1 )P ⊗AQ AQ /mAQ would be injective. On the other hand, (B/Jh−1 )P ⊗AQ AQ /mAQ = (BP ⊗AQ AQ /mAQ )/(g1 , . . . , gh−1 ), and ×Fh ⊗ 1 is the multiplication by gh , so we would conclude that g1 , . . . , gn is a regular sequence. We must then show that (B/Jh )P is flat over AQ for every h. This we will do by descending induction on h, starting from the case h = n, where we have flatness by assumption. In general, from 0 → B/Jh → B/Jh → B/Jh+1 → 0 we obtain an exact sequence A A A TorA 2 (B/Jh+1 , N ) → Tor1 (B/Jh , N ) → Tor1 (B/Jh , N ) → Tor1 (B/Jh+1 , N )

for any A-module N . Since A

Q TorA i (B/Jh+1 , N )P = Tori ((B/Jh+1 )P , NQ )

and the latter is zero by induction hypothesis, multiplication by Fh+1 gives an isomorphism ∼

TorA → TorA 1 (B/Jh , N )P − 1 (B/Jh , N )P .

40


This implies that A TorA 1 (B/Jh , N )P = mBP Tor1 (B/Jh , N )P

and hence, by Nakayama’s lemma, that A

Tor1 Q ((B/Jh )P , NQ ) = TorA 1 (B/Jh , N )P = 0 when NQ is finitely generated over AQ . This proves that (B/Jh )P is flat over AQ and concludes the proof of Lemma (5.26). 6. Mumford’s example. As we have already hinted, Hilbert schemes can be quite pathological. In this section we shall give an example, due to Mumford, of a Hilbert scheme of space curves which is everywhere non-reduced along one of its components. The general point of the offending component is a curve liying on a smooth cubic surface in P3 . In our discussion, we shall freely use some well-known facts about cubic surfaces; for a more thorough discussion of these and for proofs, we refer to [318]. Let F be a smooth cubic surface in P3 . It is a classical fact that on F there are exactly 27 lines. Denote by L one of them, and by X a hyperplane section. Using the adjunction formula, we get that ωF = O(−X) ,

(X · X) = 3 ,

(X · L) = 1 ,

(L · L) = −1 .

It is clear that H 0 (F, O(nX + mL)) vanishes when n < 0 and m ≤ 0. In fact, the assumption that m be nonpositive is not needed. This can be shown, inductively on m, using the exact cohomology sequence of 0 → O(nX + (m − 1)L) → O(nX + mL) → OL (nX + mL) → 0. This observation, coupled with Serre duality, implies that

0 h (F, O(nX + mL)) = 0 , n < 0 , h2 (F, O(nX + mL)) = 0 ,

n ≥ 0.

We can say something also on H 1 (F, O(nX + mL)). In fact, it is easily checked that |nX + mL| has no base points if n ≥ m > 0; since X is very ample, it follows that nX + mL is very ample if n > m ≥ 0. Therefore, by Kodaira vanishing and Serre duality, we get

1 h (F, O(nX + mL)) = 0 , n ≥ m ≥ 0 , h1 (F, O(nX + mL)) = 0 , 0 ≥ m > n . If n and m are integers such that n > m ≥ 0, and C is a general member of |nX + mL|, then C is a smooth curve of genus 2 m2 + m n −n − + nm + 1 g=3 2 2

§6 Mumford’s example

41

and degree d = 3n + m. Moreover, by the Riemann–Roch theorem on F , 2 m2 − m n +n − + nm . dim |C| = 3 2 2 From now on, we let n = 4 and m = 2, so that C is a smooth curve of genus 24 and degree 14, lying on a smooth cubic surface. Our purpose is p(t) to study the Hilbert scheme Hilb3 , where p(t) = 14t − 23 is the Hilbert polynomial of C. To do it, we are going to move the curve C in its p(t) linear system, and the cubic surface as well. We denote by V ⊂ Hilb3 the locus corresponding to smooth curves C belonging to |4X + 2L| for some smooth cubic surface F and some line L on F . Since cubic surfaces in P3 depend on 19 parameters, the dimension of |C| is equal to 37, and each curve C is contained in a unique cubic by degree reasons, we have dim V = 56 . It is well known, and fairly easy to prove, that monodromy acts transitively on the set of all lines lying on a given smooth cubic surface (see for instance Exercise I-1). As a consequence, V is an irreducible, p(t) locally closed subscheme of Hilb3 . Let H be an irreducible component p(t) of Hilb3 containing V . We will prove that (6.1)

dim H = dim V = 56

and that, for every point x ∈ H, (6.2)

dim Tx (H) = dim H + 1,

so that H is everywhere nonreduced. Consider the normal bundle NC/P3 , and let x be the point in H corresponding to C. We know that Tx (H) = H 0 (C, NC/P3 ) and moreover, by (5.17), that χ(NC/P3 ) = 4 · 14 = 56 . We wish to show that h0 (C, NC/P3 ) = 57 or, equivalently, that h1 (C, NC/P3 ) = 1. It follows from the exact cohomology sequence of 0 → OC (C) = NC/F → NC/P3 → NF/P3 |C = O(3X) ⊗ OC → 0 that h1 (C, NC/P3 ) = h1 (C, O(3X) ⊗ OC ). Looking at the exact sequence 0 → O(−X − 2L) → O(3X) → O(3X) ⊗ OC → 0 ,

42


one easily sees that h1 (C, O(3X) ⊗ OC ) = h2 (F, O(−X − 2L)) = h0 (F, O(2L)) = 1, proving that the dimension of H 0 (C, NC/P3 ) is equal to 57. To prove (6.1) and (6.2), we must show that the dimension of H is equal to 56. Since H contains V , it suffices to show that dim H ≤ 56 .

(6.3)

Let Γ denote a curve corresponding to a general point of H. Since H ⊃ V , we know that Γ is a smooth connected curve of genus 24 and degree 14. We also know that Γ does not lie in a plane or in a quadric. Furthermore, we can assume that Γ does not lie on a cubic, otherwise H would coincide with the closure of V , and we would be done. In fact, as the Picard group of a smooth cubic surface is discrete, Γ would belong to |4X + 2L| for some line L on the surface, since this is the case for the curves which correspond to points of V . Since h0 (P3 , O(4)) = 35 and h0 (Γ, OΓ (4)) = 33, the curve Γ lies in a pencil of quartics, all of which we may assume to be irreducible. The quartics of this pencil meet in Γ and in a residual conic Γ . By Bertini’s theorem, the singular points of a general member G of this pencil of quartics lie on Γ ∪ Γ . Since this reducible curve can have at most triple points, it follows that G is smooth. In fact, G is a K3 surface, and hence, denoting by |Γ|G the linear system on G to which Γ belongs, dim |Γ|G = g(Γ) = 24. Now let us consider the variety

Y =

G a smooth quartic, Γ ⊂ G a smooth curve (G, [Γ]) : of degree 14 and genus 24, with [Γ] ∈ H

.

We just proved that the projection π : Y → H is dominant and that its fiber dimension is at least equal to 1. To prove (6.3), it then suffices to show that dim Y ≤ 57. But this is obvious. In fact, the quartics involved in the definition of Y are quartics containing a conic, and these quartics depend on 34–9+8=33 parameters (there are ∞8 conics in P3 , and there are 9 conditions to impose for a quartic to contain one of them). Thus, dim Y ≤ 33 + dim |Γ|G = 33 + 24 = 57 . This completes the proof of (6.1) and (6.2). It is interesting to interpret the nonreducedness of H in terms of first-order deformations. Let us look at the exact diagram 0

w H 0 (C, NC/F )

w H 0 (C, NC/P3 ) α w H 0 (C, O(3X) ⊗ OC ) u β H 0 (P3 , O(3))

w0

§7 Variants of the Hilbert scheme

43

The 37-dimensional space H 0 (C, NC/F ) represents the first-order deformations of C in F . The 57-dimensional space H 0 (C, NC/P3 ) represents the first-order deformations of C in P3 , i.e., is the tangent space to H at the point corresponding to C. The kernel of β is one-dimensional and generated by an equation of the cubic F containing C. Therefore, the image of β is 19-dimensional and represents the first-order deformations of F in P3 . Any element in H 0 (C, NC/P3 ) whose image under α lies in β(H 0 (P3 , O(3))) ⊂ H 0 (C, O(3X)⊗OC ) represents a first-order deformation of C in P3 for which the deformed curve still lies on a cubic. On the other hand, any element in H 0 (C, NC/P3 ) whose image under α falls out of β(H 0 (P3 , O(3))) ⊂ H 0 (C, O(3X)⊗OC ) represents a first-order deformation of C in P3 for which the deformed curve does not lie on any cubic. These are the first-order deformations that cannot be integrated to an actual deformation. The corresponding tangent vectors to H are the ones that do not belong to T[C] (Hred ). 7. Variants of the Hilbert scheme. The existence of the Hilbert schemes parameterizing subschemes of projective spaces makes it relatively easy to perform several related constructions. In this section we shall briefly sketch a few of these. The first generalization that comes to mind is that of a Hilbert p(t) scheme HilbX parameterizing subschemes with Hilbert polynomial p(t) of a fixed projective scheme X ⊂ Pr . More generally, given a scheme S and a closed subscheme X of Pr × S, it is possible to construct a p(t) Hilbert scheme HilbX/S parameterizing couples (s, Y ), where s is a point of S, and Y is a closed subscheme of the fiber of X → S over s, with Hilbert polynomial p(t). Formally, the problem is to represent the functor p(t) hilbX/S from schemes over S to sets defined by

p(t)

hilbX/S (T /S) =

closed subschemes Y ⊂ X ×S T , flat over T with fibers having p(t) as a Hilbert polynomial

.

p(t)

Clearly, HilbPr ×S/S is nothing but H × S, where H stands for Hilbp(t) r . p(t)

In general, we shall realize HilbX/S as a closed subscheme of H × S. We let X ⊂ Pr × H be the universal family over H and set X = X × S. Likewise, we let X be the inverse image of X under the projection from Pr × H × S to Pr × S. We claim that there is a closed subscheme Z of H × S such that a morphism T → H × S factors through Z if and only p(t) if X ×H×S T is a subscheme of X ×H×S T . But then HilbX/S is just Z, since for a morphism T → H × S of schemes over S, we have that X ×H×S T = X ×H T and X ×H×S T = X ×S T . That a subscheme Z as above exists is a special case of the following simple result. Lemma (7.1). Let A be a scheme, and let B and C be closed subschemes of Pr ×A. Assume that B is flat over A. Then there is a closed subscheme

44


D of A such that any morphism T → A factors through D if and only if B ×A T is a subscheme of C ×A T . In down-to-earth terms, the lemma asserts the existence of a subscheme D of A parameterizing points a of A such that the fiber of B over a is a subscheme of the corresponding fiber of C. Proof. We shall denote by π the projection of Pr × A to A; for any scheme T , we shall also write O instead of OPr ×T when no confusion is likely. Let ϕ : F → G be a homorphism of coherent sheaves on A. The regular functions on A which are locally of the form λ(ϕ(s)), where s is a local section of F , and λ is a local section of the dual of G, make up a sheaf of ideals in OA . We shall refer to the corresponding subscheme of A as the scheme of zeros of ϕ. We apply this construction to the homomorphisms ϕn : π∗ IC (n) → π∗ OB (n) obtained by composing the inclusions π∗ IC (n) → π∗ O(n), where IC stands for the ideal sheaf of C, with the restriction homomorphisms π∗ O(n) → π∗ OB (n). We claim that the subscheme of zeros of ϕn does not depend on n for large enough n; this will be the subscheme D we are looking for. Since the claim is local on A, in proving it we are allowed to take A as small as necessary. Now denote by In the ideal sheaf of the subscheme of zeros of ϕn and look at the diagram π∗ O(1) ⊗ π∗ IC (n)

id ⊗ϕn

μn

u π∗ IC (n + 1)

ϕn+1

w π∗ O(1) ⊗ π∗ OB (n) y νn u w π∗ OB (n + 1)

If λ isa local section of the dual of π∗ OB (n + 1), then λ ◦ νn is of the form σi ⊗ λi , where the σi are sections of the dual of π∗ O(1) and the λi are sections of the dual of π∗ OB (n). For large n, the map μ n is onto, and hence any section s of π∗ IC (n + 1) is of the form μn ( tj ⊗ sj ), where the tj are sections of π∗ O(1) and the sj are sections of π∗ IC (n). Thus, σi (tj )λi (ϕn (sj )) λ(ϕn+1 (s)) = i,j

is a section of In . This proves that In+1 ⊂ In as soon as μn is onto. It remains to show that, conversely, In+1 ⊃ In for large n. If a is a point of A, we shall denote by Ba the fiber of B → A over a. Let σ be a linear form on Pr , and H the corresponding hyperplane; set H = H × A. There is an integer n0 such that, for any n ≥ n0 , π∗ OB (n) is locally free, R1 π∗ OB (n) = 0, and in addition H 0 (Ba ∩ H, OBa ∩H (n)) = π∗ OB∩(H×A) (n) ⊗ k(a), H j (Ba ∩ H, OBa ∩H (n)) = 0 for any j > 0 and any a ∈ A. Now fix a closed point a0 in A and choose σ so that H does


45

not contain any components of Ba0 , including the embedded ones. Thus the homomorphism H 0 (Ba0 , OBa0 (n)) → H 0 (Ba0 , OBa0 (n + 1)) given by σ is injective for any n. Shrinking A, if necessary, we may then assume that H 0 (Ba , OBa (n)) → H 0 (Ba , OBa (n + 1)) is injective for any a ∈ A and for n = n0 , . . . , n0 + r. By our choice of n0 , for these values of n, the sequence 0 → H 0 (Ba , OBa (n)) → H 0 (Ba , OBa (n+1)) → H 0 (Ba ∩H, OBa ∩H (n+1)) → 0 is exact for any a; it follows in particular that h0 (Ba ∩ H, OBa ∩H (n + 1)) is independent of a. This means that the values of the Hilbert polynomial hBa ∩H (t) at t = n0 + 1, . . . , n0 + r + 1 do not depend on a. Hence, hBa ∩H is independent of a, since its degree is not greater than r. In particular, h0 (Ba ∩ H, OBa ∩H (n + 1)) does not depend on a for any n ≥ n0 . By dimension reasons it then follows that H 0 (Ba , OBa (n)) → H 0 (Ba , OBa (n + 1)) is injective for any a ∈ A and for any n ≥ n0 . What the preceding argument proves is that, when n is at least n0 , π∗ OB (n) is a subsheaf of π∗ OB (n + 1), and the quotient π∗ OB (n + 1)/π∗ OB (n) is locally free. As a consequence, any section of the dual of π∗ OB (n) over a sufficiently small open set extends to a section of the dual of π∗ OB (n + 1). Now look at the diagram π∗ IC (n)

ϕn

w π∗ OB (n)

jn u u ϕn+1 π∗ IC (n + 1) w π∗ OB (n + 1) in

where in and jn are multiplication by σ. If s and λ are sections of π∗ IC (n) and of the dual of π∗ OB (n), respectively, then λ is locally of the form λ ◦ jn for some section λ of the dual of π∗ OB (n + 1). Hence, λ(ϕn (s)) = λ (jn (ϕn (s))) = λ (ϕn+1 (in (s))) is a section of In+1 . This proves that In ⊂ In+1 for large n, as desired. We may now complete the proof of the lemma. Let f : T → A be a morphism. Denote by η the projection of Pr × T onto T and by F the morphism id ×f : Pr × T → Pr × A. Set B = B ×A T , C = C ×A T . Now look at the diagram f ∗ π∗ IC (n) un 0

u

w η∗ IC (n)

w f ∗ π∗ O(n) vn

u w η∗ O(n)

w f ∗ π∗ OC (n)

w0

wn

u w η∗ OC (n)

w0

For large n, the two rows of the diagram are exact, and moreover vn and wn are isomorphisms, by Lemma (3.8) and by the remark that

46


OC = F ∗ OC . Thus, un is onto for large n. Since η∗ OB (n) = f ∗ π∗ OB (n), this implies that, for n large enough, f ∗ π∗ IC (n) → f ∗ π∗ OB (n) is zero if and only if η∗ IC (n) → η∗ OB (n) is. This means that f lands in D if and only if η∗ IC (n) → η∗ OB (n) vanishes for all sufficiently large n, that is, if and only if IC → OB is zero or, put otherwise, B = B ×A T is a Q.E.D. subscheme of C = C ×A T . This concludes the proof of (7.1). An immediate consequence of (4.10) and of the construction of HilbpX/S is the following. Lemma (7.2). HilbpX/S → S is proper. It goes without saying that, for fixed p, the Hilbert scheme HilbpX , or more generally HilbpX/S , depends on the projective embedding of X. What is independent of the embedding is the infinite union HilbX/S =

HilbpX/S ,

p

which, however, is not of finite type. When S is a point, we shall write HilbX instead of HilbX/S . Contrary to what happens when X = Pr , the Hilbert scheme HilbpX is in general not connected. For instance, when p(t) = t + 1 and X is a smooth cubic surface in three-space, HilbpX parameterizes lines lying on X. As we know, there are exactly 27 of these, so in this case, HilbpX consists of 27 points; we shall presently see that it is also reduced. If X is any closed subscheme of Pr and p(t) is a rational polynomial, p(t) Proposition (5.9) describes the tangent spaces to HilbX . If h is a point p(t) of HilbX corresponding to a subscheme Y of X, then the tangent space p(t) to HilbX at h is (7.3)

p(t)

Th (HilbX ) = H 0 (Y, NY /X ) .

In the special case where X is a smooth cubic in P3 and Y is a line in X, the adjunction formula for Y , coupled with the fact that the canonical bundle on X is OX (−1), shows that the self-intersection of Y equals −1. This means that the degree of NY /X = OY (Y ) is −1, which, by (7.3), p(t) p(t) implies that Th (HilbX ) = 0. Thus, HilbX is reduced at h. The formation of the Hilbert scheme is compatible with base change, in the sense that, for any morphism T → S, there is a canonical isomorphism HilbX×S T /T HilbX/S ×S T . This follows at once from the universal property of the Hilbert scheme.


47

Remark (7.4). The argument above shows that HilbZ/T exists also when Z → T is just an analytic family of projective schemes, at least when Z is flat over T . The question is local on T , so by (4.9) we may suppose that Z = X ×S T for some morphism T → S, where X → S is an algebraic family. Then HilbX/S ×S T is the sought-for space, since it clearly possesses the required universal property. A very important construction that can be carried out thanks to the existence of general Hilbert schemes is the one of the scheme (not of finite type) HomS (X, Y ), where X ⊂ Pr × S and Y ⊂ Pt × S are closed subschemes, and X is flat over S. This is a scheme representing the functor h(T /S) = HomT (X ×S T, Y ×S T ) . In fact, we shall see that there is an open subscheme of HilbX×S Y /S representing h. We begin by observing that, given schemes Z and W over T , associating to a morphism its graph gives a one-to-one correspondence between morphisms Z → W of schemes over T and closed subschemes Γ of Z ×T W which project isomorphically onto Z. The next remark is that, for a subscheme of a fibered product, being a graph is an open condition. Formally, we have the following result. Lemma (7.5). Let A, B, and C be schemes, and let α : A → C and β : B → C be flat projective morphisms. Let f : A → B be a morphism of schemes over C. Assume that there is a closed point c ∈ C such that fc : Ac → Bc is an isomorphism. Then there is a neighborhood U of c such that f maps α−1 (U ) isomorphically onto β −1 (U ). To prove the lemma, we may argue as follows. Let M be a line bundle on A that is very ample relative to f , and let b0 be a point of Bc . Using (3.6) and replacing M with a power, if necessary, we may assume that, for any m ≥ 1, the map f∗ M m ⊗ k(b) → H 0 (f −1 (b), M m ⊗ Of −1 (b) ) is an isomorphism for any b ∈ B and the map f∗ M ⊗f∗ M m → f∗ M m+1 is onto. Since Ac → Bc is an isomorphism, H 0 (f −1 (b0 ), M ⊗Of −1 (b0 ) ) is onedimensional, and hence f∗ M has rank 1 at b0 . Let V be a neighborhood of b0 such that the rank of f∗ M is not greater than 1 at any point of V . Then, for any b ∈ V , the dimension of H 0 (f −1 (b), M m ⊗ Of −1 (b) ) is at most 1 for any m ≥ 1, so f −1 (b) consists of at most one point. By compactness it follows that, possibly after shrinking C, we may assume that f −1 (b) consists of at most one point for any b ∈ B. Thus, if L is a line bundle on B that is very ample relative to β, its pullback to A is ample relatively to α; replacing L with a suitable power, we may even assume that f ∗ L is relatively very ample. Since A and B are flat over C, we may find an integer n0 such that, for n ≥ n0 , β∗ (Ln ) and α∗ f ∗ (Ln ) are locally free and in addition α∗ f ∗ (L) ⊗ α∗ f ∗ (Ln ) → α∗ f ∗ (Ln+1 ) is onto. Since Ac and Bc are isomorphic, β∗ (Ln ) ⊗ k(c) → α∗ f ∗ (Ln ) ⊗ k(c) is an isomorphism

48


for any n ≥ n0 ; in particular β∗ (Ln ) and α∗ f ∗ (Ln ) have the same rank. Another consequence is that there is a neighborhood U of c over which β∗ (Ln0 ) → α∗ f ∗ (Ln0 ) is an isomorphism. By our choice of n0 , β∗ (Lhn ) → α∗ f ∗ (Lhn ) is onto for any h ≥ 1 and hence an isomorphism, by dimension reasons. If follows that α−1 (U ) and β −1 (U ) are isomorphic. Lemma (7.5) enables us to conclude that h is represented by the open subset of HilbX×S Y /S consisting of the couples (s, Γ), where s ∈ S and Γ is the graph of a morphism from Xs to Ys . When not only X, but also Y , is flat over S, in addition to HomS (X, Y ), there is also the scheme HomS (Y, X). The intersection of HomS (X, Y ) and HomS (Y, X) inside HilbX×S Y /S parameterizes isomorphisms between fibers of X → S and the corresponding fibers of Y → S; we shall denote it by IsomS (X, Y ). Clearly, IsomS (X, Y ) represents the functor which associates to each scheme T over S the set of all isomorphisms, as schemes over T , from X ×S T to Y ×S T . Remark (7.6). By Remark (7.4), HomS (X, Y ) and IsomS (X, Y ) exist also, as analytic spaces, when X → S and Y → S are flat analytic families of projective schemes. Another useful space that can be constructed by means of the general Hilbert scheme parameterizes nested pairs of subschemes in the fibers of a family of projective schemes. Let S be a scheme, and let X be a closed subscheme of Pr × S. Let p1 (t) and p2 (t) be rational polynomials. Denote p1 (t) by H1 the Hilbert scheme HilbX/S and by Y1 ⊂ Pr × H1 the universal p (t)

family on it. Then HilbY21 /H1 parameterizes pairs (Y1 , Y2 ), where Y1 ⊃ Y2 are closed subschemes of a fiber of X → S and the Hilbert polynomial of Yi is pi (t), i = 1, 2. More generally, given rational polynomials p1 (t), . . . , pn (t), iterating this construction yields a “flag Hilbert scheme” parameterizing n-tuples (Y1 , Y2 , . . . , Yn ) such that Y1 ⊃ Y2 ⊃ · · · ⊃ Yn are subschemes of a fiber of X → S with Hilbert polynomials p1 (t), . . . , pn (t). In a slightly different direction, the same kind of construction makes it possible, given p1 (t), . . . , pn (t) as above and an additional polynomial p(t), to construct a Hilbert scheme parameterizing (n+1)-tuples (Y ; Z1 , . . . , Zn ), where Y is a subscheme of a fiber of X → S with Hilbert polynomial p(t), and Zi is a subscheme of Y with Hilbert polynomial pi (t) for i = 1, . . . , n. Here is another construction that can be performed thanks to the existence of general Hilbert schemes. Let S, X, p1 (t), p2 (t), H1 , and p2 (t) p (t) Y1 be as above. Set H2 = HilbX/S and H = HilbY21 /H1 . There are natural morphisms H → Hi . Let Z ⊂ X × S be a closed subscheme, flat over S and such that the Hilbert polynomial of every fiber of Z → S is p2 (t). Let α : S → H2 be the corresponding map. Then the fiber product H ×H2 S parameterizes couples consisting of a point s of S and of a subscheme of Pr with Hilbert polynomial p1 (t), containing the fiber

§8 Tangent space computations

49

of Z → S over s and contained in the fiber of X → S over s. More precisely, H ×H2 S represents the functor ⎧ ⎫ ⎨ closed subschemes Y of X ×S T , flat over T , ⎬ h(T /S) = containing Z ×S T , and such that the Hilbert . ⎩ ⎭ polynomial of every fiber of Y → S is p1 (t) As a special case, given a closed subscheme Z of Pr and a rational polynomial p(t), this construction provides a Hilbert scheme parameterizing closed subschemes of Pr with Hilbert polynomial p(t) and containing Z as a subscheme. Exercise (7.7). Let X and Y be closed subschemes of Pr × S, flat over S. Let Z1 , . . . , Zn be closed subschemes of X, and W1 , . . . , Wn closed subschemes of Y , all flat over S. Generalizing the construction of IsomS (X, Y ), show that there exists a Hilbert scheme IsomS ((X; Z1 , . . . , Zn ), (Y ; W1 , . . . , Wn )) parameterizing pairs (s, ϕ), where s ∈ S and ϕ is an isomorphism Xs → Ys carrying Zi,s isomorphically to Wi,s for i = 1, . . . , n. 8. Tangent space computations. As we observed, the formation of the Hilbert scheme is compatible with base change. In particular, this has the following implication. Let S be a scheme, and let X be a closed subscheme of Pr × S. If s is a closed point of S and, as usual, we write Xs for the fiber of X → S at s, and similarly for the one of HilbX/S at s, then (HilbX/S )s = HilbXs It follows that, for any closed point h = [Y ] ∈ HilbXs , the sequence of tangent spaces (8.1)

0 → Th (HilbXs ) → Th (HilbX/S ) → Ts (S)

is exact. In particular, (8.2)

dimh (HilbX/S ) ≤ H 0 (Y, NY /Xs ) + dims (S) .

We warn the reader that the rightmost homomorphism in (8.1) is generally not onto, even when X → S and Y are “nice” (see, for instance, Example (8.21) below). We know from (5.9) that, when S is a point, Th (HilbX/S ) = Th (HilbX ) is just H 0 (Y, NY /X ). We wish to show that it is possible to describe Th (HilbX/S ) along similar lines for arbitrary S. As usual, we write Σ to indicate Spec C[ε], where C[ε] is the ring of dual numbers. We

50


also denote by π the projection from X to S, and by Y the subscheme of Xs corresponding to h. An element of Th (HilbX/S ) consists of: (8.3) - a morphism ϕ : Σ → (S, s); - a subscheme Y of X ×S Σ, flat over Σ, extending Y . Via the inclusion X ×S Σ ⊂ X × Σ, the subscheme Y corresponds to an element of H 0 (Y, NY /X ) = HomOX (J , OY ), where J is the ideal sheaf of Y in X. On the other hand, if we denote by m the ideal sheaf of s in S, ϕ corresponds to an element of Hom(m, OS /m) = Ts (S). Lemma (8.4). Th (HilbX/S ) is the subspace of HomOX (J , OY ) ⊕ Hom(m, OS /m) consisting of all pairs (v, w) such that (8.5)

v(π∗ (u)) = w(u)

for all sections u of m.

We denote by ϕ the morphism Σ → (S, s) corresponding to w, and by Y the subscheme of X × Σ, flat over Σ and extending Y , corresponding to v. We must show that Y is a subscheme of X ×S Σ if and only if (8.5) is valid. The ideal of X ×S Σ in X × Σ is generated by the functions π ∗ (u) ⊗ 1 − 1 ⊗ ϕ∗ (u) ,

(8.6)

where u runs through all sections of m. Recall that ϕ∗ (u) = w(u)ε for any section u of m. Since w(u) is a constant, (8.6) is equal to π ∗ (u) ⊗ 1 − w(u) ⊗ ε . On the other hand, the ideal of Y consists of all functions of the form f ⊗ 1 − f ⊗ ε such that the reduction of f modulo J is equal to v(f ). Thus (8.6) belongs to the ideal of Y if and only if v(π ∗ (u)) = w(u). This proves (8.4). Corollary injective.

(8.7). The

projection

Th (HilbX/S ) → H 0 (Y, NY /X )

is

It is not difficult to describe the tangent spaces to all the variants of the Hilbert scheme we have introduced in the previous section; here we limit ourselves to the one that we will most often encounter in the sequel. Let η : X → S be a projective morphism, and denote by H the Hilbert scheme parameterizing (n + 1)-tuples (Y ; Z1 , . . . , Zn ) of subschemes of the fibers of η such that Zi ⊂ Y for i = 1, . . . , n. There are morphisms πi : H → HilbX/S , i = 0, . . . , n, given by π0 (Y ; Z1 , . . . , Zn ) = Y , πi (Y ; Z1 , . . . , Zn ) = Zi for i = 1, . . . , n, and (π0 , . . . , πn ) identifies H to a closed subscheme of HilbX/S n+1 . If W is a closed subscheme of X, we denote by IW its ideal sheaf. Moreover, given a tangent vector v to HilbX/S at W , we denote by αv the corresponding homomorphism of 2 to OW . Given a point h of H, corresponding OX -modules from IW /IW to an (n + 1)-tuple (Y ; Z1 , . . . , Zn ), set hi = πi (h) for i = 0, . . . , n. Then the tangent space to H at h is described by the following result, which is an immediate consequence of Lemma (8.4) and Corollary (8.7).


51

Lemma (8.8). The tangent space Th (H) is the subspace of Th0 (HilbX/S )⊕ Th1 (HilbX/S ) ⊕ · · · ⊕ Thn (HilbX/S ) consisting of the (n + 1)-tuples (u; v1 , . . . , vn ) such that the diagrams IY /IY2

σi

αu

u OY

ρi

w IZi /IZ2 i u

α vi

w OZi

commute for i = 1, . . . , n, where σi : IY → IZi and ρi : OY → OZi are the obvious maps. In other words, Th (H) = Hom(C • , D• ) , where C • and D• are the complexes C • = (IY /IY2 → ⊕i (IZi /IZ2 i )) ,

D• = (OY → ⊕i OZi ) .

Exercise (8.9). Let H be the Hilbert scheme parameterizing (n + 1)tuples (Y ; p1 , . . . , pn ), where Y is a closed subscheme of Pr , and the pi are points of Y . Let h = (Y ; p1 , . . . , pn ) be a point of H, and let h0 be the point of HilbPr corresponding to Y . Suppose the pi are distinct. i) Use Lemma (8.8) to show that there is an exact sequence 0 → ⊕i Tpi (Y ) → Th (H) → Th0 (HilbPr ). ii) Show that, if the pi are smooth points of Y , then the rightmost homomorphism in this sequence is onto. iii) Show with an example that the conclusion of ii) may not be valid if at least one of the pi is not a smooth point of Y . We now return to sequence (8.1), with the goal of interpreting it in cohomological terms, under the assumption that X is flat over S. As above, we indicate by J the ideal sheaf of Y in OX , and by m the ideal of s in OS . In addition, we write I for the ideal of Xs in OX , and K = J /I for the ideal of Y in OXs . From the exact sequence of OXs -modules (8.10)

0 → I/IJ → J /IJ → K → 0 ,

= HomOX (J , OY ) and observing that HomOXs (J /IJ , OY ) HomOXs (I/IJ , OY ) = HomOX (I, OY ), we get an exact sequence (8.11) 0 → HomOXs(K, OY ) → HomOX(J , OY ) → HomOX(I, OY ) → Ext1OXs(K, OY ).

52


The second and third terms from the left in this sequence are H 0 (Y, NY /Xs ) = Th (HilbXs ) and H 0 (Y, NY /X ). Since X is flat over S, we have that I = m ⊗OS OX and I 2 = m2 ⊗OS OX , and hence that I/I 2 = Ts (S)∨ ⊗C OXs . It follows that HomOX (I, OY ) = HomOXs (I/I 2 , OY ) = Ts (S) ⊗C H 0 (Y, OY ) , and (8.11) becomes (8.12) 0 → H 0 (Y, NY /Xs ) → H 0 (Y, NY /X ) → Ts (S) ⊗ H 0 (Y, OY ) → Ext1OXs (K, OY ). By Lemma (8.7), Th (HilbX/S ) is a subspace of H 0 (Y, NY /X ). Lemma (8.4) can then be rephrased as saying that an element of H 0 (Y, NY /X ) belongs to Th (HilbX/S ) if and only if its image in Ts (S) ⊗C H 0 (Y, OY ) belongs to Ts (S) ⊗ 1 Ts (S). In particular, this says that Th (HilbX/S ) = H 0 (Y, NY /X ) when Y is reduced and connected. Example (8.13). Here is a concrete example of an element of H 0 (Y, NY /X ) which does not come from a tangent vector to HilbX/S . We choose as S the affine line with coordinate t, and the origin as s. We also set X = P1 × S and denote by π the projection of X to S. Finally, we let z be an affine coordinate on P1 (minus one point) and choose as Y the subscheme with the equations t = 0 and z 2 = 0. Let Y ⊂ X × Σ be the subscheme defined by the equations z 2 = t − εz = 0. For this choice of Y , we have that J /J 2 = (z 2 , t)/(z 4 , z 2 t, t2 ) is a free OY -module generated by the classes [t] and [z 2 ], while H 0 (Y, NY /X ) = Hom(J /J 2 , OY ) = H 0 (Y, OY )u + H 0 (Y, OY )v , where u, v is the dual basis of [t], [z 2 ]. The image of Y in H 0 (Y, NY /X ) under the characteristic map is zu, and its image in Ts (S) ⊗ H 0 (Y, OY ) ∂ ⊗ z, which does not belong to Ts (S) ⊗ 1. is ∂t As is clear from its construction, the homomorphism Th (HilbX/S ) → Ts (S) in (8.1) agrees with the one obtained by restricting to Th (HilbX/S ) the homomorphism H 0 (Y, NY /X ) → Ts (S) ⊗ H 0 (Y, OY ) in (8.12). We can thus summarize what we have proved in the following statement. Proposition (8.14). Let X be flat and projective over S. Let s be a closed point of S, let Y be a closed subscheme of Xs , and denote by h the corresponding point of HilbX/S . Then the exact sequence (8.1) extends to an exact sequence (8.15)

0 → Th (HilbXs ) → Th (HilbX/S ) → Ts (S) → Ext1OXs (K, OY ),

where K stands for the ideal sheaf of Y in Xs . Moreover, under the natural identifications and inclusions Th (HilbXs ) = H 0 (Y, NY /Xs ), Th (HilbX/S ) ⊂ H 0 (Y, NY /X ), Ts (S) ⊂ Ts (S) ⊗ H 0 (Y, OY ), the homomorphisms in this sequence agree with those in (8.12). Finally, when Y is reduced and connected, Th (HilbX/S ) = H 0 (Y, NY /X ).


53

Let v be a tangent vector to S at s. We shall say that v is obstructed at h if it maps to a nonzero element of Ext1OXs (K, OY ). In other words, v is obstructed at h if it cannot be lifted to a v ∈ Th (HilbX/S ), that is, to a first-order family of subschemes in the fibers of X → S. The exact sequence (8.15) can be refined somewhat. Comparing the exact sequence of OY -modules 0 → I/(I ∩ J 2 ) → J /J 2 → K/K2 → 0 with (8.10), we get a commutative diagram with exact top and bottom rows

whence an exact sequence (8.16)

0 → Th (HilbXs ) → Th (HilbX/S ) → V → Ext1OY (K/K2 , OY ) ,

where V stands for the intersection of Hom(I/(I ∩ J 2 ), OY ) and Ts (S) inside HomOX (I, OY ) = Ts (S) ⊗ H 0 (Y, OY ). It may well happen that an element of Ts (S) is obstructed simply because it does not belong to V , as the following example shows. Example (8.17). Let X ⊂ P3 × C be the locus {([x0 : · · · : x3 ], t) : tx20 + x21 + x22 + x23 = 0}, and let π be the projection of X to S = C. The fibers of π are all smooth quadrics, except for X0 which is a cone with vertex p at t = x1 = x2 = x3 = 0. We choose {p} as Y0 , and, as usual, we denote by h the corresponding point of HilbX/S . It is easy to show that every nonzero element of Ts (S) is obstructed at h. To see this, we pass to the affine coordinates yi = xi /x0 , i = 1, 2, 3. In these coordinates, the equation defining X near p is t + y12 + y22 + y32 = 0. Thus J = (y1 , y2 , y3 ), and t ∈ J 2 . Since I is generated by t, it follows that I ⊂ J 2 and hence that V = {0}. This proves our claim. When Y ⊂ Xs is a regular embedding, sequence (8.16) is particularly nice, as the following refinement of (8.14) shows. Theorem (8.18). Let X be flat and projective over S. Let s be a closed point of S, let Y be a regularly embedded closed subscheme of Xs , and denote by h the corresponding point of HilbX/S . Then the exact sequence (8.1) extends to an exact sequence (8.19)

0 → Th (HilbXs ) → Th (HilbX/S ) → Ts (S) → H 1 (Y, NY /Xs ) .

54


The first step in proving the proposition is to observe that, under its assumptions, I ∩ J 2 = IJ , and hence HomOY (I/(I ∩ J 2 ), OY ) = HomOX (I, OY ), so that V = Ts (S). To this end, pick a point x of Y and write A for OS,s , B for OX,x , mA and mB for their maximal ideals, and I and J for Ix and Jx . We must show that I ∩ J 2 = IJ. By assumption, the ideal J/I in B/I is generated by a regular sequence f1 , . . . , fh . For each i, choose Fi ∈ B mapping to fi , and set L = (F1 , . . . , Fh ); clearly, J = I + L. By Lemma (5.23), B/L is A-flat, and hence the sequence 0 → L ⊗A A/mA → B ⊗A A/mA → B/L ⊗A A/mA → 0 is exact. This means that L/IL = L⊗A A/mA is the kernel of B/I → B/J. On the other hand, this kernel is also equal to (I + L)/I L/(I ∩ L), and hence IL = I ∩ L. But then I ∩ J 2 = I ∩ (IJ + L2 ) = IJ + I ∩ L2 ⊂ IJ + I ∩ L ⊂ IJ . For the second step of the proof, recall that there is a spectral sequence abutting to Ext•OY (K/K2 , OY ) whose E2 term is E2p,q = H p (Y, ExtqOY (K/K2 , OY )) . On the other hand, since Y → Xs is a regular embedding, K/K2 is a locally free OY -module by (5.24), and hence the sheaves ExtqOY (K/K2 , OY ) all vanish for q > 0. It follows that ExtpOY (K/K2 , OY ) = H p (Y, HomOY (K/K2 , OY )) = H p (Y, NY /Xs ) for all q. This concludes the proof of (8.18). An upper bound for the dimension of HilbX/S is given by (8.2), while Theorem (8.18) gives the lower bound h0 (NY /Xs ) − h1 (NY /Xs ) + dim Ts S for the dimension of the tangent space to the Hilbert scheme at h, in the case of regular embeddings. The following powerful generalization to relative Hilbert schemes of the lower bound (5.12) shows that a similar bound holds also for the dimensions of the spaces involved. Theorem (8.20). Let X be flat and projective over S, let s be a point of S, and assume that S is equidimensional at s. Let Y be a regularly embedded closed subscheme of Xs , and write h to indicate the corresponding point of HilbX/S . Then every component of HilbX/S at h has dimension at least h0 (Y, NY /Xs ) − h1 (Y, NY /Xs ) + dims S . As we did for (5.12), we refer to [440] or [625] for a proof. Here we just want to give a simple example of an obstructed situation which shows, among other things, that this lower bound is sharp.


55

Example (8.21). As we mentioned, the homomorphism Th (HilbX/S ) → Ts (S) in (8.1) is in general not onto, even when the morphism X → S is very nice, say, for instance, smooth with S smooth, and the subscheme corresponding to h is also smooth. A slight variation on Example (8.17) provides what is probably the simplest instance of this phenomenon. We take as S the affine line with coordinate t and let X be the locus in P4 × S defined by the two equations x1 x2 − x3 x4 = 0 and x3 + x4 = tx0 , where x0 , . . . , x4 are homogeneous coordinates in P4 . Thus, X → S is a family of quadrics in 3-space, all smooth except the one at t = 0, which is a cone over a smooth conic. The family X → S is obtained from X → S by blowing up the vertex of this cone. Formally, let λ and μ be homogeneous coordinates in P1 , and let X be defined, inside P4 × P1 × S, by the equations x1 x2 − x3 x4 = 0 ,

λx1 = μx3 ,

μx2 = λx4 ,

x3 + x4 = tx0 .

It is immediate to check that X → S is smooth, and that X is isomorphic to X except for the fact that the point x1 = x2 = x3 = x4 = 0 is replaced by a smooth rational curve E. Set zi = xi /x0 for i = 1, . . . , 4, ζ = λ/μ, and ξ = μ/λ, and let U be a neighborhood of E. Then ζ, z1 , z4 are local coordinates for X on the complement, inside U , of the locus μ = 0. Likewise, ξ, z2 , z3 are coordinates on the complement of λ = 0 in U . A generator for NE/X0 away from μ = 0 is eμ = ∂/∂z1 − ζ∂/∂z4 , and one away from λ = 0 is eλ = ξ∂/∂z3 − ∂/∂z2 . Since eμ = ζ 2 eλ , the normal bundle to E in X0 is OE (−2), and the self-intersection of E is −2. In particular, E cannot be “moved” to any nearby fiber since a smooth quadric does not contain curves with negative self-intersection. Neither can E be moved inside X0 , since H 0 (E, NE/X0 ) = 0. Summing up, the point corresponding to E is isolated in HilbX/S . In particular, in this case the lower bound given by (8.20) is an equality, since h1 (E, NE/X0 ) = 1. This, in itself, is not sufficient to conclude that Th (HilbX/S ) → Ts (S) is not onto, since in principle E could still move to first order transversely to HilbX0 . To prove our claim, we must show that Th (HilbX/S ) = H 0 (E, NE/X ) is actually zero. For this, it suffices to look at the exact sequence (8.15), which in this case reduces to δ

→ H 1 (E, OE (−2)) = H 1 (E, NE/X0 ), 0 → H 0 (E, NE/X ) → H 0 (E, OE ) − since H 0 (E, NE/X0 ) vanishes. All we have to show is that the coboundary map δ is not zero. We do this by explicit computation. The unit generator of H 0 (E, OE ) corresponds to the tangent vector ∂/∂t at 0 ∈ S, viewed as a normal vector field to X0 along E. A lift of ∂/∂t to a normal field to E for μ = 0 is ∂/∂z4 ; another lift for λ = 0 is ∂/∂z3 . The difference between these liftings is a representative of the coboundary δ(∂/∂t) and equals ∂ ∂ − = ζ −1 eμ = ξ −1 eλ . ξ ∂z1 ∂z4

56 Thus δ(∂/∂t) is H 1 (E, OE (−2)).


the

standard

generator

of

H 1 (E, NE/X0 )

=

9. C n families of projective manifolds. Up to now, we have concentrated on algebraic families of schemes or on analytic families of analytic spaces. However, in many contexts, it is useful to be able to deal also with families of complex manifolds that depend only continuously or differentiably, in an appropriate sense, on parameters. In particular, we will use continuous families of curves in our approach to Teichm¨ uller’s theorem. We begin by explaining what we mean by continuous families of complex manifolds and by establishing a few basic facts about them. Let α : X → B be a morphism of C m manifolds, where m can be 0, 1, . . . , ∞. A relative C m atlas for α : X → B is a collection of open subsets Vi of X and C m diffeomorphisms ϕi : Ui × Si → Vi , where Ui is an open set in some Rk and Si is an open subset of B, such that the Vi cover X and the ϕi are compatible with the projections to B. We shall often refer to the Ui -coordinates as the vertical coordinates on Vi and to the derivatives with respect of them as vertical derivatives. Consider relative C m atlases having the additional property that the coordinate changes are C ∞ in the vertical coordinates and, moreover, all their vertical derivatives, of any order, are C m as functions of all the coordinates. Any such atlas is contained in a unique maximal one. We will say that a maximal atlas defines a structure of C m family of differentiable manifolds on α : X → B and that α : X → B, together with the atlas, is a C m family of differentiable manifolds. For brevity, the charts of the atlas will be said to be adapted (to the given structure of family of differentiable manifolds, of course). If α : X → B and α : X → B are two C m families of differentiable manifolds, a morphism from the first to the second is a pair (f, F ) of C m morphisms fitting in a commutative square X (9.1)

F

α

u B

f

w X α u w B

such that derivatives of arbitrary order of the coordinates on X with respect to vertical coordinates on X are C m functions of all coordinates, vertical or not. In particular, taking X = R and B a point, we can speak of C m families of differentiable functions on α : X → B; these families will also be called adapted functions. Clearly, we can also speak of adapted functions on open subsets of X. There are other objects for which the notion of being adapted makes sense, for example, relative forms; we will say that such a form is adapted when, written in adapted coordinates,

§9 C n families of projective manifolds

57

it has components which are adapted functions. More generally, there is an obvious notion of C m family of differentiable vector bundles on a C m family of differentiable manifolds, and one can speak of adapted metrics on these and of adapted sections and vector-valued relative forms. When working with adapted “objects,” the main fact to keep in mind is that the class of these is closed under the four operation, under functional composition, and, crucially, also under vertical differentiation of arbitrary order. In particular, the class of C m families of differentiable vector bundles is closed under direct sum, tensor product, and pullback via morphisms of families. We shall denote the sheaf of adapted functions

on X by A X , or simply by A. Global adapted objects can often be constructed by gluing together local ones by means of suitable partitions of unity. Let W be an open covering of a neighborhood of α−1 (b0 ) for some point b0 ∈ B. An adapted partition of unity on a neighborhood of α−1 (b0 ) subordinated to W is a finite collection {χi } of compactly supported adapted functions on X such that χi = 1 on a neighborhood of α−1 (b0 ) and that the support of each χi is contained in an element of W. Lemma (9.2). When α : X → B is a proper map, for any open covering W of α−1 (b0 ), there exist adapted partitions of unity on a neighborhood of α−1 (b0 ) subordinated to W. To prove the lemma, cover X, possibly after shrinking B, with finitely many adapted charts ϕi : Ui × B → Vi such that each Vi is contained in one of the elements of W. For each i, choose real-valued nonnegative C ∞ compactly supported functions μi and μi on Ui and B, denote by μi the function μi × μi on Ui × B, and call λi the function on X which equals μi ◦ ϕ−1 on Vi and is zero on X − Vi . Clearly, λi is an adapted i λi is strictly positive function. We may choose λi in such a way that on all of α−1 (b0 ), and hence on α−1 (A), where A is a small enough ball centered at 0 ∈ B. Choose another ball D centered at 0 whose closure is contained in A and a nonnegative C ∞ function σ onB which is zero on D and strictly positive outside A. If we set λ = σ + λi , then χi = λi /λ will do. There is a version of the inverse function theorem for adapted functions, that is, a version “with parameters.” Proposition (9.3). Let U be an open subset of Rn , V an open subset of R , f : U × V → Rn a continuous function, and m a nonnegative integer. Write x = (x1 , . . . , xn ) to indicate the standard coordinates in U and t = (t1 , . . . , t ) to indicate the standard ones in V . Suppose that: i) the function f is C ∞ in x for any fixed t; ii) the function f and all its derivatives, of any order, with respect to the x variables are C m functions of x and t; iii) the Jacobian ∂f ∂x (x, t) is nonsingular at a point (x0 , t0 ) ∈ U × V .

58


Set F (x, t) = (f (x, t), t). Then there is an open neighborhood A of (x0 , t0 ) such that F (A) is open in Rn × R and F induces a homeomorphism from A to F (A). Moreover, writing the inverse of this function under the form (x, t) → (g(x, t), t), the function g is C ∞ in x, and g and all its derivatives, of any order, with respect to the x variables are C m functions of x and t. ∂F Proof. When m > 0, the Jacobian matrix ∂(x,t) is nonsingular at (x0 , t0 ); hence the standard inverse function theorem shows that F has a local inverse of class C m . To obtain the same result when m = 0, we must retrace the proof of the standard inverse function theorem. We can U and V , we assume that x0 = 0 ∈ Rn and t0 = 0 ∈ R . Shrinking ∂f −1 ∂f may suppose that ∂x is invertible and that ∂x is bounded by some positive constant C on all of U × V . We may also suppose that U = B(0, r) ⊂ Rn . We first prove the theorem under the following additional assumptions: (9.4) ⎧ ⎨ f (0, t) = 0 for all t ∈ V ; ⎩ ∂f (0, t) = I for all t ∈ V , where I = In is the n × n identity matrix. ∂x

Shrinking U and V , we may assume that 1 ∂f ∂x − I < 4 everywhere. We then set Ty (x, t) = x + y − f (x, t) and notice that Ty (x) = x if and only if f (x, t) = y. Since ∂f ∂ Ty = I − , ∂x ∂x we find that Ty (x, t) − Ty (x , t)
0, vanish and the multiplication map H 0 (Pr , O(1)) ⊗ H 0 (Xb , IXb (n)) → H 0 (Xb , IXb (n + 1)) is onto. This implies, among other things, that the dimension of H 0 (Xb , OXb (n)) is equal to p(n) and that the map H 0 (Pr , O(n)) → H 0 (Xb , OXb (n)) is onto for every b ∈ B.

64


X (n) is a locally free module Lemma (9.12). Suppose n ≥ n0 . Then α∗ O m

X (n) is → α∗ O over the sheaf of C functions on B, and ρn : π∗ O(n) surjective as a homomorphism of vector bundles. Let b0 be a point of B. Pick sections s1 , . . . , sp(n) of OPr (n) which map to independent elements of H 0 (Xb0 , OXb0 (n)), and points q1 , . . . , qp(n) of Xb0 such that the matrix ⎛ ⎞ s1 (q1 ) ... sp(n) (q1 ) ⎜ ⎟ .. .. ⎝ ⎠ . . s1 (qp(n) ) . . . sp(n) (qp(n) ) is nonsingular. Choose local C m sections σ1 , . . . , σp(n) of X → B passing through the points q1 , . . . , qp(n) . By continuity, ⎛

s1 (σ1 (b)) ⎜ .. A(b) = ⎝ . s1 (σp(n) (b))

... ...

⎞ sp(n) (σ1 (b)) ⎟ .. ⎠ . sp(n) (σp(n) (b))

stays nonsingular for b in a neighborhood of b0 .

section of O(n) on a small neighborhood of Xb0 , unique way, s = ai (b)si . The ai are given by ⎞ ⎛ ⎛ s(σ1 (b)) a1 (b) ⎟ ⎜ .. .. −1 ⎜ ⎠ = A(b) ⎝ ⎝ . . ap(n) (b)

Therefore, if s is a we can write, in a ⎞ ⎟ ⎠

s(σp(n) (b))

and hence are C m functions of b. This proves the lemma. We may now conclude the proof of (9.11). Lemma (9.12) implies

that ker ρn is a C m vector subbundle of the trivial bundle π∗ O(n) whose 0 fiber over b ∈ B is H (X , I (n)). Hence the map to the Grassmannian b X b − p(n), H 0 (Pr , O(n))) it defines is of class C m . Since this is the G( n+r r composition of ψ with the inclusion of the Hilbert scheme Hilbp(t) in the r Grassmannian, the result follows. 10. Bibliographical notes and further reading. The Hilbert scheme was introduced and constructed by Grothendieck in [330]. Further general references on the Hilbert scheme are Chapter 5 of [243] by Nitsure, Sernesi [624,625], Kollár [439], and Huybrechts– Lehn [379]. Our treatment here is largely patterned on the one in [624]. The connectedness of the Hilbert scheme of projective space is due to Hartshorne [354]. The infinitesimal theory of the Hilbert scheme, in the form of the theory of the characteristic series, goes back at least to the early days of

§11 Exercises

65

the Italian school of algebraic geometry. The problem of the completeness of the characteristic series, i.e., the one of deciding whether an embedded scheme admits a deformation over a smooth base whose characteristic map is onto, has been a guiding theme throughout much of the development of algebraic geometry. The Italian school extensively studied the problem and struggled with it without coming to a satisfactory conclusion (see, for instance, Enriques [212], Severi [630,632,635,636,637], Segre [620]; another classical source is the book by Coolidge [129]). A very extensive discussion of the problem is contained in Chapter V of Zariski’s book [690] and is updated in Mumford’s appendix to the same chapter. A satisfactory criterion for the completeness of the characteristic system was finally found by Kodaira and Spencer [436] (for weaker results, see also Kodaira [430,431]). A standard reference for all these questions is Mumford’s book [550] (see, in particular, Lecture 22). Hilbert schemes exhibit a bewildering variety of pathological behaviors. That Hilbert schemes of points are in general reducible was first proved by Iarrobino [380]; an overview of the theory of Hilbert schemes of points can be found in Iarrobino [381]. Mumford’s example of a nonreduced component of the Hilbert scheme first appeared in [547]; of course, this same example provides an instance of non-completeness of the characteristic series. Further pathological phenomena were discovered by others, in particular, by Martin-Deschamps and Perrin [501] and by Vakil [667]; this last paper provides an overview of the subject. Variants of the Hilbert scheme are commonly used in the literature, and are discussed in many of the general references given above. Flag Hilbert schemes were first studied by Kleppe [424]. C n families of projective manifolds are treated by the first two authors in [29]. 11. Exercises. 1. Let A

w A

u B

u w B

be a commutative diagram of homomorphisms of commutative rings, and let M be a B-module. i) Show that, if M is A-flat and B is B-flat, then M ⊗B B is A -flat (Hint: observe that, if P is any A-module, then (M ⊗B B ) ⊗A (P ⊗A A ) = B ⊗B (M ⊗A P )). ii) Assume that B → B is a local homomorphism of local rings, that B is noetherian, and that M is finitely generated as a B-module. Show that, if M ⊗B B is A -flat, A is A-flat, and B is B-flat, then M is A-flat.

66


A. Linearity of the characteristic map The following set of exercises sketches a proof of the linearity of the characteristic map (5.10). We treat the affine case first. The line of reasoning closely parallels the first half of the proof of (5.8). We let R and A be noetherian C-algebras, m a maximal ideal in A, and we wish to look at a flat family of subschemes of Spec R parameterized by Spec A. This corresponds to an ideal J ⊂ A ⊗ R such that (A ⊗ R)/J is a flat A-module; here and in what follows, when we omit mention of the ring over which we are tensoring, we mean that we are tensoring over C. The fiber of this family over the point corresponding to m is Spec(R/I), where R/I = A/m ⊗A (A ⊗ R)/J . A-1. Show that I is naturally isomorphic to J/(J ∩ (M ⊗ R)) A-2. Show that J is flat over A (hint: look at the Tor sequence obtained by tensoring (11.1)

0 → J → A ⊗ R → (A ⊗ R)/J → 0

with an arbitrary A-module and use the flatness of A ⊗ R and (A ⊗ R)/J). A-3. Show that J ∩ (m ⊗ R) = mJ (hint: tensoring (11.1) with A/m, show that I J/mJ). A-4. Given i ∈ I, let ϕ(i) be the class of j − 1 ⊗ i modulo mJ, where j ∈ J is a lift of i. Show that this defines a homomorphism of R-modules ϕ : I → (m ⊗ R)/mJ = m ⊗A (A ⊗ R)/J . A-5. Show that ϕ(I 2 ) ⊂ m(m ⊗A (A ⊗ R)/J) , so that ϕ defines an R-module homomorphism ψ : I/I 2 → m/m2 ⊗A (A ⊗ R)/J . A-6. Show that the homomorphism ψ is functorial under base change and that, when A = Spec C[ε], it agrees with the homomorphism I/I 2 → R/I constructed in the proof of (5.8). A-7. Let v ∈ Hom(m/m2 , C) be a tangent vector to Spec A at the point corresponding to m. Let A → Spec C[ε] be the corresponding ring homomorphism. Pull back to Spec C[ε] the family of subschemes of

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Spec R defined by J. Applying the construction in Lemma (5.8) to the pulled-back family yields a homomorphism ψ : I/I 2 → R/I . Show that ψ is the composition of ψ with the homomorphism v ⊗ 1 : m/m2 ⊗A (A ⊗ R)/J → C ⊗A (A ⊗ R)/J = R/J . A-8. Show that the characteristic map (5.10) is linear.

B. Conics in P3 In this series of exercises we describe the Hilbert scheme parameterizing plane conics in P3 and use this description to answer an enumerative problem. B-1. Let X ⊂ P3 be any subscheme of dimension 1 and degree 2, that is, with Hilbert polynomial p(t) = 2t + c for some c. Show that c ≥ 1. B-2. Now let X ⊂ P3 be any subscheme with Hilbert polynomial p(t) = 2t + 1. Show that the Hilbert function qX (t) = h0 (X, O(t)) of X is equal to p(t) and hence that X is a complete intersection of a plane and a quadric. B-3. Using the preceding result, show that the Hilbert scheme H = ∨ Hilb32t+1 is a P5 -bundle over the dual projective space P3 ; specifically, it is the projectivization of the symmetric square ∨ Sym2 (S ∨ ) of the dual of the universal subbundle S on P3 . B-4. Using the results of Chapter VII of Volume I, calculate the cohomology ring of the Hilbert scheme H. B-5. Let L ⊂ P3 be a line, and let DL ⊂ H be the locus of conics C such that C ∩ L = ∅. Show that DL ⊂ H is a divisor and find its fundamental class α ∈ H 2 (H, Z). B-6. Combining the results of the last two exercises, find the number of smooth conic curves meeting each of eight general lines L1 , . . . , L8 ⊂ P3 . (This involves three things: calculating the eighth power α8 ∈ H 16 (H, Z) ∼ = Z, showing that, for general L1 , . . . , L8 , the divisors DLi ⊂ H intersect transversely and that all these points of intersection correspond to smooth conics.) B-7. Now let H ⊂ P3 be a plane, and let EH ⊂ H be the closure of the locus of smooth conics C ⊂ P3 tangent to H. Show that this is a divisor and find its fundamental class β ∈ H 2 (H, Z).

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B-8. Find the number of smooth conics in P3 meeting each of seven general lines L1 , . . . , L7 ⊂ P3 and tangent to a general plane H ⊂ P3 . More generally, find the number of smooth conics in P3 meeting each of 8 − k general lines L1 , . . . , L8−k ⊂ P3 and tangent to k general planes H1 , . . . , Hk ⊂ P3 for k = 1, 2, and 3. B-9. Why do not the methods developed here work to calculate the number of smooth conics in P3 meeting each of 8 − k general lines L1 , . . . , L8−k ⊂ P3 and tangent to k general planes H1 , . . . , Hk ⊂ P3 for k ≥ 4? What can you do to find these numbers? C. Twisted cubics in P3 In this series of exercises we will look at the geometry of the Hilbert scheme Hilb33t+1 , one component of which parameterizes twisted cubic curves. C-1. Let X ⊂ P3 be any subscheme of dimension 1 and degree 3. Show that the Hilbert polynomial of X is of the form 3t + c with c ≥ 0 and that c = 0 if and only if X is a plane cubic curve. C-2. Let H = Hilbp3 be the Hilbert scheme parameterizing subschemes of P3 with Hilbert polynomial p(t) = 3t + 1. Show that H has two irreducible components: one component H0 whose general point corresponds to a twisted cubic curve and one component H1 whose general point corresponds to the union of a plane cubic and a point. For the next few exercises, consider the following three subschemes of P3 : a. the subscheme defined by the square of the ideal of a line, for example, C1 = V (X 2 , XY, Y 2 ); b. a planar triple line with a spatial embedded point of multiplicity 1, for example, C2 = V (X 2 , XY, XZ, Y 3 ); c. a planar triple line with a planar embedded point, for example, C3 = V (X, Y 3 Z, Y 4 ). (Note that the tangent space to C2 at the embedded point [0, 0, 0, 1] is three-dimensional, whereas in the case of C3 it is two-dimensional; hence the names spatial and planar embedded point.) C-3. Show that all three have Hilbert polynomial p(t) = 3t + 1; that is, they correspond to points of the Hilbert scheme H.

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C-4. Find the Hilbert function qCi (t) = h0 (Ci , O(t)) of all three and show that the Hilbert function of C3 is minimal, that is, if X is any subscheme of P3 with Hilbert polynomial 3t + 1, we have qX (t) ≥ qC3 (t) for all m. C-5. For each i = 1, 2 and 3, does the corresponding point [Ci ] ∈ H lie in H0 , H1 , or both? D. Dimension of the Hilbert schemes of twisted cubics and other curves For the following, let U ⊂ Hilb33t+1 be the open subset parameterizing twisted cubics. D-1. Prove that U is irreducible of dimension 12 by using the fact that a twisted cubic is residual to a line in an intersection of two quadric surfaces. D-2. Prove that U is irreducible of dimension 12 by using the fact that a twisted cubic is the image of P1 under a Veronese map. D-3. Prove that if p1 , . . . p6 ∈ P3 are any six points with no four coplanar, there exists a unique twisted cubic curve passing through all six; and use this to prove that U is irreducible of dimension 12. D-4. Let A(X) be a 2 × 3 matrix whose entries are general linear forms Li,j (X) on P3 . Show that the locus {X ∈ P3 : rank(A(X)) = 1} is a twisted cubic and use this to prove yet again that U is irreducible of dimension 12. For the following, we define the restricted Hilbert scheme to be the union of those components of the Hilbert scheme whose general point corresponds to an irreducible and nondegenerate variety. D-5. Using the techniques introduced in the above problems (mainly in problems D-1 and D-2), classify the irreducible components of the restricted Hilbert scheme parameterizing curves of degree 4 in P3 and find their dimensions. D-6. Similarly, classify the irreducible components of the restricted Hilbert scheme parameterizing curves of degree 5 in P3 and find their dimensions.

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D-7. What is the smallest degree d such that the restricted Hilbert scheme parameterizing curves of degree d in P3 has a component of dimension other than 4d? D-8. What is the smallest degree d such that the restricted Hilbert scheme parameterizing curves of degree d in P3 has two or more irreducible components whose general points correspond to curves of the same genus? D-9. Are there any components of the restricted Hilbert scheme parameterizing curves in Pr whose general point corresponds to a singular curve?

E. Chow varieties The Chow variety Cn,k,d is a variety parameterizing subvarieties X ⊂ Pn of dimension k and degree d; it has a natural compactification parameterizingcycles of dimension k and degree d, that is, formal linear combinations ai Xi with Xi ⊂ Pn irreducible of dimension k, ai ∈ N, and ai deg(Xi ) = d. It is cruder than the Hilbert scheme—it does not see scheme structures or components of dimension less than k—but sometimes that’s a good thing. While it has been largely superseded by the Hilbert scheme, it is still useful in some contexts; for example, the construction of moduli spaces: taking the quotient of the Chow variety by P GL(n + 1) often yields a different result from taking the quotient of the corresponding components of the Hilbert scheme. The following exercises will establish some of the basic facts about the Chow construction. For the following, G will stand for the Grassmannian G(n − k, n + 1) of (n − k − 1)-planes in Pn , and PN = PH 0 (G, OG (d)) will denote the projective space parameterizing hypersurfaces of degree d in G. E-1. Let X ⊂ Pn be a variety of pure dimension k and degree d. Show that the subvariety ΦX = {Λ : Λ ∩ X = ∅} ⊂ G is a hypersurface in G. E-2. Let X ⊂ Pn and ΦX ⊂ G be as above. Show that ΦX is a hypersurface of degree d, that is, as a divisor on G, it is linearly equivalent to d times the hyperplane class on G. P

N

The open Chow variety is defined to be the locus Cñ,k,d ⊂ |OG (d)| = of hypersurfaces arising in this way; the Chow variety is its closure

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in PN . The next few exercises will be devoted to a proof that C˜ is locally closed in PN . E-3. Show that the following are closed subvarieties: {(x, Λ) : x ∈ Λ} ⊂ Pn × G and {(Λ, Φ) : Λ ∈ Φ} ⊂ G × PN . E-4. For any point p ∈ Pn , let Gp ⊂ G be the sub-Grassmannian of planes containing p. Now let Φ ⊂ G be any hypersurface of degree d. Show that XΦ = {p ∈ Pn : Gp ⊂ Φ} is a closed subvariety of Pn of dimension at most k. Show moreover that if Φ is reduced and irreducible, and dim XΦ = k, then deg XΦ = d. E-5. Show that Θ = {(p, Φ) : Gp ⊂ Φ} is a closed subvariety. E-6. Let U ⊂ PN be the open subset of reduced and irreducible hypersurfaces. Using the preceding problems, show that the locus of Chow forms of irreducible varieties X ⊂ Pn is closed in U . (A similar argument works to show that Cn,k,d is locally closed in all of PN , but it is notationally messier.) E-7. Let B be a smooth, one-dimensional variety, and let X ⊂ B × Pn be a family whose general fiber Xt ⊂ Pn is a reduced and irreducible variety of dimension k and degree d. Suppose that the schemetheoretic special fiber X0 of the family is a union X0 = ∪Yi of irreducible components Yi , with Yi a scheme of multiplicity Mi . Show that the limit of the Chow forms FXt of the general fiber is lim FXt = FYmi i . t→0

(Hint: reduce to the case k = 0.) E-8. We would like to make a statement to the effect that if X ⊂ B × Pn is a closed subvariety such that the fibers Xb = X ∩ {b} × Pn are varieties of dimension k and degree d, the map B → Cn,k,d obtained by sending b ∈ B to the Chow form of Xb is a regular map. What hypotheses do we need on X?

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F. Punctual Hilbert schemes. These exercises deal with the Hilbert schemes Hilbdn parameterizing zero-dimensional subschemes of Pn of degree d (that is, schemes with Hilbert polynomial the constant d). This Hilbert scheme always contains one irreducible component H0 , of dimension dn, whose general point [X] corresponds to a reduced scheme X ⊂ Pn , that is, d general points. The point of these exercises is to show that, in general, Hilbdn may contain many other components as well. The method in each case is to exhibit families of such schemes of dimension strictly greater than dn. F-1. Start with P3 and consider, for any point p ∈ P3 , the family of schemes X whose ideals are sandwiched between successive powers of the maximal ideal m at p, that is, for some k and , we have mk+1 ⊂ IX ⊂ mk with dim(mk /IX ) = . Calculate the degree d of such a scheme X and the dimension of the family of such schemes; conclude that for k large enough and ∼ k2 /2, the Hilbert scheme Hilbd3 is reducible. F-2. What is actually the smallest d for which the above argument shows that Hilbd3 is reducible? F-3. On to Pn , and this time consider specifically schemes X of degree n + 1 + δ with m3 ⊂ IX ⊂ m2 . is Show that for n large enough, the Hilbert scheme Hilbn+4 n reducible. What is the smallest n for which you can prove this? F-4. Going even further, consider now subschemes X ⊂ Pn of degree d that are supported at a point p, contained in a subspace Λ = Pr , and whose ideals in that Pr are sandwiched between the schemes defined by the square and the cube of the maximal ideal at p, that is, such that IΛ + m3 ⊂ IX ⊂ IΛ + m2 . Using these, show that the schemes Hilbdn may be reducible even when d < n. We should say that the actual number of components of Hilbdn , or their dimensions, is completely unknown; nor is it known exactly for which pairs (d, n) the scheme Hilbdn is reducible. Even worse, given a zero-dimensional subscheme X ⊂ Pn of degree d, we know of no effective way of determining whether or not it belongs to the principal component H0 of Hilbdn , that is, whether it is a limit of d distinct

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points. Moreover, this uncertainty about punctual Hilbert schemes spills over into Hilbert schemes of curves, since these will in general have components whose general member is the disjoint union of a curve and a zero-dimensional scheme. Thus, for example, it is impossible to say (except in very special cases) how many components the Hilbert scheme parameterizing curves of degree d and genus g will have, or Hilbdt−g+1 3 what their dimensions might be.

G. Sections of the Hilbert scheme Note: for these problems, we are concerned with sections of the universal family of rational normal curves; since we are only interested in rational sections, it does not matter if we take the Chow variety or Hilbert scheme as our parameter space. G-1. Let Hilb22t+1 be the Hilbert scheme of plane conics; let X ⊂ B×P2 → B be the universal family over B. Show that X → B has no rational section. G-2. Let B ⊂ Hilb33t+1 be the locus, in the Hilbert scheme, of twisted cubic curves; let X ⊂ B × P3 → B be the universal family over B. Show that X → B does have a rational section and show how to construct one. G-3. More generally, for any n, let B ⊂ Hilbnt+1 be the locus of rational n normal curves in Pn , and again let X ⊂ B ×Pn → B be the universal family over B. For what n does X → B have a rational section?

H. Special classes of curves These exercises deal with various classes of curves in P3 . In each case, the degree and arithmetic genus of the curves in question can be calculated readily; in addition, it can be shown that the curves in question form a single irreducible family, whose dimension also can be calculated. The problem then is to say whether the locus of such curves is open in the Hilbert scheme or, in other words, whether they are dense in an irreducible component of H. Note: series (iii) is a direct generalization of series (i); series (iv) is somewhat harder than the first three; and Problem 6 is open. i. Complete Intersections. A curve C ⊂ P3 is called a complete intersection of type (d, e) if it is scheme-theoretically the intersection of a surface S of degree d and a

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surface T of degree e; by Noether’s AF + BG theorem, this says that the homogeneous ideal IC is generated by the polynomials defining S and T . H-1. Find the degree and genus of a complete intersection C ⊂ P3 of type (d, e) by applying Bézout’s theorem and the adjunction formula. H-2. Now find the degree k and genus g of C by calculating the Hilbert polynomial. H-3. Assuming d ≤ e, find the dimension of the family of such curves and show that the family is irreducible. H-4. Now let Hilb3kt−g+1 be the Hilbert scheme parameterizing curves of degree k and genus g in P3 , and let U ⊂ Hilb3kt−g+1 be the locus of complete intersection curves of type (d, e). Show that U ⊂ Hilb3kt−g+1 is open. H-5. Show by example that it is not closed. H-6. Now let Hs ⊂ H be an open subset of smooth curves, and let U s = U ∩ Hs be the locus of smooth complete intersection curves of type (d, e). Is U s closed in Hs ? ii. Curves on quadrics In this series, we deal with curves of type (a, b) on a quadric, that is, curves C ⊂ Q ⊂ P3 that lie on a smooth quadric Q and are linearly equivalent on Q to a lines of one ruling plus b lines of the other (equivalently, the zero locus of a bihomogeneous polynomial of bidegree (a, b) on Q ∼ = P1 × P1 ). H-7. Find the degree k and genus g of a complete intersection C ⊂ P3 by applying the adjunction formula. H-8. Now find the degree and genus of C by calculating the Hilbert polynomial. H-9. Assuming that a ≤ b, find the dimension of the family of all curves of type (a, b) on quadrics and show that it is irreducible. (Note that the quadric is not fixed!) H-10. Show that, in the Hilbert scheme Hilb3kt−g+1 , the locus V ⊂ Hilb3kt−g+1 of curves of type (a, b) on quadrics is open if a and b are both at least 3, but not otherwise. H-11. Analogously to problem H-6 above, let Hs ⊂ H be the open subset of smooth curves, and let V s = V ∩ Hs be the locus of smooth curves of type (a, b) on quadrics. Show that V s is closed in Hs if |b − a| > 1, but not otherwise.

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iii. Quasi-Complete Intersections We call a curve C ⊂ P3 a quasi-complete intersection of type (d, e, m) if it is residual to a plane curve of degree m in a complete intersection of type (d, e). For our purposes, we will take that to mean that there is a plane curve D of degree m whose support has no component in common with the support of C such that C ∪D = S∩T for surfaces S and T of degrees d and e (though the notion of two curves C and D being residual in a complete intersection can be extended to the case where their supports may have components in common). H-12. Find the degree and genus of a quasi-complete intersection C ⊂ P3 of type (d, e, m) by applying Bézout’s theorem and the adjunction formula. H-13. Now find the degree and genus of C by calculating the Hilbert polynomial. H-14. Assuming d ≤ e, find the dimension of the family of such curves and show that the family is irreducible. H-15. Show that the locus of quasi-complete intersection curves of type (d, e, m) is open in the Hilbert scheme. iv. Determinantal Curves To keep the notation from getting completely out of hand, we will restrict ourselves here and say that a curve C ⊂ P3 is determinantal of type n if it is the rank n − 1 locus of an n × (n + 1) matrix A of linear forms on P3 ; that is, its ideal is generated by the maximal minors of A. H-16. Find the degree and genus of a determinantal curve of type n. H-17. Find the dimension of the locus in H of determinantal curves of type n. H-18. Is this locus open in the Hilbert scheme?

I. More on Mumford’s example In this series of exercises, we will go through an analysis of the parameterizing curves of degree 14 and genus Hilbert scheme Hilb14t−23 3 24 in P3 , describing all the components of H. We will consider only the restricted Hilbert scheme, that is, the union H of those components of whose general point corresponds to a smooth irreducible curve. Hilb14t−23 3

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For the following, we will assume throughout that [C] is a general point of its component of H. I-1. Let F be a smooth cubic surface in P3 . known (cf., for instance, [318]):

The following are well

i) If {E1 , . . . , E6 } is any set of six disjoint lines on F , then there are six points {p1 , . . . , p6 } ∈ P2 not lying on a conic, no three of which are collinear, such that there is an isomorphism between F and the blow up of P2 at {p1 , . . . , p6 } under which the lines {E1 , . . . , E6 } correspond to the exceptional curves of the blow-up. ii) Let {p1 , . . . , p6 } and {E1 , . . . , E6 } be as above. Denote by Hi,j the proper transform of the line through pi and pj and by Gi the proper transform of the conic through p1 , . . . , pi , . . . , p6 , where the hat indicates omission. Then the set consisting of all the Ei , all the Hi,j , and all the Gi is the set of all lines lying on F . Moreover, the sets of six disjoint lines on F are the following: (1) {E1 , . . . , E6 }; (2) {G1 , . . . , G6 }; (3) all the sets of the form {Ei , Ej , Eh , Hk,l , Hk,m , Hl,m }, where i, j, k, l, m, n are distinct; (4) all the sets of the form {Gi , Gj , Gh , Hk,l , Hk,m , Hl,m }, where i, j, k, l, m, n are distinct; (5) all the sets of the form {Ei , Gi , Hj,h , Hj,k , Hj,l , Hj,m }, where i, j, k, l, m, n are distinct. Show that the monodromy group of the family of all smooth cubics in P3 acts transitively on the set of lines lying on F . I-2. Show that C cannot lie in a plane, a quadric surface, a cubic cone, or a cubic with a double line, because there simply do not exist smooth curves of degree 14 and genus 24 on any of these surfaces. I-3. Suppose now that C lies on a cubic surface S and that S is either smooth or has isolated double points. Show that the linear system |C| on S has dimension exactly 37. Deduce that C cannot lie on a cubic surface with isolated double points, that is, S must be smooth, because such a curve is a specialization of a curve on a smooth cubic surface and C was taken to be general. I-4. Now suppose that C lies on a smooth cubic surface S. Show that C will also lie on a sextic surface T not containing S; we can then write the intersection S ∩ T as the union of C with a curve D of degree 4. Verify in order the following intersection numbers (of

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divisors on S): (C · ωS ) = −14; (C · C) = 60; (C · D) = 24; (D · D) = 0. Deduce that D is a curve of arithmetic genus −1 and hence that D is linearly equivalent either to the union of two disjoint conics or to the disjoint union of a line and a twisted cubic; in these cases we will say that C is of type 1 or of type 2, respectively. I-5. Continuing the preceding problem, show that curves C of type 1 form an irreducible, 56-dimensional family, and the same for curves of type 2. I-6. With the conventions of the last two problems, show that h0 (NC/P3 ) = 57 if C is of type 1, while h0 (NC/P3 ) = 56 if C is of type 2. I-7. Suppose now that C lies on no cubic surface. We know C lies on a pencil of quartic surfaces; argue (using Bertini’s and Bézout’s theorems) that the general such quartic is smooth along C. Then, using a series of calculations analogous to those of Problem 3 above, deduce that C is residual to a plane conic in a complete intersection of quartic surfaces. (We will say such a curve C is of type 3.) I-8. Finally, show that curves of type 3 form an irreducible family of dimension 56; deduce that curves of type 1 are dense in a component of the Hilbert scheme and that this component is everywhere nonreduced. I-9. One last note: if you tried to do the dimension count of the preceding problem by looking at the map from the locus of curves of type 3 to the space P34 of quartic surfaces and estimating the fiber dimension, you got the wrong answer. Why?

Chapter X. Nodal curves

1. Introduction. A nodal curve is a complete algebraic curve such that every one of its points is either smooth or is locally complex-analytically isomorphic to a neighborhood of the origin in the locus with equation xy = 0 in C2 . Of course, we are interested in families of nodal curves. There are two ways of defining what a family of nodal curves over a base S is, and they are both very useful. The first one is to say that it is a proper surjective morphism of schemes, or analytic spaces, ϕ : C → S such that ϕ is flat and every fiber of ϕ is a nodal curve. The second is to say that it is a proper surjective morphism ϕ : C → S of analytic spaces such that for any p ∈ C, either ϕ is smooth at p with one-dimensional fibers, or else, setting s = ϕ(p), there is a neighborhood of p which is isomorphic, as a space over S, to a neighborhood of (0, s) in the analytic subspace of C2 × S with equation xy = f , where f is a function on a neighborhood of s in S whose germ at s belongs to the maximal ideal of OS,s . The equivalence of these definitions is somewhat subtle, and establishing it takes a good part of Section 2. The rest of the section is used to recall the elementary theory of nodal curves, including the definition of dualizing sheaf and the statement of the Riemann–Roch theorem. We conclude the section by discussing relative Kähler differentials and the relative dualizing sheaf for a family of nodal curves. An n-pointed nodal curve consists of the datum (C; p1 , . . . , pn ) of a nodal curve C together with n distinct smooth point of C. We set D = p1 + · · · + pn . In Section 3 we introduce the concept of n-pointed stable curve. This is an n-pointed nodal curve whose automorphism group, as a pointed curve, is finite. It turns out that this is equivalent to requiring that the pullback of the log-canonical sheaf ωC (D) be positive on any component of the normalization of C. An n-pointed nodal curve is said to be semistable if it satisfies a condition similar to the one expressed by the last sentence, but with “nonnegative” substituted for “positive.” The main object of study in this book, to be formally introduced in Chapter XII, is the moduli space of n-pointed stable curve of given genus g. It is denoted by M g,n and, as a set, is simply the set of isomorphism classes of n-pointed genus g stable curves. We also denote by Mg,n the set of isomorphism classes of smooth n-pointed genus g stable curves. It will turn out that Mg,n has the structure of a (3g − 3 + n)-dimensional E. Arbarello et al., Geometry of Algebraic Curves, Grundlehren der mathematischen Wissenschaften 268, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-540-69392-5 2,

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quasi-projective variety and that M g,n is a projective compactification of it. The fact that Mg,n is not compact depends on the fact that a family of smooth curves over a noncomplete base S cannot, in general, be extended to a family of smooth curves over a completion of S. This remains true even if finite base changes are permitted. To complete the family, one has to allow singular fibers of some sort. It is a remarkable fact that one can get by with fibers having at worst nodes. This is a consequence of the stable reduction theorem (4.11), which is the main result proved in Section 4. The compactness of M g,n follows directly from this theorem. In Section 5 we study how the automorphism group of a stable pointed curve varies as the curve moves in a family. More generally, we look at two families of stable pointed curves α : X → U and β : Y → U , and prove that IsomU (X, Y ) is proper over U . This result will be instrumental in proving that M g,n is Hausdorff. We also prove that IsomU (X, Y ) is unramified over U ; as we will see in Chapter XII, this will turn out to be one of the properties that makes the stack counterpart of M g,n a Deligne–Mumford stack. In Section 6 and in the following two, we discuss a number of constructions involving families of nodal curves which will be continuously used throughout the book. The first construction goes under the name of passing to the stable model. Let g and n be such that stable n-pointed genus g curves exist. This means that 2g − 2 + n > 0. Let (C; x1 , . . . , xn ) be a semistable npointed curve of genus g. Then there is a canonical way of constructing, out of this semistable curve, a stable one, which is called the stable model of (C; x1 , . . . , xn ). For this, one notices that the irreducible components which prevent (C; x1 , . . . , xn ) from being stable are precisely those components which are smooth rational and contain just two points which are either marked or nodes. The connected components of their union are chains of smooth rational curves, the so-called exceptional chains. The n-pointed nodal curve (C ; x1 , . . . , xn ) obtained by collapsing to a point each exceptional chain is clearly stable and is, by definition, the stable model of (C; x1 , . . . , xn ). Our aim in this section is to show that the procedure we just described, producing the stable model of a semistable curve, can be performed simultaneously and consistently for all fibers of any family of semistable curves. The stable model makes it possible to easily perform two other operations, which usually go under the name of contraction and projection. Let us start with projection. Let (C; x1 , . . . , xn ) be a stable n-pointed curve. Remove from it, say, the nth point. The resulting (n − 1)-pointed curve is semistable but may not be stable. If we pass to its stable model, the result is a stable (n − 1)-pointed curve. By our general construction, this operation can be performed in families, and this leads to the so-called

§1 Introduction

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projection map π : M g,n → M g,n−1 . The operation of contraction is strictly linked to the one of projection. It basically consists in keeping track of the nth point while performing the projection. Starting from a family of stable n-pointed genus g curves, this procedure produces a family of stable (n − 1)-pointed, genus g curves, plus an extra section which, however, may meet one or more of the marked sections, or go through singular points of fibers. In Section 8 we show that the contraction operation has an inverse which, of course, can be consistently performed on families. This operation will be used in Chapter XI in constructing Kuranishi families of n-pointed curves out of Kuranishi families of (n − 1)-pointed ones. In Section 7 we describe the so-called clutching operation. Here we give the two main examples. In the first one we start from a pair consisting of a stable (a + 1)-pointed genus p curve and a stable (b + 1)pointed genus q curve, and we produce a stable (a + b)-pointed curve of genus p+q by identifying the (a+1)st marked point of the first curve with the (b + 1)st marked point of the second one. In the second example we start from a stable (n + 2)-pointed curve of genus g − 1, and we produce a stable n-pointed genus g curve by identifying the (n + 1)st and (n + 2)nd point of the given curve. Of course, one may imagine more complicated operations. The important fact is that, again, all these operations can be consistently performed in families. The clutching operation is crucial in describing the boundary of moduli space ∂Mg,n = M g,n Mg,n . For example, the two simple clutching operations we described above will lead to two fundamental boundary morphisms ξp,a : M p,a+1 × M q,b+1 → M g,n , ξirr : M g−1,n+2 → M g,n .

g = p + q, n = a + b;

The last section is devoted to the so-called Picard–Lefschetz transformation. To illustrate it in simple terms, we consider a family C → Δ of genus g curves, parameterized by the disk Δ = {t ∈ C : |t| < 1}, and whose fibers are smooth except for the central one, which has a single node. Now look at a simple loop γ : [0, 1] → Δ {0} based at the point a ∈ Δ. We pull back the given family of curves to the interval [0, 1] and choose a C ∞ trivialization F : γ ∗ C → Ca × [0, 1] of this pulled-back family. We then get a diffeomorphism ϕγ = F0 F1−1 : Ca → Ca . One can verify that the isotopy class of ϕγ only depends on the homotopy class of γ. This leads to the following general picture. Suppose that π : X → B is an algebraic or an analytic family of stable genus g curves. Let B ∗ ⊂ B be the locus of points in B corresponding to smooth fibers of π. Given a ∈ B ∗ , there is a natural group homomorphism PL : π1 (B ∗ , a) → Diff(Ca )/ Diff 0 (Ca ) [γ] → [ϕγ ]

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10. Nodal curves

where Diff 0 (Ca ) is the group of diffeomorphisms of Ca that are isotopic to the identity. The group Diff(Ca )/ Diff 0 (Ca ) is denoted by Γg and is called the Teichm¨ uller modular group, while the homomorphism P L is called the Picard–Lefschetz transformation. If there were a universal family over Mg one could apply this procedure to get a map from the uller modular group and even fundamental group of Mg to the Teichm¨ hope that this is an isomorphism. It actually turns out that, as a space, Mg is simply connected. However, we will learn that it is more useful to think of Mg as a stack (which we denote by Mg ). It then turns out that the fundamental group of the stack Mg is exactly Γg . In fact, in uller Chapter XV we will construct a contractible space Tg , the Teichm¨ space, which is the parameter space for a family of smooth genus g curves, on which Γg acts properly discontinuously, with finite stabilizers, and such that the moduli map Tg → Mg is the quotient of Tg by Γg . The fact that Mg is the quotient of a contractible space modulo the action of a discrete group Γg , acting with finite stabilizers, tells us that the rational cohomology of Γg is the same as the rational cohomology of Mg , and one traditionally refers to this picture by saying that Mg is a rational K(Γg , 1). Another way of describing the same picture is to say that, as a stack, Tg is the universal cover of Mg . Whatever the fundamental group of the stack Mg may be, it is intuitively clear that the Picard–Lefschetz transformation gives a homomorphism from this group to Γg . But it is far from clear why this homomorphism should be an isomorphism and, in particular, why it should be surjective. The reason why it is possible to give an answer to this question is that the Picard–Lefschetz transformation can be described in very explicit terms. Let us go back to the family C → Δ of smooth curves acquiring a node at the central fiber C0 . One can think that the node of C0 is produced by contracting a smooth simple closed curve c on a smooth fiber Ca . The cycle [c] ∈ H1 (Ca ; Z) is called the vanishing cycle of the family. Now, one can explicitly compute the diffeomorphism ϕγ : Ca → Ca and verify that it is a Dehn twist around the cycle c. By this we mean the homeomorphism δc (well defined up to isotopy) obtained by choosing an orientation on c, cutting the surface Ca along c, rotating the right edge c of c by 180o in the positive direction, rotating the left edge c of c by 180o in the negative direction, and gluing the two edges together again. It is a theorem in the topology of surfaces that the group Γg is generated by Dehn twists, so that any element in Γg is in the image of a Picard–Lefschetz transformation. This is why the fundamental group of the stack Mg surjects onto Γg . We can then identify Sp(H1 (Ca , Z)) with Sp2g (Z), and we have a homomorphism (1.1)

χ : Γg −→ Sp2g (Z) h → h∗

§2 Elementary theory of nodal curves

83

As we shall see in Chapter XV, this homomorphism is surjective. It is instructive to observe that the image of a Dehn twist under this homomorphism has a very simple form, namely that δc∗ (d) = d + (d · c)c, where (d · c) is the intersection number of the two cycles d and c. Another aspect of the Picard–Lefschetz theory has to do with vanishing cycles. Let us go back to a family of stable curves π : X → B, and let us assume that B is a “small enough” polydisc. We prove that the total family X contracts to the central fiber X0 and that the injection of a smooth fiber Xa into X induces a surjective homomorphism H1 (Xa , Z) → H1 (X , Z) = H1 (X0 , Z) whose kernel is generated by the vanishing cycles. This result is intuitively clear, but its proof requires some computation. In fact we will give explicit formulae describing the retraction from X to X0 . To do so, it will be convenient to make use of real blow-ups. For example, let us look at the one-dimensional family the C → Δ and blow up the disk at the origin. In the blown-up disk Δ, 1 origin is substituted with a copy of S . Now look at C0 and normalize it. In the normalization N there are two points, s and t, corresponding has to the node of C0 . Blow up N at s and t. The resulting surface N 1 a boundary formed by two copies of S , call them S and T . Fix a point p in S and a point q in T . One can construct a (real-analytic) family which coincides with the original family over of smooth surfaces X → Δ Δ {0} and such that, over each point ϑ of the exceptional S 1 in Δ, there lies the surface obtained from N by identifying S and T in such a way that the angle between p and q is equal to ϑ. All of this can be contracts to its done explicitly, and one then sees that the family C → Δ 1 restriction to S . Blowing everything down gives the desired retraction of C to C0 . Passing from a disk to a polydisc is notationally a bit more involved but presents no conceptual difficulty. 2. Elementary theory of nodal curves. For future use, it is convenient to treat the elementary theory of nodal curves both from the analytic and the algebraic point of view. One says that a singular point of a one-dimensional analytic space is a node if it has a neighborhood which is complex-analytically isomorphic to a neighborhood of the origin in the locus with equation xy = 0 in C2 . A nodal curve is a complete algebraic curve such that every one of its points is either smooth or a node. More generally, a family of nodal curves over a base Y is a proper surjective morphism of schemes or analytic spaces ϕ:C→Y such that: i) ϕ is flat; ii) every geometric fiber of ϕ is a nodal curve.

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An equivalent, and often more manageable, characterization of families of nodal curves is provided by the following result. Proposition (2.1). A proper surjective morphism π : X → S of analytic spaces is a family of nodal curves if and only if the following holds. For any p ∈ X, either π is smooth at p with one-dimensional fibers, or else, setting s = π(p), there is a neighborhood of p which is isomorphic, as a space over S, to a neighborhood of (0, s) in the analytic subspace of C2 × S with equation (2.2)

xy = f ,

where f is a function on a neighborhood of s in S whose germ at s belongs to the maximal ideal of OS,s . To prove the proposition, we first need a differential characterization of nodes. Lemma (2.3). Let f be a holomorphic function on a neighborhood of 0 ∈ C2 , and suppose that f (0) = 0. Then the analytic space defined by f has a node at the origin if and only if the first-order partials of f vanish at the origin and the Hessian of f is nonsingular at the origin. One implication is obvious; if f = zw in suitable coordinates z and w, then the partials of f vanish for z = w = 0, and the Hessian is nonsingular. To prove the converse, notice that, if we choose coordinates x, y on C2 and assume that ∂f /∂x and ∂f /∂y vanish at 0 and that the Hessian is non-singular there, then on a neighborhood of the origin we may write f (x, y) = a(x, y)x2 + 2b(x, y)xy + c(x, y)y 2 , where a, b, and c are holomorphic, and ac − b2 does not vanish for x = y = 0. We may assume, possibly after a linear change of coordinates, that a(0, 0) = 0. We then set x1 = x + (b/a)y, y1 = y, and notice that f = a1 x21 + c1 y12 with a1 (0, 0) and c1 (0, 0) different from zero. Choose new coordinates x2 and y2 by setting x2 = x1 α, y2 = y1 γ, where α and γ are square roots of a1 and c1 , respectively. Thus, f = x22 + y22 . The last step √ consists in choosing as final coordinates x3 = x2 + and y3 = x2 − −1y2 . With this choice of coordinates, f = x 3 y3 . This finishes the proof of the lemma.

√ −1y2


85

Now we can prove (2.1). For any s ∈ S, we write Xs to denote π −1 (s). Suppose first that π : X → S is a family of nodal curves, and let p be a point of Xs . We may realize an open neighborhood of p in X as a locally closed subspace of Cr × S in such a way that the restriction of the projection to the second factor agrees with π and that p corresponds to (0, s). The Zariski tangent space to Xs at p has dimension h, where h can equal 1 or 2. Choose a linear projection Cr → Ch in such a way that the induced map Tp (Xs ) → Ch is an isomorphism. Since the tangent space to X at p sits in an exact sequence 0 → Tp (Xs ) → Tp (X) → Ts (S) , the composite map Tp (X) → Cr ⊕ Ts (S) → Ch ⊕ Ts (S) is injective. This means that there is a neighborhood U of p in X such that U → Cr × S → Ch × S identifies U to a locally closed subspace of Ch × S. In other words, we may choose r equal to h. If h = 1, this shows that π is smooth at p. If h = 2, we are essentially dealing with a family of plane curves. In this case we may view Xs , or rather a neighborhood of p in it, as a subspace of C2 ; under this identification, p corresponds to 0 ∈ C2 . Since Xs has no embedded components, it is locally defined by a single equation f = 0. Since we are assuming that X is flat over S, the local ring OX,p is the quotient of OC2 ×S,(0,s) modulo a principal ideal with a generator F which reduces to f on C2 × {s} = C2 . Thus X is locally defined, inside C2 × S, by the single equation F = 0. We may realize a neighborhood of s in S as an analytic subspace of some Ck with s identified to the origin. Let z = (z1 , . . . , zk ) be the coordinates in Ck , and extend F to an element of the local ring of C2 × Ck at the origin, which we will also denote by F . Our goal is to put F in as simple a form as possible, by a judicious change of coordinates. The argument we shall use is just a version “with parameters” of the proof of Lemma (2.3). We begin by considering the pair of functions ∂F/∂x, ∂F/∂y. Their Jacobian matrix is just the Hessian of F with respect to x and y, which by Lemma (2.3) is nonsingular for x = y = 0 and z = 0. The implicit function theorem then says that there are functions ϕ(z), ψ(z) such that the locus of points where ∂F/∂x and ∂F/∂y vanish is given, near the origin of C2 × Ck , by x = ϕ(z), y = ψ(z). Replacing the coordinates x and y with x − ϕ(z) and y − ψ(z), we may then suppose that the partials ∂F/∂x and ∂F/∂y vanish identically for x = y = 0. This means that we may write F (x, y, z) = −f (z) + a(x, y, z)x2 + 2b(x, y, z)xy + c(x, y, z)y 2 , where a, b, and c are holomorphic functions such that ac − b2 does not vanish for x = y = 0 and z = 0. We will operate on Q = ax2 + 2bxy + cy 2 ,

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mimicking what we would do to diagonalize a quadratic form in x and y. Possibly after a linear change of coordinates, we may assume that a(0, 0, 0) = 0. We then set x1 = x + (b/a)y, y1 = y, and notice that Q = a1 x21 + c1 y12 with a1 (0, 0, 0) and c1 (0, 0, 0) different from zero. Choose new coordinates x2 and y2 by setting x2 = x1 α, y2 = y1 γ, where α and γ are square roots of a1 and c1 , respectively. Thus, Q = x22 + y22 . Then set x3 = x2 + coordinates,

√ √ −1y2 and y3 = x2 − −1y2 . With this choice of F = x3 y3 − f (z) .

This proves the “only if” part of the proposition. The proof of the converse is simpler. First of all, if π is smooth at p, then it is also flat at p, and Xπ(p) is smooth. On the other hand, any space of the form (2.2) is flat over S. To prove this, it suffices to deal with the special case where S is the affine line, and f = t is a linear coordinate on it, since the general case can be obtained by base change from the special one. Away from x = y = t = 0, the projection from xy = t to the t-coordinate is smooth. At x = y = t = 0, the generator xy of the ideal of the central fiber in OC2 ,0 extends by definition to the generator xy − t of the ideal of X in OC2 ×C,0 . Moreover, any relation among generators of the ideal of the central fiber extends to a relation among generators of the ideal of X, since there are no relations at all. The conclusion follows from Lemma (5.1) in Chapter IX. Recall that a morphism f : X → Y of schemes or of analytic spaces is said to be a local complete intersection morphism (a l.c.i. morphism for short) if it factors, locally on X, as a regular embedding followed by a smooth morphism. Clearly, from the analytic point of view, a family of nodal curves is a l.c.i. morphism. We wish to show that the same is true in the scheme setup. More generally, we shall prove the following. Lemma (2.4). Let f : X → Y be a morphism of schemes (of finite type over C). Then f is a l.c.i. morphism if and only if the corresponding morphism of analytic spaces is l.c.i. In the proof, we shall rely on standard properties of l.c.i. morphisms, for which we refer to Section 6.3 of [480]; the results we shall use are stated and proved there only for morphisms of schemes, but they remain true, with essentially the same proof, for morphisms of analytic spaces. The next lemma summarizes the standard property we need.


87

Lemma (2.5). Let f : X → Y be a l.c.i. morphism. Suppose that f is the composition of an embedding α : X → Z and a smooth morphism Z → Y . Then α is a regular embedding. To prove (2.4), we just have to show that, if f is l.c.i. analytically, then it is l.c.i. also scheme-theoretically. Since the statement is of a local nature, we may assume that f factors into an embedding α : X → Y ×CN followed by the projection of Y × CN to Y . Lemma (2.5) says that α is a regular embedding of analytic spaces. We must show that it is also a regular embedding of schemes. In other words, we are reduced to proving the following. Lemma (2.6). Let h : W → Z be an embedding of schemes of finite type over C. Then h is a regular embedding if and only if the corresponding morphism of analytic spaces is a regular embedding. Let z be a closed point of W . We denote by A the local ring of Z at z, and by I the ideal of W in A. We also denote by B the local ring at z of the analytic space underlying Z. Clearly, A is a subring of B. The ideal in B of the analytic subspace corresponding to W is just BI. Now, one of the main technical results of Serre’s GAGA [626] is that (A, B) is a flat pair, so that in particular B is A-flat. As the inclusion A → B is a local homomorphism, B is actually faithfully flat over A. Recall in fact that a module M over a commutative ring R is said to be faithfully flat when any complex C • of R-modules is exact if and only if C • ⊗R M is exact; equivalently, when M is flat and, for any R-module N , N ⊗R M is zero if and only if N is. Faithful flatness of B over A is then a consequence of the following standard result. Lemma (2.7). Let R → R be a local homomorphism of local rings. If R is R-flat, then it is faithfully flat. The proof is straightforward. Let M be a nonzero R-module, and let L be a submodule of M generated by a nonzero element; thus L R/J for some proper ideal J. By flatness, L ⊗R R is a submodule of M ⊗R R ; on the other hand, L ⊗R R does not vanish since, again by flatness, R /J ⊗R R ∼ = R /JR , and JR is contained in the maximal ideal of R . Returning to the proof of (2.6), let b1 , . . . , bn be a minimal system of generators of I. Obviously, the bi also generate BI; we claim that, in fact, they are a minimal system of generators. For each fixed j, denote by Ij the ideal of A generated by all the bi except bj and look at the exact sequence Ij → I → Q → 0 , where Q = I/Ij . Since b1 , . . . , bn is a minimal system of generators of I, Q = 0. Tensoring with B gives an exact sequence BIj → BI → Q ⊗A B → 0 ,

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since I ⊗A B = BI and Ij ⊗A B = BIj , by flatness. On the other hand, Q ⊗A B cannot vanish, by faithful flatness. This proves that b1 , . . . , bn is a minimal system of generators of BI; by assumption and by Lemma (5.22) it is thus a regular sequence in B. To conclude the proof, we must show that it is a regular sequence in A as well, that is, that the sequence ×bi+1

(b1 , . . . , bi ) → A −−−−→ A/(b1 , . . . , bi ) is exact for all i. Tensoring with B gives an exact sequence since b1 , . . . , bn is a regular sequence in B; hence the conclusion follows from faithful flatness. This concludes the proof of Lemma (2.6) and hence of Lemma (2.4). Given a nodal curve C and a set S of nodes of C, let β : X → C be the partial normalization of C at S. We may associate to this setup a graph. There is one vertex for each connected component of X, and the half-edges issuing from a given vertex are the points of the corresponding component of X mapping to nodes of C belonging to S. A pair {q, q } of distinct half-edges constitutes an edge when q and q map to the same node of C. To each vertex v we assign an integer weight gv , the arithmetic genus of the corresponding component of X. We shall sometimes write GraphS (C) to indicate the graph we have just described, equipped with the genus weight function. When S is the set of all nodes of C, we shall write Graph(C) for GraphS (C); this is the so-called dual graph of C. Since β is finite, for any coherent sheaf F on X, the higher direct images Ri β∗ F vanish, and hence, by the Leray spectral sequence, H i (X, F ) = H i (C, β∗ F ) for any i. It then follows from the exact sequence Cp → 0 0 → OC → β∗ OX → p∈S

that the genera of X and C are related by pa (C) = pa (X) + |S| . If X1 , . . . , Xv are the connected components of X, the above formula can also be rewritten in this form: pa (Xi ) + 1 − v + |S| pa (C) = (2.8) = pa (Xi ) + 1 − χ(GraphS (C)) . We denote by α:N →C the normalization of C, by C1 , . . . , Ct the irreducible components of C, and by ν the number of nodes of C; we shall also denote by Ni the


89

normalization of Ci . As we have just seen, the genera of N and C are related by pa (C) = pa (N ) + ν = pa (Ni ) + 1 − t + ν . × We now discuss the group Pic(C) = H 1 (C, OC ) of isomorphism classes of line bundles on C. An essentially complete description of this group can be extracted from the exact sequence × × → α∗ ON − → 1 → OC e

C× p → 1,

p∈Csing

where e is defined as follows. For each singular point p of C, let × α−1 (p) = {q, r}. Then for any section f of α∗ ON , the Cp -component of e(f ) is f (q)/f (r). Of course, this definition depends on the choice of an ordering of the points of α−1 (p). The exact cohomology sequence associated to the exact sheaf sequence above is α∗

1 → (C× )s → (C× )t → (C× )ν → Pic(C) −−→ Pic(N ) → 0 , where s is the number of connected components of C, and α∗ is the pullback map. The sequence may be interpreted as saying: i) To give a line bundle L on C, we have to specify its pullback ˜ = α∗ L to N , plus “descent data,” that is, we have to specify L ˜ is the pullback of a section of L. More when a section of L precisely, for any node p of C, we have to give an identification ˜q→ ˜ r , where α−1 (p) = {q, r}. ϕp : L ˜L ˜ ii) When L is trivial, a choice of trivialization identifies each ϕp with a well-defined nonzero complex number. ˜ two choices of descent data define the same iii) Given a trivial L, element of Pic(C) exactly when they are obtained one from another ˜ that is, multiplying the given by a change of trivialization for L, trivialization by a nonzero constant on each component. If L is a line bundle on C, the multidegree of L is the t-tuple (d1 , . . . , dt ), where di = degCi (L) is the degree of the pullback of L to Ni ; the total degree (or simply degree) of L is the sum of the di ’s and is written deg(L). If d = (d1 , . . . , dt ) and e = (e1 , . . . , et ) are multidegrees, we shall write d ≥ e to mean that di ≥ ei for every i, and d > e to mean that d ≥ e and di > ei for at least one i. The Jacobian of C, which we shall denote by Pic0 (C) or J(C), is the subgroup of Pic(C) consisting of line bundles with zero multidegree, so that we have the exact sequence 1 → (C× )ν−t+s → J(C) → J(N ) → 0 .

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Thus the Jacobian of C is a C× -extension of an abelian variety. From this point of view a rational irreducible g-nodal curve represents one extreme in the sense that J(N ) = 0, while J(C) = (C× )g ; at the other extreme, we have curves C satisfying any one of the following equivalent conditions: a) J(C) is compact. b) The sum of the geometric genera of the components of C is g. c) Every component of C is smooth, and the dual graph of C is a tree. We will call such a curve of compact type. In complete analogy with the smooth case, the Riemann–Roch theorem states that, for any line bundle L on C, one has χ(L) = deg(L) + 1 − pa (C) . This formula is a straightforward consequence of the Riemann–Roch theorem for the line bundle α∗ (L) on N and of the exact sequence 0 → L → α∗ α∗ (L) →

Cp → 0 .

p∈Csing

The Serre duality theorem (cf., for instance, Hartshorne [356]) says, in our case, that there exist on C a dualizing sheaf ωC and a “trace” homomorphism (2.9)

ξC : H 1 (C, ωC ) → C

with the property that, for any coherent sheaf F on C, the pairing (2.10)

ξC

H 1 (C, F ) × Hom(F , ωC ) → H 1 (C, ωC ) −−→ C

is a duality. Furthermore, since C is, locally, a complete intersection, it is also the case that ωC is invertible, and, for any coherent sheaf F , the pairing (2.11)

ξC H 0 (C, F ) × Ext1 (F , ωC ) → Ext1 (OC , ωC ) ∼ = H 1 (C, ωC ) −−→ C

is a duality. That Ext1 (OC , ωC ) is isomorphic to H 1 (C, ωC ) is a special case of the following simple observation. Let F be a coherent sheaf on a scheme X. There is a spectral sequence abutting to Ext• (OX , F ) whose E2 term is E2p,q = H p (X, Extq (OX , F )). When F is locally free, the higher sheaf Ext’s Extq (OX , F ), q > 0, all vanish, and hence Extp (OX , F ) ∼ = H p (X, Hom(OX , F )) ∼ = H p (X, F )


91

for all p. It follows in particular that, for a locally free F on a nodal curve C, the duality pairings (2.10) and (2.11) yield a duality between H q (C, F ) and H 1−q (ωC ⊗ F ∨ ) for q = 0, 1, where F ∨ = Hom(F , OC ) is the dual of F . In the case at hand, ωC and ξC can be described quite explicitly. Let {q1 , q1 }, . . . , {qν , qν } be the preimages in N of the nodes p1 , . . . , pν of C. Then ωC is the invertible subsheaf (qi + qi ) ωC ⊂ α∗ ωN defined by the following prescription. A section ϕ of α∗ (ωN ( (qi + qi ))), viewed as a section of ωN ( (qi + qi )), is a section of ωC if and only if (2.12)

Resqi (ϕ) + Resqi (ϕ) = 0 ,

i = 1, . . . , ν .

In particular, when C is smooth, ωC is nothing but the canonical sheaf. The residue theorem is still valid on C in the following version. Let ϕ be a meromorphic section of ωC which is holomorphic at the nodes of C; then the sum of the residues of ϕ vanishes. Taking into account (2.12), this is an immediate consequence of the ordinary residue theorem applied to the form α∗ ϕ on N . To define ξC , choose a divisor D = r1 +· · ·+rh consisting of h distinct smooth points of C, with the property that any component of C contains at least one of the ri ’s. We shall see in a moment that H 1 (C, ωC (D)) vanishes. Assuming this and looking at the exact cohomology sequence · · · → H 0 (C, ωC (D)) → H 0 (C, ωC (D)/ωC ) → H 1 (C, ωC ) → H 1 (C, ωC (D)) , we find that any class ϕ in H 1 (C, ωC ) lifts to a section ϕ of ωC (D)/ωC . Set √ ξC (ϕ) = 2π −1 Resri ϕ . Notice that the right-hand side does not depend on the choice of the lifting ϕ, nor on the choice of D, by the residue theorem. It remains to show that H 1 (C, ωC (D)) is zero. Let β : X → C be the partial normalization of C at the nodes pi1 , . . . , pil . It is a useful general observation that, by the very definition of ωC , one has two exact sequences on C: (2.13)

0 → β∗ ωX → ωC → Cpi1 ⊕ · · · ⊕ Cpil → 0 , 0 → ωC → β∗ ωX (qij + qij ) → Cpi1 ⊕ · · · ⊕ Cpil → 0 .

We apply this to the full normalization α : N → C. Tensoring the first exact sequence with OC (D) and passing to cohomology, we get a surjection H 1 (C, α∗ ωN (D)) w w H 1 (C, ωC (D)) .

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The cohomology group on the left equals H 1 (N, ωN (D)), where D is viewed as a divisor on N . On the other hand, this group vanishes by degree considerations since every component of N meets D. Suppose that C is smooth and view the elements of H 1 (C, ωC ) as represented by (1,1)-forms via the Dolbeault isomorphism. It is a simple exercise, which we leave to the reader, to check that in this special case ξC is just integration over C. It follows in particular that for a smooth C, the duality pairings (2.10) and (2.11) agree with the ones that we have been using so far. As is the case for smooth curves, the duality theorem for nodal curves has several useful consequences. Here we record only one. Lemma (2.14). Let C be a connected nodal curve, and let M be a line bundle on C. Let d be the multidegree of M , and e the one of ωC . If d > e, then H 1 (C, M ) = 0. In fact, duality says that the vanishing of H 1 (C, M ) is equivalent to the one of H 0 (C, ωC M −1 ). On the other hand, by assumption, the degrees of ωC M −1 on the various components of C are all nonpositive, and one among them is strictly negative. Thus, any section of ωC M −1 vanishes identically on one component and is at best constant on the remaining ones. Since C is connected, it must then vanish identically. In analogy with what happens for smooth curves, ampleness of a line bundle on a nodal curve depends only on the multidegree. Lemma (2.15). Let C be a nodal curve, and let M be a line bundle on C. Then M is ample if and only if degD M > 0 for every irreducible component D of C. The proof is an easy exercise using (2.14) and is left to the reader. Alternatively, the lemma can be viewed as the simplest case of Seshadri’s ampleness criterion (Proposition (9.11) in Chapter XI). The partial normalization X of a nodal curve at a set S of nodes carries a distinguished finite set of smooth points, consisting of those points of X which map to nodes in S. It is important, from a technical point of view, to consider not just nodal curves, but also objects of this sort, that is, pairs (C; D) consisting of a nodal curve C plus a finite set D of smooth points of C (often referred to as the marked points). We shall call such a pair a nodal curve with marked points, and we shall often view D as a divisor on C. The invertible sheaf ωC (D) will be referred to as the log-canonical sheaf of (C; D). As is the case for ordinary curves, we can associate a graph to a nodal curve with marked points. This seems a good place to specify what kind of graphs we shall be dealing with and to establish some terminology and notation.


93

Definition (2.16). A graph Γ is the datum of: -

a finite nonempty set V = V (Γ) (the set of vertices); a finite set L = L(Γ) (the set of half-edges); an involution ι of L; a partition of L indexed by V , that is, the assignment to each v ∈ V of a (possibly empty) subset Lv of L such that L = v∈V Lv and Lv ∩ Lw = ∅ if v = w.

A pair of distinct elements of L interchanged by the involution is called an edge of the graph. A fixed point of the involution is called a leg of the graph. The set of edges of Γ is denoted by E(Γ). A dual graph is the datum of a graph together with the assignment of a nonnegative integer weight gv to each vertex v. The genus of a dual graph Γ is defined to be gv + 1 − χ(Γ) . g= v∈V (Γ)

A graph (or a dual graph) endowed with a one-to-one correspondence between a finite set P and the set of its legs will be said to be P -marked, or numbered if P is of the form {1, . . . , n} for some nonnegative integer n. As the reader will have noticed, we allow free half-edges, that is, half-edges which are not part of an edge; these are the legs. It is also important to observe that there may be multiple edges connecting two given vertices, and there may be loops, that is, edges going from one vertex to itself. Moreover, each edge {, } carries two possible orientations corresponding to the ordered pairs (, ) and ( , ). The notion of isomorphism between graphs (or dual graphs, or P marked graphs) is the obvious one. An isomorphism is the datum of a bijection between vertices and a bijection between half-edges respecting the relevant structure. The geometric realization |Γ| of a graph Γ is the one-dimensional CW-complex obtained by realizing each half edge ∈ Lv as a segment, denoted by ||, having a point |v| as one of its endpoints, and then identifying the free endpoint of a segment || with the free endpoint of a segment | | if and only if ι() = ; the resulting segment is then called an edge of |Γ|. The set of edges of |Γ| is denoted by E(|Γ|) and is usually identified with E(Γ). A graph Γ and its realization |Γ| determine each another. A graph Γ is said to be connected if its realization |Γ| is. Notice that χ(Γ) = χ(|Γ|). Clearly, an isomorphism of graphs induces an isomorphism of the respective geometric realizations. Let us go back to curves with marked points. Let C be a nodal curve, and let D be a finite set of smooth points of C. To associate a dual graph Graph(C; D) to (C; D), we proceed essentially as for ordinary curves. As in that case, there is a vertex for each component of the

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normalization of C, and its weight is the genus of the component. The half-edges issuing from a vertex are the points of the corresponding component which map either to a node of C or to a marked point. As for plain curves, the edges of the graph are the pairs of half-edges mapping to the same node of C. The legs are the half-edges coming from the marked points. It follows from formula (2.8) that the genus of the graph Graph(C; D) as defined above is just the genus of C. Most often, when considering curves with marked points, we will want the latter to be ordered. If n is a nonnegative integer, we then define an n-pointed nodal curve to be the datum of a nodal curve C together with an ordered set of n smooth points of C. More generally, if P is a finite set, a P -pointed nodal curve is the datum of a nodal curve C together with an injective map from P to the smooth locus of C; when P = {1, . . . , n}, we recover the notion of n-pointed nodal curve. The dual graph of a P -pointed curve (C; {xp }p∈P ) is naturally P -marked and will be denoted Graph(C; {xp }p∈P ). Figure 1 below illustrates a 3-pointed genus 8 nodal curve and its graph; vertices are represented by small circles bearing in the middle the genus of the normalization of the corresponding component.

Figure 1. A 3-pointed curve (left) and its graph (right) As is the case with unpointed curves, we can also associate a P marked dual graph GraphS (C; {xp }p∈P ) to the datum of a P -pointed nodal curve (C; {xp }p∈P ) and a set S of nodes of C. The vertices of this graph are the connected components of the partial normalization X of C at S, the weight gv of a vertex is the arithmetic genus of the corresponding component of X, the edges correspond to the nodes in S, and the half-edges are the marked points or the points of X mapping to nodes in S. The genus of GraphS (C; {xp }p∈P ) is equal to the genus of C. It is possible to classify the nodes of a connected P -pointed nodal curve (C; {xp }p∈P ) of genus g according to the combinatorics of the partial normalization of (C; {xp }p∈P ) at the node. Let q be a singular point of C, and let e be the corresponding edge of the dual graph Γ = Graph(C; {xp }p∈P ). We shall say that q is a nonseparating node or


95

that e is a nondisconnecting edge of Γ if the partial normalization of C at q is connected, i.e., if the graph obtained from Γ by removing e is connected. In this case the partial normalization of (C; {xp }p∈P ) at q is a connected nodal curve of genus g − 1 with |P | + 2 marked points, all but the two coming from the normalization process indexed by P . The other possibility is that the normalization of C at q has two connected components C1 and C2 of genera a and b adding to g. In this case the set P is partitioned in two complementary subsets A and B indexing, respectively, marked points on C1 and marked points of C2 . We shall then say that q is a separating node (or that e is a disconnecting edge) of type P, where P = {(a, A), (b, B)}. We shall sometimes refer to such a P as a bipartition of (g, P ). In practice, we shall usually say that q is a separating node of type (a, A) (or, equivalently, a separating node of type (b, B) = (g − a, P A)). Naturally, we can also speak of families of P -pointed curves. Formally, a family of P -pointed nodal curves is the datum of a family α : X → S of nodal curves, plus disjoint sections σp : S → X, p ∈ P , which do not meet the singular locus of any one of the fibers of α. If β : X → S , {σp : p ∈ P } is another family of P -pointed nodal curves, a morphism from the first family to the second is a cartesian square X

H

α

u S

h

w X α u w S

such that H ◦ σp = σp ◦ h for every p ∈ P . The composition of morphisms is the composition of cartesian squares. When dealing with families of P -pointed curves, we shall sometimes use the symbols we use to denote sections also to denote the corresponding divisors. For instance, we shall often write O( σp ) instead of O( σp (S)). The notion of dualizing sheaf, in a sense, provides a generalization of what, for smooth curves, is the sheaf of holomorphic 1-forms. There is, however, another more direct generalization, given by the notion of K¨ ahler differentials. Recall that, for any morphism of schemes, or of analytic spaces, ψ:X→Y , the sheaf of relative Kähler differentials Ω1X/Y (or Ω1ψ ) is defined to be the pullback to X, via the diagonal map, of the ideal sheaf I of the diagonal in X ×Y X. Denoting by π1 and π2 the projections of X ×Y X onto the two factors, the differentiation operator d : OX → Ω1X/Y

96 defined by

10. Nodal curves

d(h) = π1∗ (h) − π2∗ (h)

(mod. I 2 )

is OY -linear and satisfies Leibniz’ rule. The sheaf Ω1X of K¨ ahler differentials on X is nothing but the sheaf Ω1X/Y for Y a point. If x, y are points of X and Y such that ψ(x) = y, the stalk of Ω1X/Y over x is the module of differentials ΩOx /Oy . More generally, for any morphism Y →Y, Ω1X /Y p∗ Ω1X/Y , where X = X ×Y Y , and p : X → X is the projection to the first factor. We refer to Matsmura’s book [503] or to Chapter 6 of Qing Liu’s book [480] for the properties of modules of differentials. Here we wish only to record, for future reference, just a few of their consequences. The first is that to any commutative diagram X

ψ N N Q

wY N

S of schemes over S there is associated a canonical sheaf homomorphism ψ∗ Ω1Y /S → Ω1X/S fitting into an exact sequence ψ ∗ Ω1Y /S → Ω1X/S → Ω1X/Y → 0 . In particular, when S is just a point, this reduces to (2.17)

ψ ∗ Ω1Y → Ω1X → Ω1X/Y → 0 .

A second consequence is as follows. Suppose Z is a closed subscheme (or analytic subspace) of X, and J is its sheaf of ideals. Then Ω1Z/Y = Ω1X/Y /M , where M stands for the sub-OX -module generated by JΩ1X/Y and by the differentials of sections of J. In other words, there is a canonical exact sequence d

→ Ω1X/Y ⊗OX OZ → Ω1Z/Y → 0. J/J 2 − As an example of application, suppose that (2.18)

ϕ:C→Y


97

is a family of nodal curves, locally of the form xy = f ; it then follows that Ω1C is locally generated by dx, dy, and ϕ∗ Ω1Y , subject to the only relation ydx + xdy = df , while Ω1C/Y is generated by dx and dy, modulo the relation ydx + xdy = 0 . Like the sheaf of K¨ ahler differentials, the dualizing sheaf also has a relative counterpart ωC/Y (or ωϕ ) for a family of nodal curves (2.18). In fact, any l.c.i. morphism h : X → Y of schemes or analytic spaces admits a relative dualizing sheaf, an invertible OX -module denoted ωh or ωX/Y . A convenient reference for the construction and elementary properties of the relative dualizing sheaf is Section 6.4 of [480]. Here we just want to recall what ωh looks like locally. Let U be an open subset of X over which h factors into the composition of a regular embedding ι : U → Z and a smooth morphism π : Z → Y . The sheaf of relative K¨ ahler differentials Ω1π is locally free of rank equal to the fiber dimension r of π, while the normal sheaf NU/Z is locally free of rank equal to the codimension k of U in Z. Then there is a canonical isomorphism (2.19)

ωh |U ∧r ι∗ (Ω1π ) ⊗ ∧k NU/Z .

For a family of nodal curves ϕ : C → Y , this means the following. Near a smooth point of a fiber, ϕ is smooth, so (2.19) just says that ωC/Y is the same as Ω1C/Y . Near a singular point p of a fiber, instead, C is locally of the form xy = f . Using our previous notation, a neighborhood U of p in C is viewed as the closed set F = 0 in Z = C2 × Y , where F = xy − f , and the normal bundle of U in Z is nothing but OU ((F )). Hence a local generator of ωϕ is the class of F −1 dx∧dy modulo F , where F = xy − f . The identification between ωC/Y and Ω1C/Y away from the singularities of the fibers extends to a homomorphism ρ : Ω1C/Y → ωC/Y , given locally near p by ∧ dF modulo F, ρ(α) = class of F −1 α where α is any relative differential on C2 × Y → Y restricting to α; keep in mind that here the exterior differentiation symbol stands for relative differentiation /Y , so that for instance dF = xdy + ydx. In particular, ρ(dx) = x

dx ∧ dy , F

ρ(dy) = −y

dx ∧ dy . F

In a sense, then, one may interpret the standard local generator F −1 dx∧dy dy dy dx as dx x or − y , and say that ωC/Y is locally generated by x and y , subject to the relation dx dy + = 0. x y

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The formation of the relative dualizing sheaf is compatible with base change. In particular, taking into account the local description above, the restriction of ωϕ to a fiber ϕ−1 (y) is just the dualizing sheaf ωϕ−1 (y) , as described in the previous section. Look at the exact sequence (2.20)

ρ

→ ωC/Y → coker ρ → 0. 0 → ker ρ → Ω1C/Y −

Clearly, both ker ρ and coker ρ are concentrated along the locus of those points of C which are singular in their fiber. Write ρ(Ω1C/Y ) = IωC/Y , where I is an ideal in OC , and denote by S the corresponding subspace of C. With this notation, coker ρ = ωC/Y ⊗ OS . In a local representation xy = f of C, the image of ρ is generated by x F −1 dx∧dy and y F −1 dx∧dy, so I = IS is locally generated by x and y. It is not as straightforward to describe the kernel of ρ. Look at a point where C is of the form xy = f . Without being too pedantic about it, we shall sketch what the possibilities are for ker ρ. Let us start with the easy case where Y is a smooth curve and f = t, where t is a local parameter on Y . An element αdx + βdy belongs to the kernel of ρ if and only if αx − βy = 0. Since x and y are local parameters ˜ and that on the smooth surface C, it follows that α = y α ˜ , β = xβ, ˜ ˜ xy(˜ α − β) = t(˜ α − β) = 0. As C is smooth, it follows that α ˜ = β˜ and 1 therefore that αdx + βdy is zero in ΩC/Y . The above argument suggests how to produce nonzero elements in ker ρ whenever there exists a nonzero g ∈ OY such that f g = 0; one such element is, for instance, hgydx, where h is any function on C. This situation may present itself in two cases only, when f is a zero divisor in OY or when f is zero. This last case corresponds to the situation where the family is locally a product near the singularity under examination (so that, in particular, the generic curve of the family is singular). When Y is a single point, i.e., when we are dealing with a single nodal curve, the kernel of ρ is the one-dimensional complex vector space generated by the class of xdy = −ydx. As a consequence of this discussion, we may assert that, when the general fiber of ϕ is smooth and Y is reduced and irreducible, ρ is injective. We conclude this section with a quick glimpse at a very particular case of relative duality. We recall that a key ingredient of Serre duality on a nodal curve is the trace homomorphism (2.9). When dealing with a family ϕ : C → Y of nodal curves, this globalizes to a sheaf homomorphism ξ : R1 ϕ∗ ωC/Y → OY . When the fibers of ϕ are connected, ξ is an isomorphism. Suppose now that L is a line bundle on C and suppose further that both ϕ∗ L and R1 ϕ∗ L are vector bundles on Y . Then relative duality asserts that the pairing

ξ → OY ϕ∗ L ⊗ R1 ϕ∗ ωC/Y ⊗ L−1 → R1 ϕ∗ ωC/Y −

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99

identifies R1 ϕ∗ ωC/Y ⊗ L−1 with the dual of ϕ∗ L. Exercise (2.21). Let C ⊂ Pr be a nodal curve in projective space, and let I be its ideal sheaf. Show that the sequence d

0 → I/I 2 − → Ω1Pr ⊗ OC → Ω1C → 0 is exact. 3. Stable curves. As we explained in the introduction to this chapter, singular curves of some sort are necessary if we want to take “limits” of families of smooth curves over noncomplete schemes. In the next two sections we shall prove that nodal curves suffice. What is immediately clear is that, on the other hand, nodal curves are “too many.” Consider in fact the following setup. Let f : X → S be a family of smooth curves over, say, a smooth curve S. We blow up the surface X at a point p. The result is a new family which is identical to the old one except for the fiber at f (p), which is now singular. In short, limits are not unique if arbitrary nodal curves are allowed. Put otherwise, “moduli spaces” constructed using unrestricted nodal curves are of necessity highly nonseparated. To restore uniqueness of the limit, one must restrict the class of nodal curves used. Let (C; D) be a connected nodal curve with n marked points. One says that (C; D) is stable if it has a finite automorphism group. In the same way, when P is a finite set, one defines the notion of stable P -pointed curve. Attached to a P -pointed stable curve (C, D) is the log-canonical sheaf ωC (D). This sheaf is intrinsic in the sense that, given an isomorphism ϕ : (C, D) → (C , D ) between P -pointed stable nodal curves, there is a canonical isomorphism between ωC (D ) and ϕ∗ ωC (D). There is a simple combinatorial characterization of stability which is often easier to work with. To state it, let Γ be a connected genus g graph with n legs, and, as customary, for any vertex v, let gv and lv be the weight (the “genus”) of v and the number of half-edges emanating from v. One says that Γ is a stable graph if 2gv − 2 + lv > 0 for every vertex v. Lemma (3.1). The following are equivalent: a) (C; D) is stable; b) the graph of (C; D) is stable; c) for any component Y of the normalization of C, the pullback of the log-canonical sheaf ωC (D) to Y has strictly positive degree. The proof is straightforward. There is a natural homomorphim from Aut(C; D) to Aut(Γ), whose kernel we denote by K. Since Aut(Γ) is

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finite, the finiteness of Aut(C; D) is equivalent to the one of K. On the other hand, K is the direct product of the groups Kv , where v ranges over all vertices of Γ, and Kv stands for the automorphism group, as an lv -pointed curve, of the component of the normalization of C corresponding to v. Now, if (X; E) is a smooth curve of genus g with n marked points, the group Aut(X; E) is finite unless g = 0 and n ≤ 2, or g = 1 and n = 0, i.e., unless 2g − 2 + n ≤ 0. Applying this remark to all the components of the normalization of C proves the equivalence of a) and b). As for condition c), it is just a rephrasing of b). In fact, if Y is the component of the normalization of C corresponding to vertex v of the graph of (C; D), then the degree of the pullback of ωC (D) to Y is exactly equal to 2gv − 2 + lv . An immediate consequence of the lemma is that n-pointed stable curves of genus g exist if and only if g > 1, or g = 1 and n ≥ 1, or else g = 0 and n ≥ 3. As we explained in Section 2, the nodes of a stable P -pointed curve of genus g come in various flavors, nonseparating or separating of type P, for some bipartition P = {(a, A), (b, B)} of (g, P ). However, not all bipartitions can occur, but only the “stable” ones, that is, those allowed by the stability condition. To see what these are, recall that the partial normalization at a separating node produces two nodal curves, one of genus a with |A| + 1 marked points and one of genus b with |B| + 1 marked points, which must both be stable. Thus the bipartitions which are not “stable,” and hence cannot occur, are those such that a = 0 and |A| < 2, or b = 0 and |B| < 2. A useful generalization of the notion of stability is the one of semistability. We shall say that a connected graph Γ with n legs as above is semistable if 2gv − 2 + lv ≥ 0 for every vertex v. Likewise, a nodal curve with n marked points (C; D) is said to be semistable if its graph is. In practice, a curve is semistable if and only if at least one of the following conditions is satisfied: either it is irreducible of genus one with no marked points, or else every one of its smooth rational components contains at least two points which are either singular or marked. The following characterization of semistability needs no proof. Lemma (3.2). The following are equivalent: a) (C; D) is semistable; b) for any component Y of the normalization of C, the pullback of ωC (D) to Y has nonnegative degree. To be more precise, the pullback of ωC (D) to any component of the normalization of a semistable C which is smooth rational and contains just two marked points is the trivial line bundle, while the pullback to

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any other component has strictly positive degree, with one exception: ωC (D) is trivial if n = 0 and C is smooth of genus 1. Remark (3.3). Let (C; D) be a nodal curve with n marked points, and let ν : N → C be the partial normalization of C at t nodes. Let E be the set of those points of N which map either to marked points of C or to nodes where the normalization has taken place. Then (N ; E) is a nodal curve with (n + 2t) marked points. It is clear from the description of the dualizing sheaf of a nodal curve that ωN (E) = ν ∗ (ωC (D)) . It is then an immediate consequence of (3.1) and (3.2) that, when (C; D) is connected, it is stable (resp., semistable) if and only if every connected component of (N ; E) is. There is an obvious extension of the notion of stability (or semistability) to families. A family of n-pointed nodal curves is said to be a family of stable (resp., semistable) P -pointed curves if every one of its fibers is stable (resp., semistable). A useful consequence of Lemma (3.1) is that, in families of nodal curves, stability of fibers is an open condition. Incidentally, this is true also for semistability, but this will be proved later (cf. Corollary (6.6)). Lemma (3.4). Let f : X → S, σp : S → X, p ∈ P , be a family of P pointed nodal curves. Then the set of s ∈ S such that (Xs ; {σp (s)}p∈P ) is stable is Zariski open in S. In proving the lemma we may assume, without loss of generality, that the fibers of f are connected. The proof is immediate. It suffices to notice that, by Lemma (3.1), stability is equivalent to the ampleness of the log-canonical sheaf, and that ampleness itself is an open condition. It is possible to generalize to the stable case the notion of hyperelliptic curve. We would like to call hyperelliptic those stable curves of genus g > 1 which are “limits” of smooth hyperelliptic ones, that is, those stable curves which appear as fibers of families of stable curves over irreducible bases whose general fibers are smooth hyperelliptic. As this definition is a bit unwieldy and indirect, one of our first tasks will be to replace it with an equivalent one which is more intrinsic. This is modeled on the standard characterization of smooth hyperelliptic curves as double covers of P1 . In the stable case the projective line is replaced by a genus zero nodal curve, i.e., a nodal curve whose graph is a tree and whose irreducible components are all copies of P1 . We shall say that a genus g > 1 stable curve C is a hyperelliptic stable curve if there is an order two automorphism σ of C with isolated fixed points such that the quotient P = C/σ is a genus zero nodal curve. The requirement that the fixed points of σ be isolated insures that the quotient morphism

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C → P has degree two above each component of P . The automorphism σ is the analogue, in the stable case, of the hyperelliptic involution, and our main goal here is to show that, as in the smooth case, it is unique. A proof that hyperelliptic stable curves, as defined above, are indeed the “limits” of ordinary hyperelliptic curves will be given in Lemmas (6.14) and (6.15) of Chapter XI (see also Exercise (4.14) in the same chapter). Before we prove the uniqueness of the hyperelliptic involution, let us remark that the nodes of C can be classified into different types, depending on their behavior under σ. A node p of C can be fixed under σ, or not. In the first case, σ may or may not interchange the branches of C at p. If it does, then p maps to a smooth point of P and is not a separating node. In fact, if it were separating, C would consist of two copies of P joined at p, contradicting the stability of C. We shall say that a node of this kind is a node of type η0 . If instead σ does not interchange the two branches of C at p, then p maps to a node q of P , and the projection from each branch of C at p to the corresponding branch of P at q has degree two. Furthermore, since p is the only point mapping to q, and q is separating, p is also separating. Such a node divides C in two pieces of genera i and j ≥ i, with i + j = g; in the terminology of Section 2, p is a separating node of type (i, ∅). We shall also say that p is a node of type δi . There remain the nodes which are not fixed under σ. If p is such a node, it maps to a node q of P , and the only other point of C mapping to q is σ(p). Normalizing C at {p, σ(p)} breaks it into two curves C1 and C2 . Observe that C1 and C2 must be connected, for otherwise they would be the union of two copies of a subcurve of P , contradicting the stability of C. In particular, p and σ(p) are nonseparating. The genera i and j of C1 and C2 add to g − 1 and are strictly positive, since otherwise C would not be stable. If h is the minimum between i and j, we shall say that {p, σ(p)} is a pair of nodes of type ηi . Summing up, while the classification of separating nodes is the same for hyperelliptic curves as for general ones, in the hyperelliptic case the nonseparating nodes fall into several finer subclasses. We now prove the uniqueness of the hyperelliptic involution. Lemma (3.5). Let C be a stable curve of genus g > 1. There exists at most one order two automorphism σ of C which has isolated fixed points and is such that C/σ is a genus zero nodal curve. Proof. Suppose that C admits an automorphism σ as in the statement of the lemma. Set P = C/σ, and let π : C → P be the quotient morphism. Let τ be another order two automorphism with isolated fixed points and with C/τ a genus zero nodal curve. We shall show that τ = σ, arguing by induction on the number of irreducible components of P . We first deal with the initial case of the induction. We thus assume that P is a P1 . If C is irreducible, C/τ is also irreducible and hence

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isomorphic to a projective line. It then follows from the classification of nodes that those of C are fixed under τ and that their branches get interchanged by τ . We wish to show that the same conclusion holds if C is reducible. In this case, C consists of two copies of P joined at three or more points. Then τ interchanges the two components of C, since otherwise C/τ would have positive genus. It follows that C/τ is irreducible, and the same argument as above goes through. Now let N be the normalization of C, and let {p1 , p1 }, . . . , {pn , pn } be the pairs of points or N mapping to the n nodes of C. The automorphism τ lifts to N and interchanges pi and pi for each i. We may distinguish several cases: i) N is irreducible of genus strictly larger than 1. Then σ = τ by the uniqueness of the hyperelliptic involution in the smooth case. ii) N is irreducible of genus 1. Then n ≥ 1, since g ≥ 2. In this case there is a unique automorphism of N interchanging p1 and p1 (the symmetry about p1 followed by translation of p1 to p1 ). This shows that σ = τ . iii) N is irreducible of genus 0. Then n ≥ 2. There is a unique automorphism of N interchanging p1 with p1 and p2 with p2 (z → a/z if we normalize things so that p1 = 0, p1 = ∞, p2 = 1, and p2 = a). Thus σ = τ . iv) N is reducible. As we observed above, N is the disjoint union of two copies N1 and N2 of P , and C is obtained by identifying points pi ∈ N1 and pi ∈ N2 for i = 1, . . . , n, n ≥ 3. Clearly, there is a unique automorphism of N interchanging pi with pi for i = 1, . . . , 3. Hence σ = τ also in this last case. There is a variant of the initial case of the induction that we will also need. Let (E, e) be a stable 1-pointed curve of genus 1. Then there is a unique order two automorphism of (E, e). If E is smooth, this is the symmetry about e. If E is obtained from P1 by identifying 0 and ∞ and if e = 1, then the automorphism comes from z → 1/z. Now we turn to the inductive step. Let P be an irreducible component of P , and set C = π −1 (P ). If C is connected and has strictly positive genus, it must be carried to itself by τ , since otherwise C/τ would contain a subcurve of strictly positive genus. Now suppose that P is a leaf of P , i.e., that it meets the union P of the other components of P at a single point. It follows, in particular, that P is connected. Set C = π −1 (P ). As we observed while classifying singularities of hyperelliptic stable curves, C and C are connected and moreover have strictly positive genus. It follows that τ carries C to itself and hence also C to itself. If C and C meet at a single point, this is fixed under τ . In this case we set E = C , E = C . If instead C and C meet at a pair {p1 , p2 } of nonseparating nodes of type ηi , then we claim that τ interchanges p1 and p2 . In fact, the only other possibility is

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that p1 and p2 are both fixed under τ . But, as they are nonseparating nodes, τ would have to interchange the two branches of both, which is not the case since τ maps C and C to themselves. In this case we let E and E be the curves obtained from C and C , respectively, by identifying p1 and p2 . It is clear that σ descends to automorphisms σ and σ of E and E . By what we have said, τ also descends to automorphisms τ and τ of E and E such that E /τ ∼ = P1 and E /τ is a rational nodal curve. By the initial case of the induction and its variant, and by induction hypothesis, τ = σ and τ = σ . Thus τ = σ, as claimed. Exercise (3.6). Let C be a nodal curve, and let γ be an involution on C with isolated fixed points. Let π : C → P be the quotient map. Show that every point of P has an open neighborhood U , in the ordinary topology, such that one of the following occurs: i) π −1 (U ) is a disjoint union U1 U2 of open sets each of which maps isomorphically to U ; ii) U is isomorphic to a neighborhood of the origin in the complex z-plane, π−1 (U ) is isomorphic to a neighborhood of the origin in the complex t-plane, and π is locally given by t = z 2 ; iii) U is isomorphic to a neighborhood of the origin in the complex t-plane, π −1 (U ) is isomorphic to a neighborhood of the origin in xy = 0, and π is locally given by t = x + y; iv) U is isomorphic to a neighborhood of the origin in zw = 0, π −1 (U ) is isomorphic to a neighborhood of the origin in xy = 0, and π is locally given by z = x2 , w = y 2 . Q.E.D. 4. Stable reduction. In the introduction to this chapter and in the previous section, we observed that the most naive approach to compactifying the moduli space of smooth genus g curves, that is, throwing in the isomorphism classes of all possible singular curves of arithmetic genus g, fails because it unavoidably produces highly nonseparated spaces. As we announced, one remedy to this is to strongly limit the class of singular curves one considers, allowing only stable curves. The main technical tools to prove that this does indeed work will be given in this section and in the next one. Before proceeding, we give a very brief overview of those facts concerning the moduli space of stable curves which can serve as a motivation for what we will be doing. We shall limit ourselves to the case of unpointed curves. First of all, the moduli space M g of stable genus g curves is, set-theoretically, just the set of isomorphism classes of stable curves of genus g. It will be proved in Chapters XII and XIV

§4 Stable reduction

105

that M g has a natural structure of normal algebraic variety of dimension 3g − 3. Given a stable curve C of genus g, we shall denote by [C] ∈ M g the isomorphism class of C. It is of course natural to ask whether M g is the parameter space of a family π : C −→ M g of stable curves such that [π−1 (x)] = x for any x ∈ M g . It turns out that such a family exists only on the Zariski open subset of M g corresponding to automorphism-free stable curves and that the presence of automorphisms prevents this family from existing over the whole of M g . On the other hand, what turns out to be true (cf. Theorem (2.9) of Chapter XII) is that there exist a normal algebraic variety Z, a finite morphism (4.1)

m : Z −→ M g ,

and a family of stable curves of genus g (4.2)

η : X −→ Z

such that, for every z ∈ Z, [η −1 (z)] = m(z). One of the main results of the first five chapters of this volume, to be proved in Chapter XIV, is that M g is a projective variety. A basic step in the proof will be to show that M g is complete. This will be checked by using the valuative criterion for completeness. Informally speaking, the valuative criterion for completeness asserts that an algebraic variety X is complete whenever any holomorphic map from a punctured disc to X which is meromorphic at the origin can be extended across the puncture. Clearly this criterion can be somewhat softened by only requiring that the extension be possible after a base change z = ζ k on the punctured disc. In applying the valuative criterion of completeness to the moduli space M g , we are thus confronted with maps from a punctured disc ˙ = {z ∈ C : 0 < |z| < ε} to M g which are meromorphic at the origin. Δ In view of the existence of the finite map (4.1), after a harmless base ˙ to Z. Then change, we can lift any such map to a map f from Δ pulling back via f the family of stable curves (4.2) produces a family of stable curves (4.3)

˙ π˙ : C˙ → Δ

˙ into moduli From this point of view, extending the original map from Δ space over the puncture corresponds to filling in the family (4.3), possibly after a base change, with a stable curve as central fiber. We notice first

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that we can fill in the family with a possibly very nasty central fiber. This is a consequence of the meromorphicity of the map ˙ →Z , f :Δ as we now explain. We begin by completing Z to a normal variety Z, so that f extends to a morphism f : Δ = {z ∈ C| |z| < ε} −→ Z . We now look at the family (4.2) over Z, take a completion X of X , and let X be the closure of X in X × Z. Clearly the natural projection η from X to Z extends the family (4.2). But then a completion of the family (4.3) is π : C = X ×Z Δ −→ Δ . From this point on we may forget how the above family was constructed and deal exclusively with the abstract situation of a proper morphism (4.4)

π : C −→ Δ

having the property of being a family of stable curves away from the origin of Δ. We wish to show that after a base change on Δ of the form z = ζ k , the central fiber of π can be replaced so as to get a family of stable curves over Δ. This process goes under the name of stable reduction. We begin by proving the slightly weaker statement that the central fiber can be replaced, after a finite base change, so as to get a family of nodal curves. We shall first deal with the case where the ˙ are smooth. Thus C is a surface whose singularities fibers of π over Δ are concentrated along the central fiber. By successive blow-ups of points in the central fiber and normalization, we may then assume that C is smooth and that the central fiber of C is a divisor with normal crossings. It is important to realize that this does not mean that the central fiber is a nodal curve, since it can very well be nonreduced. In fact, to say that π −1 (0) is a divisor with normal crossing means that, given any point p on π −1 (0), one can choose local analytic coordinates x, y on C centered at p in such a way that, in a neighborhood of p, the map π is given by either z = xc or z = xa y b for suitable positive integers a, b, and c. By compactness we may cover the central fiber with a finite number of coordinate patches for C in which π is given as above. We then


107

perform a base change z = ζ k , where k is chosen to be a multiple of c and of ab, for all these patches. So we obtain a new family π : C → Δ . Finally we normalize C and get yet another family π : C −→ Δ . We claim that the central fiber at π is now a nodal curve. To show this, we work on each of the patches of C separately. Suppose first that we are on a patch where π looks like z = xc . Then, since k is a multiple of c, we can write k = ch, so that C is given locally by the equation 0 = xc − ζ ch =

(x − ωζ h ) .

ωc =1

Normalizing breaks up C into the disjoint union of the smooth branches ωζ h = x. The other case is more interesting. In this case, π looks like z = xa y b , and we can write k = rsuv, where r and s are relatively prime, a = ru, and b = su. Hence, locally, the equation of C is 0 = xa y b − ζ k =

(xr y s − ωζ vrs ) .

ω u =1

We can think of the normalization of C to happen, locally, in two stages. The first one consists in breaking up C into the disjoint union of the branches ωζ vrs = xr y s . The second one consists in normalizing each branch separately. As one √ sees replacing ζ with ζ = vrs ωζ, each of the branches of C is a copy of ζ vrs = xr y s . We claim that the normalization of this branch is the surface in C3 with equation (4.5)

ζ v = αβ

and that the normalization map is given by (4.6)

x = αs

,

y = βr .

Checking this involves verifying two properties. The first is that equation (4.5) defines a normal surface. This is certainly so by Proposition (5.4) of Chapter II of Volume 1, since the only singular point of (4.5) is at the origin, and the surface in question is a hypersurface in affine 3-space. The second point to be checked is that the map (4.6) is birational, that is, generically one-to-one and onto. To see that it is generically

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one-to-one, assume that (α, β, ζ) and (α , β , ζ ) map to the same point. This translates into ζ = ζ , α s = α , β r = β . s

r

We can then write α = σα, where σ is an sth root of unity and β = τ β, where τ is an rth root of unity. Since αβ = α β , unless α or β vanishes, we get that στ = 1; since r and s are relatively prime, this implies that σ = τ = 1. To check the surjectivity of the map (4.6), all we have to do is the following. Given complex numbers x, y, ζ satisfying the equation xr y s = ζ vrs , we must find α and β such that αβ = ζ v and x = αs , y = β r . We can certainly find α and β satisfying the last two conditions. But then (αβ)rs = xr y s = ζ vrs , so that αβ = ξζ v , where ξ rs = 1. Since r and s are relatively prime, we may write 1 = mr + ns and replace the original α and β with αξ −mr and βξ −ns . These obviously solve our problem. We have thus proved that the surface C is locally either of the form ζ h = x or of the form ζ v = αβ, showing that C → Δ is a family of nodal curves. The preceding argument has been carried out under the simplifying hypothesis that the general fiber of π is smooth. Let us now remove this unnecessary assumption. Thus π:C→Δ is a morphism with one-dimensional fibers such that all fibers other than the central one π −1 (0) are nodal curves. Denote by Z ⊂ C the closure of the locus of nodes in the fibers of π −1 (z) for z = 0. Possibly after shrinking Δ we may assume that Z has no zero-dimensional components. Possibly after a base change of the form z = ζ k we may also assume that Z is a union of sections Z1 , . . . , Zδ . Now normalize the surface C. The resulting surface C is the disjoint union of components C1 , . . . , Cc and is fibered over Δ via a morphism π : C → Δ. Possibly after a further base change, we may assume that the preimage of each Zi in C is the disjoint union of two sections Xi and Yi : these may either lie in the same component of C or in different ones. Since all fibers of π , except the central one, are smooth, by what has already been proved, after a finite base change one may transform the family π : C → Δ into a family of nodal curves by replacing the central fiber. Let π : C → Δ be the resulting family. By abuse of language we continue to denote by Xi and Yi the sections of π corresponding to the sections of π with the same names. The idea now is to glue Xi to Yi for each i and thus to get a family of nodal curves over Δ, which, by construction, will be obtained from


109

π : C → Δ, away from the central fiber, via a base change. There are however a couple of small details to be taken care of. In fact, it could very well happen that one or more of the sections Xi , Yi pass through a node of the central fiber of π or that two of these sections meet on the central fiber; in either case the gluing procedure produces a central fiber which is not a nodal curve. We shall deal with the two problems separately. We will show below that we can assume C to be smooth. This will solve the first problem. In fact, if a section of π went through a singular point of the central fiber, it would have intersection number at least equal to two with this same fiber and hence with any other fiber; since the general fiber is smooth, this is absurd. Once we can assume C to be smooth, the second problem can be immediately solved as well by repeatedly blowing up at the intersection points of sections. Our task then is to show that the singularities of C , if any, can be resolved by successive blow-ups without destroying the property of the central fiber being reduced. To see that this is a point to worry about, just notice that blowing up the origin in a family which locally looks like xy = z produces a fiber over z = 0 which has a multiple component, namely the exceptional curve counted twice. By what we have shown, any singularity that C may have is of the form (4.7)

xy = z n+1

or, as one says, is an An singularity. To blow up such a singularity, one may proceed as follows. Let U be a neighborhood of the origin in the space of the variables x, y, and let ξ, η, ζ be homogeneous coordinates in P2 ; denote by X the subvariety of U defined by (4.7) and by f its projection to the z factor. The blow-up of U at the origin is the subvariety of U × P2 defined by the equations xη = yξ , xζ = zξ , yζ = zη . To get the total transform of X, one must add to these equation (4.7). ˆ is nothing but The blow-up of X at the origin, which we denote by X, the proper transform of X. To see what it looks like, we shall examine it separately in the open subsets {ξ = 0}, {η = 0}, and {ζ = 0} of U × P2 . Let us begin with {ζ = 0}. On this open set one can take as local coordinates z, ξ/ζ, η/ζ and write x = zξ/ζ

,

y = zη/ζ .

In these coordinates, the equation of the total transform of X then becomes ξη = z n+1 , z2 ζζ

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ˆ is so that the equation of X (4.8)

ξη = z n−1 . ζζ

ˆ ∩ {ζ = 0} is smooth except for an An−2 singularity at In other words, X x = y = z = ξ = η = 0 when n > 2 and smooth when n = 1, 2. We next look at what takes place inside {ξ = 0}. On this open set one can take as local coordinates x, η/ξ, ζ/ξ, and write (4.9)

y = xη/ξ

,

z = xζ/ξ .

In these coordinates the equation of the total transform of X is x2 η/ξ = xn+1 (ζ/ξ)n+1 , ˆ is so the equation of X η/ξ = xn−1 (ζ/ξ)n+1 . ˆ ∩ {ξ = 0} is smooth. Interchanging x with y In particular, we find that X and ξ with η shows that the same happens in {η = 0}. If we denote by ˆ → X with f , formulas (4.8) and fˆ the composition of the natural map X −1 ˆ (4.9) also show that the fiber f (0) is reduced with nodes as its only singularities. More exactly, for n > 1, it consists of the four components a) b) c)

x=z=ξ=ζ=0 , y=z=η=ζ=0 , x=y=z=ξ=0 ,

d)

x=y=z=η=0 .

The union of components a) and b) is the proper transform of f −1 (0). Components c) and d) are projective lines whose union is the exceptional ˆ → X. When n = 1, the picture is slightly different: fˆ−1 (0) divisor of X consists of the two components a) and b), plus the exceptional divisor ˆ → X, which is the smooth conic of X x = y = z = ξη − ζ 2 = 0 . When n = 1, 2, one blow-up resolves the singularity of X. In general, one ˜ the resulting must blow up n+1 times to desingularize X; denote by X 2 ˜ ˜ manifold and by f the composition of f with X → X. In any case the fiber f˜−1 (0) is reduced, has only nodes as singularities, and consists of the two branches of the normalization of f −1 (0), joined by a chain E1 + · · · + En of n smooth rational curves


111

Figure 2. ˜ → X is precisely E1 + · · · + En . The exceptional divisor of X Returning to the family π : C → Δ , the analysis of the minimal resolution of an An singularity we just completed shows that we can assume C to be smooth, and, as we already remarked, this insures that, possibly after further blow-ups at smooth points of the central fiber, gluing each section Xi to the corresponding section Yi produces a family of nodal curves (4.10)

π : C → Δ.

To sum up, what we have done so far is to replace the central fiber of the family (4.4) so as to get a family of nodal curves, possibly after a finite base change. What remains to be done is to show that one can blow down suitable smooth rational components in the central fiber so as to end up with a family of stable curves. This is a special case of a general procedure, which goes under the name of “passing to the stable model” and will be described in detail in Section 6. In the case at hand, we give a direct argument. Denote the central fiber of π : C → Δ by C; if it fails to be stable, this is because it contains smooth rational components which meet the rest of C in no more than two points. Let Γ be a connected component of the union of all these rational curves. Of necessity Γ is a chain of smooth rational curves which may either be attached to the rest of C at both ends, as in Fig. 2 above, or else be attached to the rest of C at one end only:

Figure 3. We shall refer to such a Γ as an exceptional chain.

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10. Nodal curves

Denote by C the nodal curve obtained by contracting all exceptional chains to points. By construction C is stable. Moreover the genus of C and the genus of C are equal. This follows immediately from formula (2.8). Now pick sections D1 , . . . , Dd of π such that D1 + · · · + Dd meets all components of C save those lying in exceptional chains. We claim that, possibly after shrinking Δ, the divisor D1 + · · · + Dd meets every component of any fiber different from the central one. To prove this, it is convenient to recall that C was obtained from C by gluing together sections Xi , Yi for i = 1, . . . , δ and to lift D1 , . . . , Dd to sections E1 , . . . , Ed of C . If our claim is false, there must be an entire component D of C which does not contain any one of the sections E1 , . . . , Ed . By construction this means that the central fiber of D → Δ is a connected piece Γ0 of an exceptional chain of C; thus the arithmetic genus pa (Γ0 ) vanishes. Since the arithmetic genus is constant in flat families, the fibers of D → Δ other than the central one are smooth and rational. Since the fibers of C → Δ, except the central one, are stable, there must be at least three among the section Xi , Yi which are contained in D. But then Γ0 cannot be a piece of an exceptional chain. We now denote by L the line bundle O(D 1 + · · · + Dd ) on C. We just proved that, if z = 0, the restriction Lπ−1 (z) has positive degree

on every component of π −1 (z) and hence is ample. On the other hand, denoting by λ the natural map from C to C , the restriction LC is of the form λ∗ M for a suitable line bundle M ; by construction, M is ample on C . For any large enough k and any z = 0, Lk π−1 (z) is very ample, and moreover H 1 (π −1 (z), Lk ) = 0, dim H 0 (π −1 (z), Lk ) = kd + 1 − g,

where g stands for the genus of the fibers of π. Likewise, for large k, M k is very ample, and dim H 0 (π −1 (0), Lk ) = dim H 0 (C , M k ) = kd + 1 − g ; thus, by Riemann–Roch, H 1 (π −1 (0), Lk ) = 0 . By the theory of base change in cohomology the direct image π ∗ Lk is free. This implies that we can choose a frame σ0 , . . . , σN of π ∗ Lk ; this yields a well-defined map ϕ : C → PN . Now denote by C˜ the image of C ˜ the projection to Δ. in PN × Δ via (ϕ, π) and denote by π Then π ˜ : C˜ → Δ agrees with π : C → Δ, except that now the central fiber is not C but C . On the other hand, by what has been proved, the Hilbert polynomial of π ˜ −1 (z) is independent of z. This, as we observed

§5 Isomorphisms of families of stable curves

113

in Section 2 of Chapter IX, implies that π ˜ is a flat morphism. Since the fibers of π ˜ are all stable, π ˜ : C˜ → A is a family of stable curves. The proof of stable reduction is now complete. The argument we have given contains all the ingredients needed to prove also the following stable reduction theorem for pointed curves. Theorem (4.11) (Stable reduction). Let π

→ Δ = {z ∈ C| |z| < ε} C− be a proper morphism whose fibers are (possibly nonreduced) complete curves. Let σi : Δ → C, i = 1, . . . , n, be sections of π. Assume that the restriction of the family π, together ˙ = {z ∈ Δ| |z| > 0}, is with the sections σ1 , . . . , σn , to the pointed disc Δ a family of n-pointed stable curves. Then there exist an integer k and a family of stable n-pointed curves π : C → Δ , σi : Δ → C ,

i = 1, . . . , n,

˙ is the pullback of the family π whose restriction to the pointed disc Δ k via the base change z = ζ . We leave to the reader to fill in the missing details in the pointed case. To this end, it may be of help to notice that the construction of family (4.10) only uses the fact that all fibers of π, except the central one, are nodal, and not the full strength of the stability assumption, which is used just in the blowing-down procedure. 5. Isomorphisms of families of stable curves. In Section 4 we proved the stable reduction theorem which, very loosely speaking, asserts that any family of stable curves over a punctured disc can be filled in with a stable curve. The main result of this section is that such a fill-in is unique. In the previous section we also explained that the stable reduction theorem is essentially equivalent to the assertion that the moduli space of stable curves is complete. Likewise, the uniqueness of stable reduction is essentially equivalent to the assertion that the moduli space of stable curves is separated. Consider two families α : X → S , σ1 , . . . , σn : S → X

and

β : Y → S , τ1 , . . . , τn : S → Y

of stable n-pointed curves over the same base S. We may then construct a scheme or analytic space IsomS ((X; σ1 , . . . , σn ), (Y ; τ1 , . . . , τn )) parameterizing pairs (s, ϕ), where s ∈ S and ϕ is an isomorphism Xs → Ys carrying σi (s) to τi (s) for i = 1, . . . , n (cf. Exercise (7.7) in Chapter IX). The main result we have in mind is the following.

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Theorem (5.1). Let α : X → S , σ1 , . . . , σn : S → X

and

β : Y → S , τ1 , . . . , τn : S → Y

be two families of stable n-pointed curves over S, where S is a scheme or an analytic space. Then IsomS ((X; σ1 , . . . , σn ), (Y ; τ1 , . . . , τn )) is finite and unramified over S. Proof. To keep things simple, throughout the proof we shall omit any mention of the sections in the notation for families of n-pointed curves; thus, for instance, we shall write IsomS (X, Y ) instead of In the proof, we may limit IsomS ((X; σ1 , . . . , σn ), (Y ; τ1 , . . . , τn )). ourselves to the case where the two families involved are algebraic, since any flat analytic family of projective schemes is locally the pullback of an algebraic one, by Lemma (4.9) in Chapter IX, and stability is an open condition in families. The morphism IsomS (X, Y ) → S is quasi-finite, since a stable curve has a finite automorphism group; to prove that it is a finite morphism, we must show that it is proper. By the valuative criterion, properness is a consequence of the following statement: (5.2). Let α : W → Δ, β : Z → Δ be two families of stable n-pointed curves over the disk Δ = {t ∈ C : |t| < ε}, and let W ∗ → Δ∗ , Z ∗ → Δ∗ be their restrictions to the pointed disk Δ∗ = Δ {0}. Let γ : W ∗ → Z ∗ be an isomorphism of families of n-pointed curves over Δ∗ . Suppose that γ is meromorphic as a map from W to Z. Then γ extends uniquely to an isomorphism ξ : W → Z of families of n-pointed curves over Δ. For the sake of simplicity, in proving (5.2) we shall deal only with the essential case in which the general fibers of the families involved are smooth; the general case follows by the same cut-and-paste procedures used in the previous section. The uniqueness of ξ is clear, as Z is separated; what needs to be proved is its existence. The singularities of W and Z are, in suitable coordinates, of the form xy = tn+1 , that is, they are An singularities. In particular they are normal and have a minimal resolution in which the exceptional divisor is a chain of n smooth rational curves E1 , . . . , En , as we explained in the previous section (cf. Fig. 2). Notice that the self-intersection of Ei equals −2 for each i. If we resolve all the singular points of W and Z situated on the central fibres, we may replace our original families with new ones α : W → Δ , β : Z → Δ, where W and Z are smooth along their central fibres. The price we have to pay is that now the central fibres are no longer stable, but only semistable. Notice however that the only destabilizing components are smooth rational with self-intersection equal to −2 and that they do not meet any of the marked sections; furthermore, any exceptional curve of

§5 Isomorphisms of families of stable curves

115

the first kind in the central fibers meets at least two marked sections. Let Γ ⊂ W ∗ ×Δ∗ Z ∗ be the graph of γ. By meromorphicity, Γ extends be a to an analytic subspace Γ ⊂ W ×Δ Z , proper over Δ. Let Γ desingularization of Γ. By the structure theorem for morphisms between → W is a finite sequence of blow-ups. Summing smooth surfaces, Γ , obtained from W by blowing up up, there is a smooth surface W finitely many times at points of the central fiber, such that γ extends → Z : we choose W minimal with respect to this to a morphism γ :W property. If W is different from W , it contains an exceptional curve of the first kind E which is not contracted by γ but gets contracted in → W . It follows in particular that E does not meet marked sections, W so the same is true of its image in W , which we denote by E ; as a consequence, E is not an exceptional curve of the first kind. We know that the inverse of γ is a sequence of monoidal transformations. The proper transform of E under any one of these has self-intersection not greater than the one of E ; thus the self-intersection of E is greater than or equal to −1. Since E is rational and is not an exceptional curve of the first kind, it must be singular. But then, since E is smooth, one of the above monoidal transformations must be centered at a singular point of E so that the self-intersection of E is at least equal to the self-intersection of E plus three. This leads to the absurd conclusion that Z contains curves with positive self-intersection. This contradiction establishes that γ extends to a morphism γ : W → Z . −1

also extends to a morphism The same argument establishes that γ from Z to W . Hence γ must be an isomorphism. Clearly γ sends chains of self-intersection −2 rational curves to chains of self-intersection −2 rational curves. This shows that γ extends to a morphism from Wreg to Zreg and therefore, by Hartogs’ theorem, to a morphism ξ from W to Z. As the same argument applies to γ −1 , it follows that ξ is an isomorphism. This concludes the proof of (5.2) and hence shows that π : IsomS (X, Y ) → S is proper. We now show that π : IsomS (X, Y ) → S is unramified. This means that, for any point γ ∈ IsomS (X, Y ), the map of Zariski tangent spaces dπ : Tγ (IsomS (X, Y )) → Tπ(γ) (S) is injective. To show that this is true, set s = π(γ) and observe that an element v of Tγ (IsomS (X, Y )) consists of a morphism ϕ : Spec C[ε] = Σ → (S, s) plus an isomorphism γ : ϕ∗ X → ϕ∗ Y of families of n-pointed curves over Σ extending γ; moreover, dπ(v) is just ϕ. Thus, if dπ(v) = 0, then ϕ∗ X and ϕ∗ Y are trivial families Xs × Σ and Ys × Σ, where Xs = α−1 (s) and Ys = β −1 (s). If we identify Xs and Ys via γ, then γ gets identified to a Σ-automorphism of Xs × Σ restricting to the identity on the central

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10. Nodal curves

fiber and on the marked sections, that is, to what is usually referred to as an infinitesimal automorphism of Xs . Formally, an infinitesimal automorphism of an n-pointed curve (C; p1 , . . . , pn ) is a Σ-automorphism of C × Σ which restricts to the identity on the central fiber and on the marked sections {pi } × Σ. The result is then a consequence of the following simple remark. Lemma (5.3). A stable n-pointed curve has no nontrivial infinitesimal automorphisms. To prove the lemma, denote by g the genus and by N the normalization of C, viewed as an (n + 2δ)-pointed curve, where δ is the number of singular points of C, and let N = ∪Ni be the decomposition of N into irreducible components; recall that each of the Ni is stable. Then observe that any infinitesimal automorphism of (C; p1 , . . . , pn ) lifts to an infinitesimal automorphism of N . This reduces us to the case where C is smooth; an infinitesimal automorphism is then just a vector field vanishing at the marked points, that is, an element of H 0 (C, TC (− pi )). On the other hand, the stability assumption means that 2g − 2 + n > 0, that is, that the degree of TC (− pi ) is negative. This concludes the proof of the lemma and hence of Theorem (5.1). Q.E.D. An immediate corollary of Theorem (5.1) is that one can do away with the meromorphicity assumption in (5.2). More precisely, the following result holds. Corollary (5.4). Let α:X →S,

σ1 , . . . , σn : S → X

and

β :Y →S,

τ1 , . . . , τn : S → Y

be families of stable n-pointed curves as in Theorem (5.1), where S is a reduced and irreducible normal scheme or analytic space. Let S ∗ be the complement in S of a closed proper subscheme or analytic subspace, and set X ∗ = α−1 (S ∗ ), Y ∗ = β −1 (S ∗ ). Let γ : X ∗ → Y ∗ be an isomorphism of families of n-pointed curves over S ∗ . Then γ extends uniquely to an isomorphism ξ : X → Y of families of n-pointed curves over S. Here is another immediate consequence of Theorem (5.1). Corollary (5.5). Let α : X → S , σ1 , . . . , σn : S → X

and

β : Y → S , τ1 , . . . , τn : S → Y

be families of stable n-pointed curves as in Theorem (5.1), and let ϕ, ψ : X → Y be isomorphisms of families of n-pointed curves over S. Suppose that the induced isomorphisms ϕs and ψs between Xs = α−1 (s) and Ys = β −1 (s) are equal for each (closed) point s of S. Then ϕ = ψ.

§6 The stable model, contraction, and projection

117

Proof. Clearly the corollary is trivial when S is reduced. Set Z = IsomS ((X; σ1 , . . . , σn ), (Y ; τ1 , . . . , τn )) and let π : Z → S be the natural projection. Then ϕ and ψ can be viewed as sections a, b : S → Z. Let s be a (closed) point of S, set z = a(s) = b(s), and denote by mz and ms the maximal ideals of OZ,z and OS,s . Pullback via π gives a homomorphism f : OS,s → OZ,z , and pullback via a and b homomorphisms u, v : OZ,z → OS,s . By construction, u and v are left inverses to f . We let fh : ms /mhs → mz /mhz ,

uh , vh : mz /mhz → ms /mhs

be the homomorphisms induced by f , u, and v. Clearly, uh and vh are left inverses to fh ; in particular, fh is always injective. Theorem (5.1) asserts that π is unramified and hence that f2 is onto. One then easily shows, inductively on h, that fh is onto for every h ≥ 2. The upshot is that the homomorphism of completions (5.6)

Z,z S,s → O f : O

is an isomorphism, i.e., that the projection π is étale at z, and that u and v are inverses of f and hence are equal. It follows that u and v are also equal. Q.E.D. For future reference, we record here another useful fact concerning isomorphisms of families of nodal curves, which is an immediate consequence of Lemma (7.5) in Chapter IX. Lemma (5.7). Let f : X → S and f : X → S be families of nodal curves, and let π : X → X be an S-morphism. Suppose that, for each point s ∈ S, the induced morphism Xs → Xs between fibers is an isomorphism. Then π is also an isomorphism. 6. The stable model, contraction, and projection. In this section and in the following two, we shall discuss a number of constructions involving families of nodal curves which will be crucial in the rest of the book. The first construction we wish to present is the one of the stable model, a special instance of which we already encountered in Section 4. Let g and n be such that stable n-pointed genus g curves exist, i.e., such that 2g − 2 + n > 0. Then there is a canonical way of attaching a stable n-pointed genus g curve to a semistable one (C; x1 , . . . , xn ). To see why this is the case, recall that the irreducible components which prevent (C; x1 , . . . , xn ) from being stable are precisely those components which are smooth rational and contain just two points which are either

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10. Nodal curves

marked or nodes. The connected components of their union are chains of smooth rational curves, and, as we did in Section 4, we refer to them as exceptional chains. The n-pointed nodal curve (C ; x1 , . . . , xn ) obtained by collapsing to a point each exceptional chain is clearly stable and is called the stable model of (C; x1 , . . . , xn ). Let α : C → C be the collapsing map. It is importantto notice that, since ωC is trivial on each exceptional chain, α∗ (ωC ( xi )) is canonically isomorphic to ωC ( xi ). Exceptional chains come in two flavors, depending on whether they contain no marked points or one marked point. The figure below illustrates the two kinds of chains, drawn in red, and the corresponding collapsing maps.

Figure 4. As we shall presently see, the collapsing operation which produces the stable model of a semistable curve can be performed simultaneously and consistently for all fibers of any family of semistable curves. We shall use the following simple result, which generalizes a well-known property of smooth (unpointed) curves. Lemma (6.1). Let (C; D) be a semistable curve of genus g with n marked points, and let V be the set of vertices of its graph. Suppose that 2g − 2 + n > 0. For each v ∈ V , denote by Cv the corresponding component of C, by gv the genus of its normalization, and by lv the number of the half-edges issuing from v. Let L be a line bundle on C and set dv = degCv L. Then i) H 1 (C, L) = 0 if dv ≥ 2(2gv − 2 + lv ) for all v; ii) L is base-point-free if dv ≥ 2(2gv − 2 + lv ) for all v; iii) L is very ample if C is stable and dv ≥ 3(2gv − 2 + lv ) for all v. Proof. The assumptions imply that dv ≥ 0 for every v and in fact that dv > 0 unless Cv is a smooth rational curve and lv = 2, which of course can happen only if C is not stable. Thus, if dv = 0, L is trivial on Cv . But then, denoting by (C ; D ) the curve obtained by contracting Cv to


119

a point and by ϕ : C → C the contraction map, L = ϕ∗ L for some line bundle L on C . The curve (C ; D ) and the line bundle L still satisfy the assumptions of the lemma, and moreover H i (C, L) = H i (C , L ) ,

i = 0, 1 .

Iterating the contraction procedure, we are thus reduced to proving the lemma under the additional assumption that dv > 0 for every vertex v. Property i) follows from (2.14), since dv > degCv ωC for every v. In fact, dv ≥ 2(2gv − 2 + lv ) > degCv ωC when 2gv − 2 + lv > 0, while dv > 0 = degCv ωC when 2gv − 2 + lv = 0. To prove ii), we must show that H 1 (C, Ix L) vanishes for every x ∈ C, where Ix stands for the ideal sheaf of x. We distinguish two cases. Suppose first that x is not a singular point of C, so that Ix L is a line bundle. If Cv does not contain x, then degCv Ix L = dv > degCv ωC . If x ∈ Cv , then (6.2)

degCv Ix L = dv − 1 ≥ degCv ωC .

If equality holds, we are in one of the following cases. Either 2gv − 2 + lv = 0, and hence gv = 0, lv = 2, or else 2gv − 2 + lv = 1, and Cv does not contain marked points. The latter can happen only for gv = 0, lv = 3 and for gv = 1, lv = 1. In all cases, Cv cannot be the only component of C. Thus (2.14) applies to Ix L. Suppose instead that x is a singular point of C. We let α : C → C be the partial normalization at x. If Cv is a component of C, we denote by Cv the corresponding component of C . The sheaf Ix L is of the form α∗ L , where L is a line bundle on C , and hence H 1 (C, Ix L) = H 1 (C , L ). We also recall that α∗ ωC = Ix ωC . If Cv is any component of C, then (6.3)

degCv L − degCv L = degCv ωC − degCv ωC .

Thus, degCv L ≥ degCv ωC − degCv ωC + 2(2gv − 2 + lv ) ≥ degCv ωC + 2gv − 2 + lv > degCv ωC when 2gv − 2 + lv > 0, while degCv L ≥ dv − 1 > −1 = degCv ωC when 2gv − 2 + lv = 0, and (2.14) again proves the vanishing of H 1 (C , L ) = H 1 (C, Ix L). It remains to prove iii). We must show that, for any choice of points x, y ∈ C, the group H 1 (C, Ix Iy L) vanishes. We distinguish various cases. First suppose that x and y are not singular points of C. Then Ix Iy L is a line bundle, and its degree on Cv is at least 3(2gv − 2 + lv ) − 2 ≥ 2gv − 2 + lv ≥ degCv ωC , since C is stable. Equality occurs only when both x and y belong to Cv , 2gv − 2 + lv = 1, and Cv

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does not contain marked points. This can happen only for gv = 0, lv = 3 and for gv = 1, lv = 1. In both cases, Cv cannot be the only component of C and, if Cw is any other component, the degree of Ix Iy L on it is strictly greater than degCw ωC , so that (2.14) applies. In the remaining cases, at least one among x and y is singular. Let α : C → C be the partial normalization at those points of {x, y} which are nodes of C. If Cv is a component of C, we denote by Cv the corresponding component of C . The sheaf Ix Iy L is of the form α∗ L , where L is a line bundle on C , and hence H 1 (C, Ix Iy L) = H 1 (C , L ). The second case we consider is the one where x = y and x, y are both singular. One proceeds as in the second part of the proof of ii). Formula (6.3) is valid and implies that degCv L ≥ degCv ωC + 2(2gv − 2 + lv ) > degCv ωC . We conclude by appealing to (2.14) again. A similar argument covers the case where x is singular and y is not. If y ∈ Cv , formula (6.3) is valid, and one can argue exactly as in the previous case; if y ∈ Cv , formula (6.3) gets replaced by degCv L − degCv L = degCv ωC − degCv ωC + 1, which implies degCv L ≥ degCv ωC + 2(2gv − 2 + lv ) − 1 > degCv ωC . The final case to be considered is the one where x = y is singular. Instead of (6.3), we get that deg Cv L−degCv L = 2 degCv ωC −2 degCv ωC , which gives degCv L ≥ 2 degCv ωC −2 degCv ωC +3(2gv −2+lv ) ≥ 2 degCv ωC +2gv −2+lv . Moreover, if these inequalities are equalities, then degCv ωC = 2gv − 2 + lv ; in other words, Cv does not contain marked points. Observe that degCv ωC = degCv ωC −hv , where hv equals 2 if Cv is the only component containing x, 1 if x belongs to Cv and to another component, and 0 if x ∈ Cv ; moreover, hv ≤ lv . Hence, since 2gv − 2 + lv > 0 by stability, degCv ωC + 2gv − 2 + lv is nonnegative, and vanishes if and only if gv = 0, lv = 3, and hv = 2. Thus degCv L ≥ degCv ωC , and if we have equality, then Cv contains no marked points, gv = 0, lv = 3, and hv = 2. If this happens, C is connected and has components other than Cv , as lv − hv = 1 and Cv contains no marked points. Since the degree of L on any component of C different from Cv is strictly greater than the one of ωC , (2.14) shows that H 1 (C , L ) = H 1 (C, Ix2 L) = 0. This concludes the proof. Q.E.D. Corollary (6.4). Let (C; D) be as in (6.1). Set M = ωC (D). Then H 0 (C, M 2 ) ⊗ H 0 (C, M k ) → H 0 (C, M k+2 ) is onto for every k ≥ 4.


121

This can be proved by a standard argument. Notice that M h satisfies the assumptions of parts i) and ii) of (6.1) for h ≥ 2, and those of part iii) for h ≥ 3. In particular, we can choose sections s and t of M 2 which have no common zeros. Let E be the divisor of s. By part i) of (6.1), from the exact sequence ×s

i →0 0 → M i−2 −−→ M i → M|E

for i = k + 2, k we deduce the exact sequence (6.5)

×s

k+2 )→0 H 0 (C, M k ) −−→ H 0 (C, M k+2 ) → H 0 (E, M|E

k ). Thus the composite and the surjectivity of H 0 (C, M k ) → H 0 (E, M|E mapping ×t k+2 k ) −−→ H 0 (E, M|E ) H 0 (C, M k ) → H 0 (E, M|E

is onto, since s and t have no common zeros. But then s ⊗ H 0 (C, M k ) + t ⊗ H 0 (C, M k ) maps onto H 0 (C, M k+2 ), by (6.5). This concludes the proof. Corollary (6.6). Let f : X → S, σp : S → X, p ∈ P , be a family of P -pointed nodal curves. Then the set of s ∈ S such that (Xs ; {σp (s)}p∈P ) is semistable is Zariski open in S. It does no harm to assume that f has connected fibers. Set L = ωf ( σp ). Suppose that the fiber of f at s0 ∈ S is semistable, and set xp = σp (s0 ). Let (Xs 0 ; {x (Xs0 ; {xp }p∈P ). Then p }p∈P ) be the stable model of Ls0 = ωXs0 ( xp ) is the pullback of L = ωXs ( xp ). Since the fibers 0 of Xs0 → Xs 0 are points or chains of P1 ’s, a Leray spectral sequence argument shows that H q (Xs0 , Lks0 ) = H q (Xs 0 , L s0 ) k

for all q ≥ 0 and all k. Combining this with Lemma (6.1) shows that, for high enough k, the linear system |Lks0 | has no base points, and H 1 (Xs0 , Lks0 ) vanishes. The theory of base change in cohomology (cf. Proposition (3.3) in Chapter IX) then implies, in particular, that every section of Lks0 extends to a section of Lk over a neighborhood of Xs0 . It follows that |Lks | has no base points for all s in a neighborhood of s0 . But then Lemma (3.2) implies that, when s belongs to this neighborhood, (Xs ; {σp (s)}p∈P ) is semistable. We now return to the problem of simultaneously constructing the stable models of all fibers in a family of semistable curves. Here is what we shall prove.

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Proposition (6.7). Consider a family F of semistable P -pointed curves of genus g f : X → S , σp : S → X , p ∈ P. Suppose that 2g − 2 + |P | > 0. Then there exist a family F of stable P -pointed genus g curves f : X → S ,

σp : S → X , p ∈ P,

and a morphism π : X → X such that: i) f ◦ π = f ; ii) σp = π ◦ σp for every p ∈ P ; iii) for each closed point s of S, the fiber Xs is the stable model of Xs , and πs : Xs → Xs is the collapsing map. The pair (F , π) is unique up to a unique isomorphism. ωX/S ( σp ), L = ωX /S ( σp ). Then:

Set L = k ∼

a) for every k > 0, there are canonical isomorphisms π ∗ L → Lk and k ∼ L → π∗ Lk ; q b) R π∗ Lk = 0 for all q > 0 and all k ≥ 0; k c) Rq f∗ L and Rq f∗ Lk are canonically isomorphic for every q ≥ 0 and every k ≥ 0. We shall only prove existence and uniqueness of F and π. For a proof of properties a), b) and c) the reader is referred to [426]; here we simply observe that these statements are essentially obvious when S is reduced. We begin with the proof of existence. As usual, for any s ∈ S, we write Ls for the pullback of L to the fiber Xs . Set xp = σp (s), and let (Xs ; {xp }p∈P ) be the stable model of (Xs ; {xp }p∈P ). Clearly, Ls is the pullback to Xs of L s = ωXs ( xp ), and, as we also observed in the proof of Corollary (6.6), H q (Xs , Lks ) = H q (Xs , L s ) k

for all q ≥ 0 and all k. Lemma (6.1) and the theory of base change in cohomology imply that R1 f∗ Lk = 0 for k ≥ 2 and that f∗ Lk is locally free for all k. Moreover, Corollary (6.4) implies that f∗ L2 ⊗ f∗ Lk → f∗ Lk+2 is onto as soon as k ≥ 4; in particular, ⊕k≥0 f∗ L4k is a locally finitely generated graded OS -algebra. We set F = L4 and X = Proj(⊕k≥0 f∗ F k ) .


123

The family f : X → S is flat since, for large k, f∗ (OX (k)) equals f∗ F k , which is locally free, and by part ii) of Lemma (6.1) there is a surjective map π fitting in a commutative diagram w X X' π [ ' ) [ f ^ f S We let σp be the composition of σp and π. Then, by part iii) of Lemma (6.1), the fiber of f : X → S, {σp }p∈P at s is just the stable model (Xs ; {σp (s)}p∈P ) of (Xs ; {σp (s)}p∈P ), and the fiber of π at s is just the collapsing map. We now come to uniqueness. The homomorphism (6.8)

OX → π∗ OX

is clearly injective. We claim that, in fact, it is an isomorphism. It suffices to treat the algebraic case, since any flat analytic family of projective schemes is locally the pullback of an algebraic one, by Lemma (4.9) in Chapter IX, and semistability is an open condition. Let s be a closed point of S, and let q1 , . . . , q ∈ Xs be the images of the exceptional chains of Xs . Choose an affine open subset Y = Spec A of X containing q1 , . . . , q . Denote by mi ⊂ A the maximal ideal corresponding to qi , and let T be the complement of m1 ∪ · · · ∪ m in A. We also set B = Γ(π −1 (U ), OX ). It will suffice to show that T −1 A → T −1 B is an isomorphism. Let u be a function on an open neighborhood U of π −1 (q1 , . . . , q ). Possibly after shrinking, we may assume that U is the complement of a Cartier divisor in f −1 (V ), where V is a suitable open neighborhood of s. We may thus view u as a section over f −1 (V ) of Of −1 (V ) (D), where D is a Cartier divisor not meeting π −1 (q1 , . . . , q ). The line bundle F has strictly positive degree on all components of Xs except those which belong to exceptional chains. On the latter, which are smooth rational, it has degree zero. Thus, for large enough k, the line bundle F k (−D) ⊗ OXs satisfies the assumptions of parts i), ii) of Lemma (6.1); it follows that it is base-point-free and that its higher cohomology groups vanish. By the theory of base change, then, possibly after shrinking V , the direct image via f of F k (−D) is locally free over V , and its fiber at s is H 0 (Xs , F k (−D) ⊗ OXs ); it follows that there is a section v of F k (−D) over f −1 (V ) which does not vanish at any point of π −1 (q1 , . . . , q ). We may view uv as a section of F k over f −1 (V ). If we also regard v as a section of F k , the quotient uv/v is a regular function on a neighborhood of {q1 , . . . , q } which pulls back to u via π. This shows that T −1 A → T −1 B is onto, proving that (6.8) is an isomorphism. Now suppose that f : X → S, {σp }p∈P , is another family of stable curves and that π : X → X is another collapsing map sharing with F

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and π properties i), ii), and iii). Clearly, there is a bijection of sets α : X → X such that π = α ◦ π. Since (6.8) is an isomorphism, this map actually comes from a morphism. By Lemma (5.7), this morphism is an isomorphism. This concludes the proof of uniqueness in (6.7). Summing up, we have associated to the family F a new family X u StMd(F ) = f

u S

σp , p ∈ P

plus the collapsing map CollF : X → X . The family StMd(F ) is called the stable model of F . Remark (6.9). It follows immediately from the functoriality of ωX/S ( σp ) under morphisms of families of P -pointed nodal curves that StMd(F ) and CollF depend functorially on F in the following sense. Suppose we are given two families F1 and F2 of semistable P -pointed genus g curves and a morphism

H =

between them. StMd(F2 )

X1

H

w X2

u S1

h

u w S2

Then there is a natural morphism from StMd(F1 ) to

X1 StMd(H) =

u S1

H

h

w X2 u w S2

such that H ◦ CollF1 = CollF2 ◦H, and moreover StMd(HK) = StMd(H) StMd(K) whenever HK is defined. In particular, the stable model is a functor from the category of families of semistable P -pointed genus g curves to the category of families of stable ones. The stable model makes it possible to easily perform two other operations, which usually go under the name of contraction and projection, for reasons that will become clear in Chapter XII. To explain what these are, start with a stable P -pointed curve, where P is a finite nonempty set, and remove (or better, unmark) one of the marked points, say the


125

one labeled by p ∈ P . The result is a (P {p})-pointed curve, which may not be stable but is certainly semistable. If we pass to its stable model, the result is a stable (P {p})-pointed curve. This same operation can be performed in families. Given a family Xu F = f

u S

σp , p ∈ P

of stable P -pointed genus g curves, we get a family of semistable (P {p})pointed ones by ignoring the pth section. We call this new family F and pass to its stable model StMd(F ), which we also denote by Prp (F ). By Remark (6.9), Prp is a functor from families of stable P pointed genus g curves to families of stable (P {p})-pointed ones, which we shall call the pth projection. We may also elect to keep track of the pth section of F ; composing it with the collapsing map gives an additional section δ of Pr(F ), which however may meet one or more of the marked sections or go through singular points of fibers. Associating to F the pair Contrp (F ) = (Prp (F ), δ) gives a morphism from families of stable P -pointed genus g curves to pairs consisting of a family of stable (P {p})-pointed ones plus an extra section. This will be called the pth contraction. Lemma (6.10). If p and q are distinct points of P , then Prp and Prq commute. In other words, Prp Prq (F ) and Prq Prp (F ) are canonically isomorphic. The same applies to Contrp and Contrq . The lemma is obviously true when S is a single point. In the general case, denote by X the total space of Prp (F ), by X the one of Prq (F ), and by X the one of Prp Prq (F ). There are a diagram of collapsing morphisms α w X X β

u γ X w X and a set-theoretic map η : X → X whose restriction to each fiber of X → S is the collapsing morphism associated to the projection Prq . Set-theoretically, ηα = γβ. As we observed in the proof of (6.7), the homomorphism OX → α∗ OX is an isomorphism. We thus get a sheaf homomorphism (actually, an isomorphism) OX → η∗ OX by composing OX → γ∗ OX → γ∗ β∗ OX η∗ α∗ OX η∗ OX . Thus η comes from a morphism. uniqueness of the stable model.

The lemma now follows from the

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10. Nodal curves

7. Clutching. Another construction, which is related to passage to the stable model and to projection, is the so-called clutching operation. We first describe it for a single curve and then explain how it can be generalized to families. Let P be a finite set, and let Γ be a P -marked dual graph; thus to each p ∈ P there is attached a leg p . Let V be the set of vertices of Γ. As usual, for each vertex v, we denote by Lv the set of half-edges issuing from v. Suppose that we are given an Lv -pointed genus gv nodal curve Cv for each vertex v. We may then construct a new P -pointed curve C by identifying two points of Cv if and only if they are marked points labeled by the two halves of an edge of Γ, and by labeling with p ∈ P the point originally labeled by the leg p .

Figure 5. A graph and a clutching associated to it. Clearly, the curve Cv is the partial normalization of C at the nodes introduced by the identifications. Hence the process we have just described is a sort of inverse to partial normalization. If Γ is connected and all the Cv are connected, then C is connected as well. Conversely, the connectedness of C implies the connectedness of Γ. From now on, we assume that Γ and all the Cv are connected. The genus g of C is given by formula (2.8), which can be rewritten as g= gv + 1 − χ(Γ) = gv + h1 (Γ) . v∈V

v∈V

Since one passes from C to Cv by a process of partial normalization, C is stable if and only if all the Cv are (cf. Remark (3.3)). Stability is thus equivalent to 2gv − 2 + lv > 0 , where lv = |Lv |. The curve C we have constructed is said to be obtained from the Cv by clutching along the graph Γ. The clutching procedure is readily generalized to families. Let Γ be as above, and denote by E the set of its edges. Suppose that for each v ∈ V , we are given a family Xvu Fv = fv

u S

σ , ∈ Lv

§8 Stabilization

127

of stable Lv -pointed genus gv curves, all over the same base S. We let X be the disjoint union of the Xv , and f : X → S the morphism whose restriction to Xv is fv . This gives an L-pointed family of nodal curves over S, which we denote by F . If m ∈ L, taking residues along σm yields a well-defined surjective homomorphism ωf ( σ ) → Oσm (S) , whence surjective homomorphisms ωfk (k σ ) → Oσm (S) for all k > 0. These drop down to homomorphisms (k) σ ) → OS , R : f∗ ωfk (k which are surjective for k ≥ 2, by Lemma (6.1). We may then construct homomorphisms σ ) → OSE R(k) : f∗ ωfk (k by defining the component of R(k) indexed by edge {, } to be (k) (k) R + (−1)k−1 R (here we have arbitrarily chosen an orientation on {, }). The homomorphism R(k) has constant rank; in fact, the kernel k (k of its fiber at s ∈ S can be identified with H 0 (Xs , ωX p∈P σ p (s))), s where Xs is the curve obtained from Xs by clutching along Γ, and hence its dimension is independent of s. It follows that the kernel of R(k) , which we denote by Sk , is locally free. It also follows from (6.4) that the graded OS -algebra ⊕k≥0 Sk is locally finitely generated. We set X = Proj(⊕k≥0 Sk ) . It is a consequence of (6.1) that the fiber of X → S at s is precisely the curve obtained from Xs by clutching along Γ. For each p ∈ P , we define σp : S → X to be the composition of σ p with the natural morphism X → X . The family X u F = f

σp , p ∈ P

u S is said to be obtained from F by clutching along Γ. As was the case for the stable model, it is clear from the construction that clutching is functorial. 8. Stabilization. The contraction operation produces, out of a family of stable npointed curves, a family of (n − 1)-pointed ones plus an extra section.

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We shall show that this operation has an inverse, which goes under the name of stabilization. Suppose that we are given a family of P -pointed nodal curves, consisting of a family ϕ:X→S

(8.1)

of nodal curves plus sections σp , p ∈ P , and, in addition, a further section δ:S →X. Set Q = P ∪ {q}, where q does not belong to P . Adding δ to the σp and labeling it with q does not, in general, produce a family of Q-pointed nodal curves, for two reasons; the first is that δ may well meet the other sections, and the second is that it may go through nodes of the fibers of f . As we shall see, there is however a canonical way of modifying X so as to obtain a family of Q-pointed nodal curves. If one starts with a family of stable P -pointed curves, the result is a family of stable Q-pointed ones; in addition, applying to this family the operation Prq gives back the family of P -pointed curves one started with. It is useful, though not strictly necessary, to remark that it suffices to perform the construction in the special case in which the family is the projection of a fiber product to one of the factors and the extra section is the diagonal. Consider in fact the commutative diagram X (8.2)

δ

w X ×S X = Y

ϕ u S

δ

π2 u wX

where δ = (id, δ), and πi : X ×S X → X stands for projection to the ith factor. The diagram is cartesian, and the sections σp are the pullbacks of the sections τp : (σp , id) : X → X ×S X. Hence diagram (8.2) is a morphism of families of P -pointed nodal curves. Furthermore, the extra section δ is the pullback of the diagonal section Δ : X → X ×S X. Thus, if the problem we have posed can be solved for Y → X and the sections τp , p ∈ P , and Δ, then a solution to our original problem can be simply obtained by pullback. We begin by describing the construction in rather informal and naive terms, starting with the “universal” case of Y → X and of the diagonal section Δ. What we have to do is find a morphism Y → Y and lifts of the sections τp and Δ to sections of Y → X which make the latter into a family of Q-pointed nodal curves. Clearly, nothing needs to be done to Y except where a section crosses the diagonal, or where a section meets a node. Suppose first that τp crosses the diagonal, i.e., that δ meets σp somewhere. Since ϕ is smooth at the crossing point, by the very definition

§8 Stabilization

129

of family of P -pointed curves, from the analytic point of view, X ×S X can be locally represented as U × V , where U is an open neighborhood of the origin in C2 , and V an open subset of S, with π2 the projection to the product of the second factor of C2 times S. Furthermore, if x, x are the standard coordinates in C2 , we may set things up so that section τp is {x = 0}, and Δ is {x = x }. It is then clear that we just need to perform a blow-up in the x, x coordinates and replace τp and Δ by their proper transforms. In formulas, this amounts to the following. Let λ, μ be homogeneous coordinates in P1 . Then the sought-for modification of Y = X ×S X is locally the projection {((x, x ), s, [λ : μ]) ∈ U × V × P1 : xμ = x λ} → U × V , and the proper transforms of the sections τp and Δ are, respectively, λ = 0 and

λ = μ.

Notice that this construction does not affect the fibers away from the locus x = 0; on the other hand, its effect on a fiber at a point of X where x vanishes is to add a P1 (the red line in Fig. 6 below) meeting the rest of the fiber at a single point and crossed by the proper transforms of τp and Δ at distinct points.

Figure 6. We now turn to the case where Δ (or, which is the same, δ) meets a node of a fiber. Say the fiber lies over s0 ∈ S. Near the node, X can be analytically represented as the locus with equation xy = f , where f is a function on an open neighborhood V of s0 , vanishing at s0 , and hence Y = X ×S X can be locally realized as the locus W = {((x, y, x , y ), s) ∈ U × V : xy = f = x y } , where U is a neighborhood of the origin in C4 , and Δ as the locus wih equations x = x , y = y .

130

10. Nodal curves

In this case also the solution is simple and consists in replacing W with W = {((x, y, x , y ), s, [λ : μ]) ∈ U ×V ×P1 : xy = f = x y , λx = μx, λy = μy } , and Δ with the section Δ corresponding to the locus λ = μ. The net effect on the fiber at x = y = 0, s = s0 , is to replace the node with a P1 (the red line in Fig. 7 below), meeting once each of the two branches of the former node, and crossed by Δ at a further point.

Figure 7. We now go back to our original family (8.1). We want to find a morphism π : X → X, plus sections σp , p ∈ P and δ = σq of ϕ = ϕ ◦ π : X → S which make the latter into a family of Q-pointed nodal curves. As we have observed, a solution can be obtained from Y → X, {τp }p∈P and Δ by base change via δ : S → X. To describe the answer we distinguish two cases, as above. Case 1. Suppose δ meets σp somewhere. Analytically, X can be locally represented as U × V , where U is an open neighborhood of the origin in C and V an open subset of S, and ϕ as the projection to the second factor. Furthermore, if x is the standard coordinate in C, we can arrange things so that section σp is {x = 0} and δ is {x = ξ}, where ξ is a function on V . Choose homogeneous coordinates λ and μ on P1 . Locally, X is the subspace of the product U × V × P1 with equation (8.3)

xμ = ξλ ,

and the equations of sections σp and δ are (8.4) respectively.

λ = 0 and

λ = μ,

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131

Case 2. Suppose δ meets a node of a fiber, say of ϕ−1 (s0 ). Analytically, a neighborhood of the node is isomorphic to the subspace {xy = f } of U × V , where V is a neighborhood of s0 in S, f is a function on V , vanishing at s0 , and U is a neighborhood of the origin in the C2 with coordinates x, y. The section δ is defined by x = ξ, y = η, where ξ, η are functions on V such that ξη = f . Then, letting again λ, μ be homogeneous coordinates in P1 , X is locally the locus in U × V × P1 with equations (8.5)

xy = f ,

λξ = μx ,

λy = μη ,

while the lift δ of δ is the locus (8.6)

λ = μ.

We are now confronted with two main difficulties. The first is that it must be shown that the various local constructions we have performed fit together. The second is that the constructions are analytic, and it is not a priori clear that, if we start with an algebraic family, the result will be algebraic. We shall address these two problems simultaneously. We denote by D the subspace of X corresponding to δ, by I its ideal sheaf, and by I ∨ = HomOX (I, OX ) the dual of I. There is a natural homomorphism OX → I ∨ which is injective since I contains elements which are not zero divisors, as is clear from the local analysis carried out above. We consider the “diagonal” homomorphism h → (h, h) OX → I ∨ ⊕ OX (

σi ) ,

and denote by K its cokernel. We then set X = Proj(⊕k≥0 Symk K) . To define the lifts of the sections σp to sections of X → S, consider the natural surjective homomorphism K → OX ( σi )/OX , and its pullback σp∗ (K) → σp∗ (OX (

σi )/OX )

via σp . Since σp∗ (OX ( σi )/OX ) is invertible, this defines a section of X → X along σp . We let the lifting σp be the composition of it with σp . To define the lifting δ we shall proceed along the same lines, using the following result. Lemma (8.7). (I ∨ /OX ) ⊗OX OD is an invertible OD -module.

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10. Nodal curves

Granting the lemma, the surjective homomorphism (8.8)

K ⊗OX OD → (I ∨ /OX ) ⊗OX OD

defines a section of X → X along D. Composing it with δ gives the required lifting. We will now show that this new definition of X and δ agrees, locally, with the previous constructions and, at the same time, we will prove (8.7). Notice, to begin with, that D is a Cartier divisor except where δ goes through a node of a fiber; away from these points, I ∨ /OX = OX (D)/OX = OX (D) ⊗OX OD , so the lemma is trivial. In proving our contention, we shall give separate arguments for cases 1 and 2 described above, keeping the notation introduced there, and working in the analytic setup. Things are very simple in case 1. We view 1/(ξ − x) as a section of I ∨ , and 1/x as a section of OX ( σi ), and denote by α and β their classes in K. Clearly, α and β locally generate K, and the relations among them are generated by (x − ξ)α = xβ . Thus, if we set β = α − β , the classes α and β also generate K locally, and the relations among them are generated by (8.9)

ξα = xβ .

It is now essentially obvious that X is locally just (8.3). In fact, set W = U ×V , denote by W the portion of X lying above W , and consider 2 the surjective sheaf homomorphism OW → K|W sending (f, g) to f α + gβ. 2 ) whose We can view W as the subspace of P1 × W = Proj(⊕ Symk OW homogeneous ideal sheaf is the kernel of 2 Proj(⊕k≥0 Symk OW ) → Proj(⊕k≥0 Symk K|W ) . 2 , and μ the section (0, 1). These Let λ be the section (1, 0) of OW two sections give homogeneous coordinates on the first factor of P1 × W , and map, respectively, to α and β. It then follows from (8.9) that the homogeneous ideal sheaf of W is generated by ξλ − xμ, as claimed. Furthermore, α and β have the same image under under (8.8), since β maps to zero. This shows that δ is indeed described, in coordinates, by the second equation in (8.4). One similarly checks that the first of these equations describes σp . Case 2 requires more work. Set χ = ξ − x, ρ = y − η, and notice that K locally agrees with I ∨ . What needs to be proved is the following.

Lemma (8.10). The ideal I is locally generated by χ and ρ. relations between χ and ρ are generated by ηχ = xρ ;

yχ = ξρ .

The

§8 Stabilization

133

The OX -module I ∨ is locally generated by sections α and β such that (8.11)

α(χ) = x ;

α(ρ) = η ;

β(χ) = ξ ;

β(ρ) = y .

The relations between α and β are generated by yα = ηβ ;

ξα = xβ .

Granting the lemma, and arguing as in case 1, the last two statements show that, indeed, X is locally as described by (8.5). Some parts of the lemma are obvious. For instance, it is clear that χ and ρ generate I. Assuming that the relations between them are as described in the lemma, it is also clear that (8.11) can be taken as the definition of α and β. The real work will be in showing that α and β generate I ∨ , and in determining the relations between χ and ρ, and between α and β. The lemma asserts the exactness, for any point z of D, of the two sequences G

(8.12)

Φ

2 2 −→ OX,z − → Iz → 0, OX,z 2 2 −→ OX,z −→ Iz∨ → 0, OX,z F

where F and G are the matrices ξ y F = , −x −η and Φ and Ψ are given by u Φ = uχ + vρ , v

Ψ

G=

η −x

y −ξ

,

u Ψ = uα + vβ . v

Since the completion of a local ring R is faithfully flat over R, it suffices to prove the exactness of the analogues of (8.12) with OX,z and Iz replaced by their completions. In other words, we must prove the exactness of G

(8.13)

Φ

→ I → 0, B 2 −→ B 2 − B 2 −→ B 2 −→ I ∨ → 0, F

Ψ

where B stands for the completion of the local ring of X at z, and I for the ideal in B generated by χ and ρ. Denoting by A the completion of the local ring of S at ϕ(z), we have that B = A[[x, y]]/(xy − ξη).

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10. Nodal curves

Set

H=

0 −1

1 0

,

K=

0 −1 1 0

.

Lemma (8.14). The diagram ···

w B2

F

H ···

u w B2

G

w B2 K

t

G

F

w B2 H

u w B2

t

G

K

u w B2

F

w B2

t

G

u w B2

w B2

w ···

H t

F

u w B2

w ···

is commutative with exact rows. The commutativity is straightforward, as is the fact that GF = F G = 0. To prove the exactness, we shall construct an operator Λ : B 2 → B 2 such that (8.15)

ΛF + GΛ = id = ΛG + F Λ .

This operator, however, will not be B-linear, but just A-linear. Notice that any element of B can be written uniquely in the form i≥0

a i xi +

ai y −i ,

i 0 and to (η n ) when i ≤ 0. Since ξ and η belong to the maximal ideal of A, this shows that ui = 0 for any i, finishing the proof of Lemma (8.18). Combining (8.18) with (8.16) and (8.17) proves the exactness of the sequences (8.13), modulo checking that the homomorphisms B 2 → I and B 2 → I ∨ one obtains do indeed agree, up to sign, with Φ and Ψ. This simple verification is left to the reader. Lemma (8.10) is now fully proved. Corollary (8.19). I ∨ is OS -flat. By faithful flatness, to prove the corollary, it suffices to show that I ∨ is A-flat. By Lemmas (8.14) and (8.18), (8.20)

· · · → B 2 −→ B 2 −→ B 2 −→ B 2 −→ B 2 −→ B 2 → I ∨ → 0 F

G

F

G

F

is a resolution of I ∨ by A-flat modules. On the other hand, we know that there is an A-linear operator Λ : B 2 → B 2 satisfying (8.15). This shows that tensoring (8.20) with any A-module yields an exact sequence. This proves that I ∨ is A-flat. We now finish the proof of (8.7). In the notation of Lemma (8.10), one immediately sees that β = α + 1, meaning that the difference between the homomorphisms β and α acts as multiplication by the function 1. This implies, in particular, that α

138

10. Nodal curves

and β are the same modulo OX . From the second of the exact sequences (8.12) we deduce the other exact sequence 2 2 −→ OX,z /OX,z −→ Iz∨ /OX,z → 0, OX,z F

Ψ

2 is given by where the inclusion OX,z → OX,z u u → , −u

and F and Ψ are obtained from F and Ψ by passage to the quotient. If we restrict this last exact sequence to D, F pulls back to the zero map, by the explicit form of F . The outcome is an isomorphism between 2 /OD,z , which in turn is isomorphic to OD,z . Iz∨ /OX,z ⊗OX,z OD,z and OD,z This completes the proof of (8.7). In the course of it, we observed that α and β have the same image in Iz∨ /OX,z . This shows that δ is given, in coordinates, by (8.6). The construction of X → S and of δ is now complete. The stabilization procedure functorially associates to a pair F = (family of stable P -pointed genus g curves, extra section) a new family Stab(F ) of stable (P ∪ {q})-pointed genus g curves, the section labeled with q coming from the extra section. As we loosely announced, the stabilization construction is the inverse of contraction. To make this precise, and anticipating notation that will be more formally introduced in Chapter XII, we denote by Mg,P ∪{q} the category of families of stable (P ∪ {q})-pointed genus g curves, and by C g,P the one of families of stable P -pointed genus g curves plus an extra section. Theorem (8.21). The contraction and stabilization functors Contrq : Mg,P ∪{q} → C g,P ,

Stab : C g,P → Mg,P ∪{q}

are inverse to each other up to isomorphism of functors. In particular, they are equivalences of categories. The proof uses the following general result and its corollary. Lemma (8.22). Let f : Z → W be a proper morphism of schemes or analytic spaces, G a coherent sheaf on Z, and w a point of W . Denote by mw the maximal ideal in OW,w . Suppose that the dimension of f −1 (w) is at most 1 and that H 1 (f −1 (w), G ⊗OW k(w)) = 0. Then: i) there is an open neighborhood U of w such that R1 f∗ G|U = 0; ii) for each h ≥ 0, the homomorphism (f∗ G)w → H 0 (f −1 (w), G ⊗OW OW,w /mhw ) is onto.

§8 Stabilization

139

Proof. Since R1 f∗ G is coherent, i) is equivalent to the assertion that the stalk of R1 f∗ G at w is zero. In turn, by the theorem on formal functions (see [356], Theorem 11.1 or [324], (4.2.1)), this will follow if we can show that H 1 (f −1 (w), G ⊗OW OW,w /mhw ) = 0 for h ≥ 1 . This assertion reduces to one of the assumptions for h = 1, and we shall prove it in general by induction on h. Look at the exact cohomology sequence of h+1 → G ⊗OW OW,w /mhw → 0. 0 → mhw G/mh+1 w G → G ⊗OW OW,w /mw

It is clear that, to do the inductive step, what must be shown is that (8.23)

H 1 (f −1 (w), mhw G/mh+1 w G) = 0 .

The natural homomorphism of Of −1 (w) -modules G ⊗k(w) mhw /mh+1 → mhw G/mh+1 w w G is onto, and the cohomology of its kernel vanishes in degree greater than 1 for dimension reasons. Passing to cohomology, this gives a surjection 1 −1 (w), mhw G/mh+1 H 1 (f −1 (w), G ⊗k(w) mhw /mh+1 w ) → H (f w G) .

On the other hand, the left-hand side is a direct sum of copies of H 1 (f −1 (w), G ⊗OW k(w)), which is zero by assumption, and hence we are done. We now turn to ii). Let I be the ideal sheaf of w. A piece of the higher direct image exact sequence of 0 → I h G → G → G/I h G → 0 is f∗ G → f∗ (G/I h G) → R1 f∗ (I h G). Since f∗ (G/I h G) is a skyscraper sheaf whose stalk at w is just H 0 (f −1 (w), G ⊗OW OW,w /mhw ), to prove ii), it suffices to show that R1 f∗ (I h G) vanishes in a neighborhood of w. Clearly, H 1 (f −1 (w), I h G ⊗OW k(w)) = H 1 (f −1 (w), mhw G/mh+1 w G) , so (8.23) shows that the sheaf I h G satisfies the assumptions of the lemma, whose part i) then gives the required local vanishing of R1 f∗ (I h G). Q.E.D.

140

10. Nodal curves

Corollary (8.24). Let Z, W be schemes or analytic spaces over S, and let f : Z → W be a proper S-morphism whose fibers have dimension at most equal to one. Let G be a coherent sheaf on Z, flat over S. Consider the following assumptions on G: a) H 1 (f −1 (w), G ⊗OW k(w)) = 0 for each (closed) point w of W ; b) G ⊗OW k(w) is generated by global sections for each (closed) point w ∈ W. If a) holds, then: i) R1 f∗ G = 0; ii) f∗ G is OS -flat; iii) for any morphism α : T → S, there is a canonical isomorphism f∗ G ⊗OS OT → (f × id)∗ (G ⊗OS OT ) ; If, in addition, b) is also satisfied, then iv) f ∗ f∗ G → G is onto. We shall give the proof only for schemes, leaving to the reader the task of adapting it to the analytic context. Property i) follows immediately from part i) of Lemma (8.22). Without loss of generality, we may suppose that S and W are affine. Choose a finite cover U of Z with affine open sets, and denote by C q (U, G) the sheaf of alternating q-cochains with values in G, relative to the cover U; thus, C • (U, G) is a finite resolution of G. Then f∗ C • (U, G) is a resolution of f∗ G, because of i) and since Ri f∗ G vanishes when i > 1 for dimension reasons. Since each f∗ C q (U, G) is a finite direct sum of OS -flat sheaves, it follows that f∗ G is OS -flat. In proving iii), we may suppose that T is affine. Denote by V the cover of Z ×S T consisting of the affine open sets of the form U ×S T , where U belongs to U. Then C • (V, G ⊗OS OT ) = C • (U, G) ⊗OS OT ; (f × id)∗ C • (V, G ⊗OS OT ) = (f∗ C • (U, G)) ⊗OS OT . The left-hand side of the second equality is a resolution of (f ×id)∗ (G ⊗OS OT ), while the right-hand side is a resolution of f∗ G ⊗OS OT by flatness. Property iii) follows. Now suppose that G ⊗OW k(w) is generated by global sections for each w ∈ W . This assumption, together with part ii) of Lemma (8.22), implies that f ∗ f∗ G → G ⊗OZ k(z) is onto for any z ∈ Z. Property iv) then follows from Nakayama’s lemma, and the proof of the corollary is complete. We will also need this elementary result.

§8 Stabilization

141

Lemma (8.25). Let f : X → S be a morphism of schemes or analytic spaces, let E and F be coherent sheaves on X, and let α : E → F be an OX -module homomorphism. Suppose that, for each (closed) point s ∈ S, the induced homomorphism E ⊗OS k(s) → F ⊗OS k(s) is an isomorphism. If F is OS -flat, α is an isomorphism. Proof. Let Q be the cokernel of α. The sequence E ⊗OS k(s) → F ⊗OS k(s) → Q ⊗OS k(s) → 0 is exact, and hence Qx = ms Qx for any (closed) point x above s, where ms indicates the maximal ideal in OS,s . It follows that Qx = mx Qx for any (closed) point x of X and hence that Q is zero by Nakayama’s lemma. Now let K be the kernel of α. By the flatness assumption, O Tor1 S,s (Fx , k(s)) = 0 for any x above s, and hence, 0 → K ⊗OS k(s) → E ⊗OS k(s) → F ⊗OS k(s) → 0 is exact. Arguing as above, it follows that K vanishes, which proves the lemma. Q.E.D. Proof of Theorem (8.21). It follows from the uniqueness of the stable model (cf. Proposition (6.7)) that applying stabilization and then contraction to an object in C g,P produces another object which is canonically isomorphic to the original one. Conversely, suppose that we are given a family of stable (P ∪ {q})-pointed curves consisting of a family ψ:Y →S of nodal curves plus sections τt , t ∈ P ∪ {q}. Applying the qth contraction yields a family of stable P -pointed curves consisting of a family ϕ:X→S and sections σp , p ∈ P , plus an extra section δ coming from τq . We may then apply to this setup the stabilization procedure and get a new family of stable (P ∪ {q})-pointed curves consisting of a family ϕ : X → S of nodal curves plus sections σt , t ∈ P ∪ {q}. We shall show that there is a unique isomorphism γ : Y → X making the diagram

(8.26)

Y[ ] [ π

γ

X

w X π

142

10. Nodal curves

commute, where π and π are the collapsing morphisms, and carrying section τt to σt for each t. In the proof we shall freely use the notation introduced earlier in this section while constructing X → X. Several simple, but somewhat tedious, verifications will be left to the reader. We begin by proving existence. The sheaf OY (− t∈P τt ) satisfies assumption a) in Corollary (8.24) relative to the morphism π. In fact, the positive-dimensional fibers of π are projective lines, and each one of them meets at most one of the τt with t ∈ P . It then follows from Corollary (8.24) that π∗ OY (− t∈P τt ) is S-flat and that its formation commutes with base change; in particular, its pullback to any fiber Xs = ϕ−1 (s) is the pushforward via Ys → Xs of the pullback of OY (− t∈P τt ) to Ys . By direct inspection, one easily checks that the restriction to Xs of the pullback homomorphism (8.27) OX (− σt ) → π∗ OY (− τt ) t∈P

t∈P

is an isomorphism for any s. By the S-flatness of π∗ OY (− t∈P τt ), it then follows from Lemma (8.25) that (8.27) is an isomorphism. Pullback via π gives a homomorphism I → π∗ OY (−τq ), and hence a pairing π∗ OY (τq − τt ) ⊗ I → π∗ OY (τq − τt ) ⊗ π∗ OY (−τq ) t∈P t∈P → π∗ OY (− τt ) ∼ = OX (− σt ) t∈P

t∈P

i.e., a homomorphism π∗ OY (τq −

(8.28)

τt ) → I ∨ (−

t∈P

σt ).

t∈P

We leave to the reader the straightforward task of checking, using the explicit presentation of I ∨ given by Lemma (8.10), that the restriction of this homomorphism to each fiber Xs is an isomorphism. Since I ∨ is S-flat by Lemma (8.19), Lemma (8.25) then implies that (8.28) is an isomorphism. The sequence u v → OY (τq − τ t ) ⊕ OY − → OY (τq ) → 0 , (8.29) 0 → OY (− τt ) − t∈P

t∈P

where u(a) = (a, a) and v(a, b) = a − b, is exact. By Corollary (8.24), R1 π∗ OY (− t∈P τt ) vanishes, and hence the pushforward via π of (8.29) is also exact. We thus get a commutative diagram with exact rows 0 w OX (− σt ) w I ∨ (− σt ) ⊕ OX w K(− σt ) w0 t∈P

0

∼ = u w π∗ OY (− τt ) t∈P

t∈P

∼ = u w π∗ OY (τq − τt ) ⊕ π∗ OY t∈P

t∈P

w π∗ OY (τq )

w0

§9 Vanishing cycles and the Picard–Lefschetz transformation

143

whence an isomorphism K(− t∈P σt ) → π∗ OY (τq ). Pulling this back to Y , we get a homomorphism

σt ) → π ∗ π∗ OY (τq ) → OY (τq ) (8.30) π ∗ K(− t∈P

This homomorphism is onto since OY (τq ) satisfies the assumptions of Corollary (8.24). On the other hand, X = Proj(⊕k≥0 Symk K) ∼ = Proj(⊕k≥0 Symk K ) , where K = K(− t∈P σt ). Hence (8.30) defines an X-morphism γ : Y → X . The induced morphism Ys → Xs is an isomorphism for any s ∈ S, and hence γ is an isomorphism by Lemma (5.7). Clearly, it maps τt to σt for each t ∈ P ; we leave it to the reader to check that it carries τq to σq . Having proved the existence, we turn to the uniqueness. We must show that, if ψ is an automorphism of Y over X carrying each τt to itself, then ψ is the identity. This is certainly true set-theoretically. In fact, the fibers of Y → X which do not consist of a single point are projective lines, and ψ fixes three points on each of them. That ψ is the identity morphism then follows Lemma (5.5). Q.E.D. 9.

Vanishing cycles and the Picard–Lefschetz transformation.

In this section we shall discuss the topology of families of curves. Suppose that we are given a family of nodal curves parameterized by a complex space and that the general fiber of the family is smooth. We shall be concerned with the following question, which is the central one of Picard–Lefschetz theory: how is the topology of the family of smooth curves reflected in the nature of the singular fibers? For our purposes, this is a very relevant question. In fact, in the next chapters, when compactifying moduli spaces, we will often be able to analyze the nature of boundary points (e.g., their smoothness) only by “going around them.” The fundamental notion in this circle of ideas is the one of Picard– Lefschetz transformation, which we now introduce. Let (9.1)

α:C→B

σi : B → C, i = 1, . . . , n,

be a family of smooth n-pointed genus g curves parameterized by a complex space B. Let (9.2)

η : I = [0, 1] → B

be a path, and set Ct = α−1 (η(t)), ρi (t) = σi (η(t)). Consider the pullback of C to I and take any topological trivialization of it (9.3)

F : η∗ C −→ C0 × I

144

10. Nodal curves

as a family of n-pointed curves, that is, a trivialization which maps the sections ρi to constant sections. For each choice of r, s ∈ I, this trivialization defines a homeomorphism Fr,s : Cr → Cs between the fibers of η ∗ C at r and s carrying ρi (r) to ρi (s) for every i. In particular, we get a homeomorphism F0,1 : C0 → C1 . The isotopy class of F0,1 relative to (ρ1 (0), . . . , ρn (0)), that is, the class modulo isotopies which do not move the image of ρi (0) for every i, is independent of the choice of trivialization. In fact, if G is another trivialization of η ∗ C, an isotopy between F0,1 and G0,1 is, for instance, Ht = Ft,1 ◦ G1,t ◦ G0,1 . The isotopy class of F0,1 relative to (ρ1 (0), . . . , ρn (0)) is also independent of the path η, within each homotopy class with fixed endpoints. To see this, let ϑ : I × I → B be a homotopy with fixed endpoints between η and another path ξ; thus, ϑ(t, 0) = η(t), and ϑ(t, 1) = ξ(t). We can trivialize the pullback of C via ϑ as a family of n-pointed curves; by restriction this induces trivializations of the pullbacks via η and ξ. Trivializing ϑ∗ C determines a homeomorphism M(t1 ,s1 ),(t2 ,s2 ) : α−1 (ϑ(t1 , s1 )) → α−1 (ϑ(t2 , s2 )) for any pair of points (t1 , s1 ) and (t2 , s2 ) of I × I. Then Kt = M(t,1),(1,1) ◦ M(t,0),(t,1) ◦ M(0,0),(t,0) is an isotopy between M(0,1),(1,1) ◦ M(0,0),(0,1) and M(1,0),(1,1) ◦ M(0,0),(1,0) . On the other hand, by the independence on the choice of trivialization, M(0,0),(0,1) and M(1,0),(1,1) are, respectively, isotopic to the identity on C0 and to the identity on C1 . Thus, M(0,1),(1,1) and M(0,0),(1,0) , which are, respectively, the homeomorphism from C0 to C1 given by the trivialization of ξ ∗ C and the one given by the trivialization of η ∗ C, are isotopic. The most important instance of the above construction is the following. Let b0 be a point of B, let η be a loop based at b0 , and set C = α−1 (b0 ), pi = σi (b0 ). Then the construction associates to the homotopy class with fixed endpoints of η the isotopy class relative to (p1 , . . . , pn ) of an oriented homeomorphism of (C; p1 , . . . , pn ) to itself, the so-called Picard–Lefschetz transformation associated to η. To put this into perspective, we introduce the notion of mapping class group. If S is a compact, connected, and oriented topological surface, uller modular group, is the the mapping class group ΓS , also called Teichm¨ group of all isotopy classes of orientation-preserving homeomorphism of S to itself. More generally, if q1 , . . . , qn are distinct points of S, the mapping class group of (S; q1 , . . . , qn ) is the group of all isotopy classes relative to (q1 , . . . , qn ) of orientation-preserving homeomorphism of (S; q1 , . . . , qn ) to itself, and we denote it by Γ(S;q1 ,...,qn ) , or simply by Γg,n , where g is


145

the genus of S; we shall usually write Γg for Γg,0 . Associating to the homotopy class of a loop based at b0 the corresponding Picard–Lefschetz transformation thus yields a group homomorphism (9.4)

P L : π1 (B, b0 ) → Γ(C;p1 ,...,pn )

called the Picard–Lefschetz representation. If we are given a homomorphism from the mapping class group Γ(C;p1 ,...,pn ) to some other group G, the basic Picard–Lefschetz representation (9.4) induces a representation π1 (B, b0 ) → G, which often also goes under the name of Picard–Lefschetz representation. The premier example of this situation is the one in which G is the group of automorphisms of H1 (C, R), where R is any coefficient group, and ΓC → G sends ϕ to ϕ∗ . We now introduce certain basic elements of the mapping class group, called Dehn twists. Informally, the Dehn twist δc around a smooth simple closed curve c on an oriented smooth surface S is the homeomorphism (well defined up to isotopy) obtained by choosing an orientation on c, cutting the surface S along c, rotating the right edge c of c by 180o in the positive direction, rotating the left edge c of c by 180o in the negative direction, and gluing the two edges together again.

Figure 8. A Dehn twist In formulas, δc can be defined as follows. Choose a tubular neighborhood U of c, and assume that it is parameterized by a positively oriented system of smooth coordinates y, ϑ, where −1 < y < 1, c is the locus y = 0, and the parameterization is periodic of period 2π in ϑ. Let ψ be a nonincreasing C ∞ function such that ψ(x) = 0 for all x ≤ −1/2, ψ(0) = −1, and ψ(x) = −2 for all x ≥ 1/2. Then δc is defined to be the identity outside of U and (9.5)

(y, ϑ) → (y, ϑ + ψ(y)π)

inside U . The definition of δc is clearly independent of the orientation we choose on c. It follows from formula (9.5) that δc is a diffeomorphism.

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It is easy to check that the smooth isotopy class of δc does not depend on the choices made (i.e., on the choice of a tubular neighborhood, of its coordinatization, and of a function ψ). It is also important to notice, and equally easy to check, that if c is smoothly isotopic to another smooth simple closed curve c˜, then the Dehn twists attached to c and c˜ are smoothly isotopic. For details, we refer the reader to Chapter XV, where a more thorough discussion of the mapping class group can be found. It is useful to record how Dehn twists act on homology. Looking at Fig. 8, it is immediate to see that, for a Dehn twist δc , the homomorphism δc ∗ : H1 (S, Z) → H1 (S, Z) is given by (9.6)

δc ∗ (d) = d + (d · c)c ,

where (d · c) is the intersection number of the two cycles d and c. Dehn twists and Picard–Lefschetz transformation are intimately related through the concept of vanishing cycle, which we now introduce informally. The essential picture to have in mind is the one of a oneparameter family of smooth curves degenerating to a curve with a node p. Let π : X → Δ be such a family, where Δ = {t ∈ C : |t| < R}, and the singular fiber is the one over the origin. We assume that X is smooth.

Figure 9. The fundamental topological observations about this setup, whose proofs will be quickly given below, are the following. First of all, the central fiber X0 is a deformation retract of X. Composing the retraction with the inclusion Xt0 → X, where t0 = 0, gives a map rt0 : Xt0 → X0 , and one can in fact set things up so that rt0 is a homeomorphism everywhere except along a smoothly embedded circle c which gets collapsed to the node of X0 ; one refers to c (or to any simple closed curve isotopic to it) as the vanishing cycle on Xt0 , relative to the given family. The second basic observation is that the Picard–Lefschetz representation (9.7)

P L : π1 (Δ {0}, t0 ) → ΓXt0


147

sends the positively oriented generator to the Dehn twist δc along the vanishing cycle. If X0 is stable, or even just semistable, the vanishing cycle is homotopically nontrivial in Xt0 , and this easily implies that δc is also nontrivial. A remarkable consequence is that the restriction of π : X → Δ to the punctured disk is topologically not a product family. Put otherwise, in general one cannot get rid of singular fibers because of purely topological reasons. We now proceed to prove the two claims above. In proving the existence of a deformation retraction onto X0 notice, first of all, that we may shrink Δ at will, since the family is differentiably locally trivial away from the central fiber, where all topological complications occur. We may then assume, possibly after shrinking the base of the family and rescaling the t coordinate, that there is an open neighborhood V of the node of the central fiber which is biholomorphic to {(x, y) ∈ C2 : |x| < 2, |y| < 2} and that π is locally given by t = xy. Set U = {(x, y) ∈ V : |xy| < 1, |x|2 − |y|2 < 1}. The locus |x| = |y| is fibered in circles above Δ, except over the origin, where the fiber is a single point. It will turn that these shrinking circles are the vanishing cycles in their respective fibers; the name “vanishing cycle” originates from this picture. The complement of |x| = |y| in U decomposes in the two connected components U+ = {|x| > |y|} and U− = {|y| > |x|}. We shall express each one of these as a product of {|t| < 1} with a suitable open set in the central fiber. Explicitly, a diffeomorphism between U+ ∪ U− and (X0 ∩ (U+ ∪ U− )) × {|t| < 1} is given by the prescription x 2 2 (|x| − |y| ), 0, xy when (x, y) ∈ U+ , (x, y) → |x| (9.8) y (x, y) → 0, (|y|2 − |x|2 ), xy when (x, y) ∈ U− . |y| An inverse to the first of these transformations is given by (ξ, 0, t) → (x, y) , where

(9.9)

|ξ| + |ξ|2 + 4|t|2 ξ , x= |ξ| 2 y = t/x ,

√ and a similar expression is valid for the second one. Write t = σ + −1τ . The product representation (9.8) provides canonical liftings u0 and v0 of

148

10. Nodal curves

the vector fields ∂/∂σ and ∂/∂τ . By a partition of unity argument, we can in fact find liftings u and v to all of X which agree with u0 and v0 , respectively, on a neighborhood of {|x| = |y|}. Integrating u and v yields a diffeomorphism H : (X {|x| = |y|}) → (X0 {p}) × {|t| < 1} of fiber spaces over {|t| < 1}. The trivialization we just described gives in particular a deformation retraction {hs }s∈[0,1] of X {|x| = |y|} to X0 {p} lifting the radial deformation retraction of Δ to the origin. As one can easily check using the coordinate change (9.8) and its inverse (9.9), the retraction can √ be continuously extended to all of X by setting hs (x, y) = s(x, y) for |x| = |y|. This concludes the proof that X0 is a deformation retract of X. We now show that the Picard–Lefschetz transformation attached to the positively oriented generator of π1 (Δ {0}, t0 ) is just the Dehn twist δc along the vanishing cycle. It is clear that we may assume that t0 is as close to 0 as we wish. Taking as generator of π1 (Δ {0}, t0 ) the loop λ(θ) = t0 e2π

√ −1θ

,

θ ∈ I = [0, 1] ,

we shall describe, for each θ ∈ I, a specific homeomorphism kθ , depending continuously on θ, between Xt0 and Xλ(θ) . To this end, we choose an odd nondecreasing C ∞ function χ such that χ(s) = −1 for all s ≤ −1/2 and χ(s) = 1 for all s ≥ 1/2. Outside the region U , kθ is simply defined to be the map given by the trivialization H described above. Inside U , instead, we set √ √ (9.10) kθ (x, y) = e −1π(1+χ(s))θ x, e −1π(1−χ(s))θ y , where s is essentially the logarithm of |x|, or, more exactly, log(|x|/ |t0 |) s= . log(1/ |t0 |) In terms of the coordinates s and ϕ = arg x, the map k1 is given by (9.11)

k1 (s, ϕ) = (s, ϕ + (1 + χ(s))π) ,

while it is the identity away from the vicinity of the vanishing cycle. By formula (9.5) in Chapter XV, this says that k1 is the Dehn twist associated to the vanishing cycle c, as was to be shown. Another useful way of visualizing a curve acquiring a node and the corresponding vanishing cycle, is to look at the map ϕ : Vt = {(x, y, t) : xy = t, |x| < 1, |y| < 1} −→ {z : |z| < 2}, (x, y, t) −→ x + y,


149

which √ exhibits Vt as a two-sheeted ramified cover branched at the points ±2 t. In fact, ϕ is the quotient map via the involution (x, y, t) → (y, x, t). The vanishing √ is the preimage, under the covering map, of the √ cycle segment [−2 t, 2 t]. This segment, and hence the vanishing cycle above it, shrinks to a point as t → 0. As we observed, the nontriviality of the Picard–Lefschetz representation makes it impossible to replace the central fiber of π : X → Δ with a topological surface. On the other hand, nothing prevents us from performing this operation if we restrict to a real ray issuing from the origin. The new fiber over the origin is obtained by normalizing at the node, substituting each of the points mapping to the node with a circle and gluing the two circles together; the specific gluing is dictated by the direction of the ray. This dependence on the ray is the obstruction to performing the construction over all of Δ. On the other hand, this observation suggests that the obstruction disappears if one substitutes the origin of Δ with a circle parameterizing all real directions through it. This calls for a digression on real blow-ups, which will come in handy later on, when we will need to work with a bordification of Teichm¨ uller space while presenting Kontsevich’s proof of the Witten conjecture. Real blow-ups. We only need to introduce real blow-ups in a special situation. Let X be a complex manifold, and let D ⊂ X be a reduced divisor with normal crossings. As the reader will recall, this means that, for any point p of X, there are a neighborhood U of p and local coordinates t1 , . . . , tn on U , centered at p, such that D ∩ U is of the form t1 t2 · · · tk = 0, with k possibly equal to 0. We are going to define the real (oriented) blow-up of X along D, which we will denote BlD (X). We shall do the construction locally and then show how the various local patches fit together. Let p, U , and t1 , . . . , tn be as above. We then define BlD (U ) ⊂ U × (S 1 )k to be the locus BlD (U ) = {(x, τ1 , . . . , τk ) : ti (x) = |ti (x)|τi , i = 1, . . . , k} , where we view S 1 as the unit circle in the complex plane. Equivalently, one can describe BlD (U ) as the region Re(ti τi−1 ) ≥ 0 ,

i = 1, . . . , k,

inside the locus with equations (9.12)

Im(ti τi−1 ) = 0 ,

i = 1, . . . , k .

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10. Nodal curves

Using the Jacobian criterion, it is easy to check that equations (9.12) define a real-analytic submanifold of U × (S 1 )k and that the projection πU : BlD (U ) → U −1 is a real-analytic isomorphism away from πU (D ∩ U ). The structure of BlD (U ) can be made even more explicit if U is the polycylinder {|ti | < ε, i = 1, . . . , n}; we can always reduce to this case by shrinking U . Then BlD (U ) is real-analytically isomorphic to the product of the real k-torus (S 1 )k , of k copies of the half-closed interval [0, ε), and n − k ε-disks. In fact, writing Δε for the ε-disk centered at the origin of the complex plane,

−→ BlD (U ), (S 1 )k × [0, ε)k × Δn−k ε (τ1 , . . . , τk , r1 , . . . , rk , tk+1 , . . . , tn ) → (r1 τ1 , . . . , rk τk , tk+1 , . . . , tn , τ1 , . . . , τk ), is a real-analytic isomorphism whose inverse is BlD (U ) −→ (S 1 )k × [0, ε)k × Δn−k , ε (t1 , . . . , tn , τ1 , . . . , τk ) → (τ1 , . . . , τk , |t1 |, . . . , |tk |, tk+1 , . . . , tn ) . Notice that these isomorphisms are compatible with the natural projection BlD (U ) → U = Δnε and with the “polar coordinates” map (S 1 )k × [0, ε)k × Δn−k → Δnε given by (τ1 , . . . , τk , r1 , . . . , rk , tk+1 , . . . , tn ) → ε (r1 τ1 , . . . , rk τk , tk+1 , . . . , tn ). It is clear from this description that the fiber of πU at a point q of U is an h-dimensional real torus, where h is the number of coordinates ti , i = 1, . . . , k, vanishing at q. In particular, the fiber over the central point p is a k-dimensional real torus (9.13)

−1 T = πU (p) ∼ = (S 1 )k .

We next check the independence of the definition of real blow-up on the choice of coordinates. Let q be a point of U , and V ⊂ U a neighborhood of q. Pick coordinates s1 , . . . , sn on V in such a way that D ∩ V = {s1 · · · sh = 0}, where h ≤ k. Possibly after renumbering, there are nowhere vanishing holomorphic functions a1 , . . . , ah on V such that si = ai ti , i = 1, . . . , h. We define a real-analytic map α : BlD (V ) → BlD (U ) by setting tk (x) |a1 (x)| |ah (x)| th+1 (x) . σ1 , . . . , σh , ,..., α(x, σ1 , . . . , σh ) = x, a1 (x) ah (x) |th+1 (x)| |tk (x)| Clearly, πU ◦α = πV , and α is the only continuous map with this property. Moreover, α has a real-analytic inverse on α(BlD (V )) given by a1 (x) ah (x) (x, τ1 , . . . , τk ) → x, τ1 , . . . , τh . |a1 (x)| |ah (x)|


151

The existence and uniqueness of α imply that the various local blow-ups BlD (U ) patch together to yield a well-defined real-analytic manifold with corners BlD (X), together with a projection π : BlD (X) → X. It is clear from the construction that the fiber of π at a point x ∈ X is a real k-dimensional torus, where k is the number of components of the germ of D at x. Let B be the polydisc {(t1 , . . . , tn ) ∈ Cn : |ti | < ε, i = 1, . . . , n}, and let D be the divisor t1 · · · tk = 0. Using polar coordinates, it is easy to of BlD (B). In fact, describe the universal cover B = {(r1 , . . . , rk , ϑ1 , . . . , ϑk , tk+1 , . . . , tn ) ∈ R2k ×Cn−k : 0 ≤ ri ≤ ε, |ti | ≤ ε} , B → BlD (B) is and the covering map ρ : B (r1 , . . . , rk , ϑ1 , . . . , ϑk , tk+1 , . . . , tn ) → (r1 e2π

√ −1ϑ1

, . . . , rk e2π

√ −1ϑk

, tk+1 , . . . , tn , e2π

√ −1ϑ1

, . . . , e2π

√ −1ϑk

).

Having dealt with real blow-ups, we turn to families of nodal curves. First, we consider the case of a single nodal curve C of genus g, and we let Csing = {p1 , . . . , pk } . We denote by ν : N → C the normalization map, and we set ν −1 (pi ) = {ri , si } , i = 1, . . . , k

and

R = {r1 , s1 , . . . , rk , sk } .

Consider the real oriented blow-up of N along R, τ : BlR (N ) −→ N. By definition, the fiber of τ over a point r ∈ R is a copy of S 1 . Hence, BlR (N ) is a, possibly disconnected, Riemann surface with boundary (see Fig. 10 below). We may construct an oriented topological surface Σ of genus g by giving, for each point y ∈ Csing , an identification between the two boundary components τ −1 (r) ∼ = S 1 and τ −1 (s) ∼ = S 1 , where −1 {r, s} = ν (y). Of course, in order to get an oriented surface, this identification should carry the orientation of τ −1 (r) into the opposite orientation of τ −1 (s). By construction, the surface Σ comes equipped with a system of simple closed curves {γ1 , . . . , γk }. We denote by h : BlR (N ) → Σ the quotient map and by (9.14)

ξ:Σ→C

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10. Nodal curves

the natural contraction map defined by the property: ξh = ντ .

Figure 10. Given a family f : X → B of nodal curves, we would like to perform the same operation simultaneously on all singular fibers of f . We should of course expect this process to destroy complex analyticity. Worse still, since the gluing of the circles τ −1 (r) and τ −1 (s) is noncanonical, we will be forced to base-change from B to BlD (B). In carrying out this procedure, we shall limit ourselves to families f : X → B which are transverse, in the following sense. Let b be any point of B, and let p1 , . . . , pk be the nodes of the fiber f −1 (b). Locally near pi , the family is of the form xy = ti , where ti is a function on a neighborhood of b. When the functions t1 , . . . , tk are part of a system of local coordinates t1 , . . . , t on a neighborhood of b, we shall say that the family f : X → B is transverse at b and that t1 , . . . , t is a standard system of parameters. A family will be said to be transverse if it is transverse at every point of the base. Notice that the base and the general fiber of a transverse family are automatically smooth. As we shall see, when we discuss moduli spaces of stable curves in Chapter XII, the attribute “transverse” is justified by the fact that families of this sort give rise to “subvarieties” of moduli space which are transverse to all strata of the stratification of moduli space by topological type. Let then f : X → B be a transverse family of nodal curves. We denote by D the locus in B parameterizing singular curves. Clearly, in a neighborhood of a point b ∈ B, a local equation for D is t1 · · · tk = 0, where t1 , · · · , t is a standard system of parameters at b. Hence D is a divisor with normal crossings. We will define a new family h : Z → BlD (B)


153

of compact orientable differentiable surfaces which agrees with f away from D. As was the case for real blow-ups, it will suffice to do the construction locally on B; we leave it to the reader to check that the local families we obtain patch together. We then assume that B is the polydisc {(t1 , . . . , t ) : |ti | < ε2 , i = 1, . . . , }, that the fiber f −1 (0) has k nodes p1 , . . . , pk , that the family is of the form xy = ti in a neighborhood Ui of pi , and that it is smooth outside the Ui . Thus D is the locus t1 · · · tk = 0. We shall construct Z → BlD (B) by suitably modifying the family f : X → BlD (B) obtained by pullback from f : X → B. To keep notation simple, we will assume that k = 1, and we will write t for t1 and p for p1 . The real blow-up of B is the locus in B × S 1 with equation t = |t|τ . On the other hand, X is locally isomorphic, near p, to the locus with equation xy = t in C2 × B, and hence X is locally isomorphic to a neighborhood of {x = y = 0} in

+1 1 xy = |xy|τ, |x| < ε, ×S W = (x, y, t2 , . . . , t , τ ) ∈ C , |y| < ε, |ti | < ε2

whereas the projection of X to BlD (B) can be identified with (x, y, t2 , . . . , t , τ ) → (xy, t2 , . . . , t , τ ) . Now look at (9.15) W =

(x, y, t2 , . . . , t , ξ, η) ∈ C

+1

x = |x|ξ, y = |y|η, . × (S ) |x| < ε, |y| < ε, |ti | < ε2 1 2

Since |xy|ξη = xy, (x, y, t2 , . . . , t , ξ, η) → (x, y, t2 , . . . , t , ξη) defines a map β from W to W . The map is one-to-one except where x = y = 0; the fiber above (0, 0, t2 , . . . , t , τ ) is a circle, namely the locus of those points (0, 0, t2 , . . . , t , ξ, η) such that ξη = τ . To construct Z, we just graft onto X {x = y = 0}, via the attaching map β|{x =0,y =0} , a neighborhood of the locus {x = y = 0} in W . This procedure does to a singular fiber of f exactly what we did above to a single nodal curve. First one normalizes the fiber at the node x = y = 0, then one performs real blow-ups at the two resulting points of the normalized curve, and finally one identifies the two circles thus obtained, keeping track of orientations; how the identification is made depends on the value of the angle parameter τ . When the number of nodes of f −1 (0) is greater than 1, all one has to do to construct h : Z → BlD (B) is to perform the procedure described above at every node. By construction, the fibers of h are all homeomorphic topological surfaces.

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10. Nodal curves

The composite map W → W → BlD (B) is not a smooth fibration, but we are going to show that one can put on Z a structure of realanalytic manifold with corners in such a way that Z → BlD (B) is a real-analytic fibration. While this is not strictly necessary in the sequel, it does simplify certain proofs. In the considerations that follow, for the sake of simplicity, we ignore the variables t2 , . . . , t , which behave as harmless parameters. In Z {x = y = 0}, we keep the real-analytic structure coming from the one of X . Inside W , we proceed as follows. Recall that, in our case, x = |x|ξ, y = |y|η W = (x, y, ξ, η) ⊂ C2 × (S 1 )2 . |x| < ε, |y| < ε In other words, W is just the product of two blown-up disks. Set V = {(s, r, ξ, η) : s, r ∈ R, ξ, η ∈ S 1 , 0 ≤ r < ε2 , |s| < ε − r/ε} . Define the map ϕ : W → V by (x, y, ξ, η) → (|x| − |y|, |xy|, ξ, η) and the map ψ : V → W by (s, r, ξ, η) →

s+

! √ √ s2 + 4r −s + s2 + 4r ξ, η, ξ, η . 2 2

We leave to the reader the easy verification that these maps are inverse to each other. We transplant to W the real-analytic structure of V , and we check that it is compatible with the one of Z {x = y = 0}. One immediately sees that the map rη x (x, r, η) → x, , ,η |x| |x| gives a real-analytic diffeomorphism from U = {(x, r, η) ∈ C × R × S 1 : 0 ≤ r < ε2 , r/ε < |x| < ε} to W {x = 0}, endowed with the real-analytic structure induced by the one of Z {x = y = 0}. It is equally easy to see that the two maps x r , r, ,η , (x, r, η) → |x| − |x| |x| ! √ 2 s + s + 4r (s, r, ξ, η) → ξ, r, η 2


155

between U and V ϕ({x = 0}) are inverse of each other. They are also real-analytic. In fact, the only point to worry about is the possibility of s2 + 4r vanishing; but this occurs only when s = r = 0, that is, for x = y = 0. One similarly deals with the open set W {y = 0}. This completes the construction of the real-analytic structure on Z. It is obvious from the definitions that the map h : Z → BlD (B) is a realanalytic fibration. Its fibers are compact orientable real-analytic surfaces. We may summarize much of what has been proved in the following statement. Proposition (9.16). Let f : X → B be a transverse family of nodal curves, and let D ⊂ B be the locus parameterizing singular fibers. Then there are a real-analytic fibration h : Z → BlD (B) and a surjective continuous map λ : Z → X ×B BlD (B) such that the diagram Z h

λ

w X ×B BlD (B)

wX f

u

BlD (B)

id

u

w BlD (B)

π

u wB

is commutative. Furthermore, denoting by Σ ⊂ X the locus of singular points in the fibers of f , the map λ restricts to a realanalytic diffeomorphism Z λ−1 (Σ ×B BlD (B)) → X Σ ×B BlD (B), and λ−1 (Σ ×B BlD (B)) is a real-analytic submanifold of Z, fibered in circles over Σ ×B BlD (B). Remark (9.17). In fact, the construction of h gives more than (9.16) says. It is particularly important to keep track of the detailed structure of Z above the loci of the form π−1 (b0 ), where b0 is a point of B. Of course, this is of interest only when b0 belongs to D, that is, when Xb0 = f −1 (b0 ) is singular. Let p1 , . . . , pk be the nodes of Xb0 , and let xi , yi be local coordinates on the two branches of Xb0 at pi , centered at the singular point. The locus π −1 (b0 ) is a real k-torus and can be viewed as the set of k-tuples τ = (τ1 , . . . , τk ) of complex numbers of absolute value one. The fiber h−1 (τ1 , . . . , τk ) is constructed as follows. The first step is to normalize Xb0 to obtain a smooth curve Y . We then perform real blow-ups at all point of Y mapping to nodes of Xb0 . In other words, we replace ε-neighborhoods of the origin in the coordinates xi and yi with the manifolds with boundary Mi = {(xi , ξi ) ∈ C2 : |xi | < ε, |ξi | = 1, xi = |xi |ξi } and Ni = {(yi , ηi ) ∈ C2 : |yi | < ε, |ηi | = 1, yi = |yi |ηi }, respectively. The final step is to glue together the boundaries of Mi and Ni , for each i, by identifying (0, ξi ) and (0, ηi ) when ξi ηi = τi . One thus obtains a topological surface Yτ . We denote by Ai the union of the images in Yτ of Mi and Ni , and by γi the common image of their boundaries, a circle. Loosely speaking, one can say that Yτ has been obtained from Xb0 by replacing the node pi with the circle γi for each i. The fiber of h at

156

10. Nodal curves

the point of π −1 (b0 ) corresponding to τ is Yτ , endowed with a suitable real-analytic structure. This structure coincides with the one of Xb0 away from the circles γi , that is, from the point of view of Xb0 , away from the nodes. In Ai , the real-analytic structure is obtained by transplanting the natural one of (−ε, ε) × S 1 via the homeomorphism (−ε, ε) × S 1 → Ai given by (s, ρ) → (sρ, ρ) ∈ Mi (s, ρ) → (−sτi ρ−1 , τi ρ−1 ) ∈ Ni

when s ≥ 0, when s ≤ 0.

Corollary (9.18). Let f : X → B, Z, h, and π be as in (9.16). Let b0 be a point of B, set T = π −1 (b0 ), and let k and be, respectively, the number of nodes of f −1 (b0 ) and the dimension of B at b0 . Denote by Δ the unit disk in the complex plane. Then there are arbitrarily small open neighborhoods U of b0 such that there exist diffeomorphisms α : π−1 (U ) → T × [0, 1)k × Δ −k and β : (π ◦ h)−1 (U ) → h−1 (T) × [0, 1)k × Δ −k with the property that the diagram (π ◦ h)−1 (U ) h u

π −1 (U )

β

α

w h−1 (T) × [0, 1)k × Δ −k h × id × id u w T × [0, 1)k × Δ −k

is commutative. In particular, there are compatible deformation retractions or π −1 (U ) to T and of (π ◦ h)−1 (U ) to h−1 (T). Furthermore, α, β, and the retractions can be chosen so that the latter descend to compatible deformation retractions of U to b0 and of f −1 (U ) to the fiber f −1 (b0 ). Finally, if σ1 , . . . , σn are sections of f : X → B making it into a family of n-pointed nodal curves, the product decomposition of (π ◦ h)−1 (U ) and the retractions can be chosen so as to be compatible with the sections σ1 , . . . , σn . For the proof, we may suppose that B is the polydisc {(t1 , . . . , t ) ∈ C : |ti | < 1 , i = 1, . . . , }, that b0 is the origin of C , and that D is the locus t1 · · · tk = 0. We may also take U = B, so that π −1 (U ) = BlD (B) and (π ◦ h)−1 (U ) = Z. We know that BlD (B) is real-analytically diffeomorphic to the product of T and [0, 1)k × Δ −k , where T is the set of all k-tuples (τ1 , . . . , τk ) of complex numbers of absolute value one. We indicate the ith coordinate in [0, 1)k by ri . As in the statement of (9.16), we denote by Σ the locus of singular points in the fibers of f and by λ the natural map from Z to X ×B BlD (B). Clearly, λ−1 (Σ) decomposes into connected components F1 , . . . , Fk , and Fi is fibered in circles over Ei , the locus in BlD (B) ∼ = T × [0, 1)k × Δ −k defined by the vanishing of ri . The required diffeomorphism between Z and h−1 (T) × [0, 1)k × Δ −k can be obtained by lifting to Z the vector fields on T × [0, 1)k × Δ −k corresponding to differentiation in the ti and ri variables, and successively integrating


157

them. A deformation retraction of Z ∼ = h−1 (T) × [0, 1)k × Δ −k to h−1 (T) is then, for instance, ρs (x, r, t) = (x, sr, st), 0 ≤ s ≤ 1; this lifts a similarly defined deformation retraction of BlD (B) to T, which in turn lifts the radial deformation retraction of B to b0 . The trouble is that, in general, this does not descend to a deformation retraction of X, since ρs does not necessarily map each submanifold Fi to itself. In other words, Fi does not necessarily correspond, in h−1 (T) × [0, 1)k × Δ −k , to the product of a submanifold of h−1 (T) and of the locus ri = 0 in [0, 1)k × Δ −k . The remedy is to be more careful in the choice of the liftings of the coordinate vector fields. More exactly, we choose the liftings of ∂/∂ti and ∂/∂ti to be everywhere tangent to all the Fj , and the lifting of ∂/∂ri to be everywhere tangent to all the Fj except Fi . As the reader will readily verify, this yields a product decomposition of Z in which all the Fi appear as products of submanifolds of the factors. This in turn leads to a deformation retraction of Z to h−1 (T) which descends to one of X to f −1 (b0 ). Finally, to make the product decomposition of (π ◦ h)−1 (U ) and the deformation retractions compatible with the given sections, it suffices to choose the liftings of the coordinate vector field in such a way that they are tangent to the sections. Picard–Lefschetz revisited. We now go back to the Picard–Lefschetz transformation. Our goal is to study it in the context of a general transverse family of nodal curves f : X → B, using the constructions we have just introduced. For future reference, we wish to highlight the main consequence of Corollary (9.18). Lemma (9.19). Let f : X → B be a transverse family of nodal curves. Let b0 be a point of B. Set C0 = f −1 (b0 ). Then there are arbitrary small neighbourhoods U of b0 such that there exist compatible retractions r : U → b0 and R : X|U → C0 . Moreover, if σ1 , . . . , σn are sections of f : X → B making it into a family of n-pointed nodal curves, the retractions can be chosen so as to be compatible with the sections σ1 , . . . , σn . If p1 , . . . , pk are the nodes of C0 = f −1 (b0 ), and C = f −1 (b), b ∈ U , is a smooth fiber, then R−1 (pi ) ∩ C is a circle ci . Going back to Corollary (9.18), an alternative description of ci is as follows. There is a unique point of π −1 (U ) mapping to b. Suppose that, under the diffeomorphism π −1 (U ) → T × [0, 1)k × Δ −k , this point corresponds to (τ1 , . . . , τk , r1 , . . . , rk , tk+1 , . . . , t ). Then the diffeomorphism (π ◦ h)−1 (U ) → h−1 (T)×[0, 1)k ×Δ −k maps ci to γi ×{(r1 , . . . , rk , tk+1 , . . . , t )}, where γi is the fiber over the node pi of the contraction map h−1 (τ1 , . . . , τk , 0, . . . , 0) → f −1 (b0 ) (see also Remark (9.17), where Yτ , τ = (τ1 , . . . , τk ), corresponds to h−1 (τ1 , . . . , τk , 0, . . . , 0)).

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In accordance with the discussion at the beginning of this section, by vanishing cycle on C we mean (any simple closed curve isotopic to) one of the circles c1 , . . . , ck . Occasionally, we will use the words “vanishing cycle” also to designate the homology class of one of the ci . As we know, vanishing cycles and Dehn twists are closely linked. We wish to go over this link again, in the standard situation of a transverse family of nodal curves (9.20)

f : X → B = {(t1 , . . . , t ) : |ti | < 1, i = 1, . . . , },

where t1 , . . . , t is a standard system of parameters. Thus, if p1 , . . . , pk are the singular points of the central fiber X0 , then, near pi , X is of the form xy = ti . The singular fibers of the family are parameterized by D = {t1 · · · tk = 0} ⊂ B, and hence by restriction to B ∗ = B D we get a family of smooth curves X ∗ → B ∗ . We fix a base point b0 ∈ B ∗ , set C = Xb0 , and look at the Picard–Lefschetz representation P L : π1 (B ∗ , b0 ) → ΓC . The fundamental group of B ∗ is abelian and freely generated by k loops λ1 , . . . , λk , where each λi winds simply around the divisor ti = 0 in the positive direction. In other words, using additive notation, π1 (B ∗ , b0 ) = Zλ1 ⊕ · · · ⊕ Zλk . For each singular point pi of the central fiber of f , let ci be the corresponding vanishing cycle in C. Let us show again that the Picard– Lefschetz representation assigns to each λi the Dehn twist around ci , i.e., that P L(λi ) = δci .

(9.21)

Consider the family h : Z → BlD (B) constructed in Proposition (9.16). Corollary (9.18) asserts in particular that, from a real-analytic point of view, this family is just (9.22)

h × id × id : h−1 (T) × [0, 1)k × Δ −k → T × [0, 1)k × Δ −k ,

where T is the k-dimensional real torus which replaces the origin of B in the blow-up. Restricting to T × (0, 1)k × Δ −k , we get back the family X ∗ → B ∗ , at least real-analytically. The point b0 is of the form (τ 1 , . . . , τ k , r1 , . . . , rk , tk+1 , . . . , t ) with ri > 0 for every i. An explicit form of the loop λi is then λi (t) = (τ 1 , . . . , τ i e2π

√

−1t

, . . . , τ k , r1 , . . . , rk , tk+1 , . . . , t ) .


159

In view of the product representation (9.22), to prove (9.21), it suffices to prove its analogue for the real family h−1 (T) → T. Recall that, in the notation of Remark (9.17), the fiber of this family at τ = (τ1 , . . . , τk ) is denoted by Yτ . We take τ = (τ 1 , . . . , τ k ) as base point and consider the loop μi : [0, 1] → T given by μi (t) = (τ 1 , . . . , τ i e2π

√ −1t

, . . . , τ k) .

We shall describe an explicit trivialization of the family over the unit interval obtained from h−1 (T) → T by pullback via μi , by giving explicit diffeomorphisms Ft : Yτ → Yμi (t) depending smoothly on t. By definition, the Picard–Lefschetz transformation corresponding to the Recall from Remark loop μi will be (the isotopy class of) F1 . (9.17) that Yτ is constructed starting from a fixed surface Y with 2k boundary components by identifying the boundary components pairwise; what varies with τ = (τ1 , . . . , τk ) is just how these identifications are made. More specifically, the boundary components have neighborhoods Mj = {(xj , ξj ) ∈ C2 : |xj | < ε, |ξj | = 1, xj = |xj |ξj } and Nj = {(yj , ηj ) ∈ C2 : |yj | < ε, |ηj | = 1, yj = |yj |ηj }, 1 ≤ j ≤ k, which are glued together by identifying (0, ξj ) ∈ Mj and (0, ηj ) ∈ Nj whenever ξj ηj = τj . In our situation, all components of μi (t) are independent of t, except the ith one; thus, all the gluings are independent of t, except the one between Mi and Ni . Hence, we may regard Yμi (t) as being obtained from a fixed curve Y with two boundary components by gluing the neighborhoods Mi and Ni of these as specified above. We are now in a position to describe Ft . We choose an even smooth function χ : (−ε, ε) → R which vanishes outside [−ε/2, ε/2], is identically equal to 1 on a neighborhood of 0, and is nonincreasing for positive values of s. We then define Ft to be the identity outside Mi and Ni , and √ √ (xi , ξi ) → xi eπ −1χ(|xi |)t , ξi eπ −1χ(|xi |)t in Mi , √ √ (yi , ηi ) → yi eπ −1χ(|yi |)t , ηi eπ −1χ(|yi |)t in Ni . This√ is a good√ definition since √it respects √the gluings. In fact, ξi eπ −1χ(|0|)t ηi eπ −1χ(|0|)t = ξi ηi e2π −1t = τ i e2π −1t . In Remark (9.17) we have introduced explicit real-analytic coordinates s, ρ on the open set Ai of Yτ obtained from Mi and Ni by gluing their boundaries. In terms of these coordinates, F1 is given by √ −1χ(s) (s, ρeπ √ ) if s ≥ 0 , (s, ρ) → (s, ρe−π −1χ(s) ) if s ≤ 0 . These formulas transform into (9.5) if we take as ψ the function which equals −χ for negative values of its argument and χ − 2 for positive ones. Therefore, F1 is just the Dehn twist δγi , where γi is the simple closed

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curve in Yτ which is the image of the boundaries of Mi and Ni . On the other hand, as we observed earlier, the curve γi corresponds to the vanishing cycle ci in C = Xb0 under the product decomposition (9.22). This proves (9.21). Our next goal is to compare the homology of a smooth fiber C = Xb0 of the family (9.20) with the one of the central fiber X0 . Let R be any commutative ring with unit. The inclusion j : C → X induces a homomorphism H1 (C, R) → H1 (X, R) = H1 (X0 , R). This map can be identified with r∗ : H1 (C, R) → H1 (X0 , R) ,

(9.23)

where r : C → X0 is the composition of j with the retraction map from X to X0 given by (9.18). The map r is a homeomorphism, except for the fact that it contracts each vanishing cycle ci to the corresponding node pi . Let Γ be the dual graph of X0 . We claim that the homomorphism (9.23) fits into an exact sequence (9.24)

r

∗ → H1 (C, R) −→ H1 (X0 , R) → 0 0 → H 1 (|Γ|, R) −

and that the image of H 1 (|Γ|, R) in H1 (C, R) is the subgroup generated by the homology classes of the vanishing cycles. We prove this by comparing the long exact homology sequences of the pairs (C, ∪ci ) and (X0 , ∪{pi }). Parts of the two sequences give the commutative diagram with exact rows H2 (C, ∪ci ) ∂ w H1 (∪ci ) u 0

w H1 (C) r u ∗ w H1 (X0 )

w H1 (C, ∪ci )

w H0 (∪ci )

w H1 (X0 , ∪{pi })

w H0 (∪{pi })

from which one immediately sees that r∗ is onto and that H1 (∪ci ) maps surjectively onto its kernel. All that remains to be done is to identify the cokernel of ∂ with H 1 (|Γ|). Now, H2 (C, ∪ci ) is freely generated by the fundamental classes of the closures of the connected components of C ∪ci . Denote these components by C1 , C2 , . . . and the respective fundamental classes by [C1 ], [C2 ], and so on. We denote by vi the vertex of Γ corresponding to Ci and by ei the edge corresponding to ci . Choose the generator of H1 (ci ) an orientation on each ci , and denote by [ci ] corresponding to this orientation. Then ∂[Ci ] = εij [cj ], where εij may be 1, –1, or 0. Denote by v1∗ , v2∗ , . . . and by e∗1 , e∗2 , . . . the bases of C 0 (|Γ|) and C 1 (|Γ|) which are dual to v1 , v2 , . . . and e1 , e2 , . . . , respectively. We put an orientation on |Γ| by defining the incidence number [ej : vi ] to equal εij ; this amounts to requiring that vi∗ (∂ej ) = εij . It is now a simple routine check to verify that the assignments [Ci ] → vi∗ and [ci ] → e∗i ∂

give an isomorphism of complexes between H2 (C, ∪ci ) − → H1 (∪ci ) and

§11 Exercises

161

δ

→ C 1 (|Γ|). The construction of (9.24) and the proof of its C 0 (|Γ|) − exactness are now complete. 10. Bibliographical notes and further reading. The notion of stable curve is due to Alan Mayer and David Mumford, and first appeared in the notes of a seminar by Mumford [548], included in the unpublished notes of the 1964 Woods Hole Summer Institute. In the same seminar the stable reduction theorem for curves is stated (Lemma A). A proof of the stable reduction theorem similar to the one given in the present book can be found in Alan Mayer’s seminar notes [505]. As a side remark, we may observe that stable reduction for curves is a key ingredient in de Jong’s theory of alterations [395]. In [398] Oort and de Jong prove the following extension result. Let D be a divisor with normal crossings on a scheme S. Let p : C → U = S D be a family of stable curves with locally constant topological type. Assume that p extends, as a stable curve, to the generic points of D. Then p extends to a family of stable curves over S. The basic source for the foundations of the theory of stable npointed curves, including in particular the key constructions of projection, clutching, and stabilization, is Knudsen’s paper [426]. The classical reference for the Picard–Lefschetz theory is Lefschetz’s book L’analysis situs et la géométrie algébrique [471]. A nice treatment of the theory is given by Clemens in his seminar notes [125] and also by Barth, Hulek, Peters, and Van de Ven in [52]. A thorough treatment of real oriented blow-ups can be found in the paper [373] by Hubbard, Papadopol, and Veselov. 11. Exercises. A.

Stable reduction I.

In the following exercise the reader is led, step by step, through the process of semi-stable reduction for the family of elliptic curves with the affine equation given by (11.1)

y 2 = x3 − t,

where t is the parameter. Equation (11.1) defines a smooth surface. Let C0 : y 2 = x3 be the central fiber of this family. A-1. First reduce the central fiber to a divisor with normal crossing. For this, one does three successive blow-ups at (x, y) = (0, 0) and 0 with normal crossing and four components: gets a central fiber C C0 = C + 2E + 3F + 6G, where C is the proper transform of C0 ,

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and C ∼ = E ∼ = F ∼ = G ∼ = P1 , with E · F = E · C = F · C = 0, G · E = G · F = G · E = 1.

Figure 11. A-2. Set p = C ∩ G, q = F ∩ G, r = E ∩ G. In a neighborhood of q the family looks like t = x6 y 3 . Perform the base change t → t6 to get (t2 − x2 y)(t2 ε − x2 y)(t2 ε2 − x2 y) = 0 with ε a primitive 3rd root of 1. Normalize each one of the three components via (x, y, t) → (v, u2 , ci uv) with i = 1, 2, 3 and ci a suitable constant. The result consists, locally, in 3 copies of a curve with a node. In each copy one branch of the node maps 2–1 over G with simple ramification over q, while the other branch maps, locally, 1–1 to F (see Fig. 11). A-3. With the same base change and normalization, analyze what happens near p and r. Show that, over p, one gets a single node with one branch mapping 6–1 to G with total ramification over p and the other branch mapping 1–1 to C, while, over r, one gets two nodes, and in each of these, one branch maps 3–1 to G with total ramification over r, and the other branch maps 1–1 to E (see Fig. 12).

Figure 12. C, F , E the curves lying over G, C, F, E, respectively. A-4. Denote by G, is a smooth Show that G is a smooth curve of genus 1, that C rational curve, that F is the disjoint union of three smooth rational

§11 Exercises

163

is the disjoint union of two smooth rational curves, and that E curves. If p1 is the point lying over p, if q1 , q2 , q3 are the ones lying over q and r1 , r2 are the ones lying over r, then the central fiber looks like this:

Figure 13. A-5. Show that all the nonsingular fibers of the original family (11.1) are mutually isomorphic. A-6. Show that each rational component in Fig. 13 above has selfintersection equal to –1 in the resulting family. Show that by blowing down these curves one gets a trivial family of elliptic curves. B.

Stable reduction II.

In each of the following problems, we ask you to find the stable limit of a family of curves {Ct } parameterized by a disc or spectrum of a discrete valuation ring with parameter t. B-1. Ct is smooth for t = 0; the total space C is smooth; C0 has a tacnode, e.g., (11.2)

C = {([X, Y, Z], t) ∈ P2 × U : Z 2 Y 2 = X 4 + t(Y 4 + Z 4 )}.

B-2. Ct is smooth for t = 0; the total space C is smooth; C0 has a planar triple point. B-3. Ct is smooth for t = 0; C0 has a spatial triple point. (This is not necessary, but you can assume that the total space C has the minimal singularity possible, an ordinary double point.)

C. Stable reduction III. Here again we ask you to find the limit of families of curves {Ct } parameterized by a disc. The difference is that in the preceding batch in each case the general member of the family was smooth; here the general

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10. Nodal curves

member is singular. One note: problems 5–9 may be easier to do after you have done the following series. C-1. Ct has one node pt for t = 0; C0 has a cusp at p0 = lim pt . C-2. Ct has one node pt for t = 0; C0 has a tacnode at p0 = lim pt . C-3. Ct has one node pt for t = 0; C0 has a planar triple point at p0 = lim pt . C-4. Ct has two nodes pt , qt for t = 0; C0 has a cusp at p0 = lim pt = lim qt . C-5. Ct has three nodes pt , qt , rt for t = 0; p0 = lim pt = lim qt = lim rt .

C0 has a cusp at

C-6. Let B be a smooth curve of genus g − 1, and p(t), q(t) two arcs on C with p(0) = q(0) but p(t) = q(t) for t = 0. Let Ct = B/p(t) ∼ q(t) be the curve obtained from B by identifying p(t) and q(t). What is the stable limit of the curves Ct as t → 0? C-7. Let B be a stable curve of genus g − 1, p a node of B, and q a smooth point of B. Let p(t) be an arc on C with p(0) = p. Let Ct = B/p(t) ∼ q be the curve obtained from B by identifying p(t) and q. What is the stable limit of the curves Ct as t → 0? C-8. Let B be a stable curve of genus g − 1, p, q two nodes of B, and p(t), q(t) arcs on C with p(0) = p and q(0) = q. Let Ct = B/p(t) ∼ q(t) be the curve obtained from B by identifying p(t) and q(t). What is the stable limit of the curves Ct as t → 0? C-9. Finally, let B be a smooth curve of genus g−2, and p(t), q(t), r(t), s(t) four arcs on C with p(0) = q(0) = r(0) = s(0), but with p(t), q(t), r(t), s(t) distinct for t = 0. Let Ct = B/p(t) ∼ q(t), r(t) ∼ s(t) be the curve obtained from B by identifying p(t) with q(t) and r(t) with s(t). What are all the possible stable limits of the curves Ct as t → 0? D. Stable reduction IV. Now we ask the reader to find the limits of families of pointed curves {(Ct ; p1 (t), . . . , pn (t))}. D-1. Let C be a smooth curve of genus g, and p(t), q(t) two arcs on C with p(0) = q(0) but p(t) = q(t) for t = 0. What is the stable limit of the pointed curves (C; p(t), q(t)) as t → 0? D-2. Let C be a stable curve of genus g, p a node of C, and p(t) an arc on C with p(0) = p. What is the stable limit of the pointed curves (C; p(t)) as t → 0?

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165

D-3. Let C be a smooth curve of genus g, and p(t), q(t), r(t) three arcs on C with p(0) = q(0) = r(0). What are all the possible stable limits of the pointed curves (C; p(t), q(t), r(t)) as t → 0? D-4. Suppose that C is a curve with a node and p, q ∈ C are two distinct smooth points of C. Describe the limit of the stable pointed curve (C, p, q) as p and q approach the node a. along different branches of the node; and b. along the same branch of the node. Note that in one of these cases the stable limit is determined by the information given, while in the other case it is not!

E. Other limits. E-1. Let {Ct = V (F 2 + tG) ⊂ P2 } be a general pencil of quartic plane curves specializing to a double conic. What is the stable limit of the curves Ct ? E-2. Let {Ct } be as above, and let p1 , . . . , p24 be the flex points of the quartic Ct for t = 0. What are the limits of the points pi as t → 0? E-3. Again, let {Ct } be as above, and let L1 , . . . , L28 be the bitangent lines of the quartic Ct for t = 0. What are the limits of the lines Li as t → 0? E-4. Now let {Ct = V (F 2 +tG)} be a general pencil of sextic plane curves specializing to a double cubic, or more generally still a general pencil of plane curves of degree 2d specializing to the double of a curve of degree d. We ask the same questions as above: what is the stable limit of the curves Ct ; what are the limits of their flex points, and what are the limits of their bitangent lines.

F. Miscellaneous exercises on nodal curves. F-1. It is a classical fact that a smooth curve of genus g ≥ 2 has at most 84(g − 1) automorphisms. Is this true for stable curves as well? F-2. Show that for g ≥ 4, the singular locus of Mg is exactly the locus of curves with nontrivial automorphisms. (Note: this is not true for g = 2 and 3. Why not?) F-3. Show that it is not the case for any g that the singular locus of M g is the locus of stable curves with nontrivial automorphisms.

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F-4. Describe ΩC/U and ωC/U , where C = {([X, Y, Z], t) ∈ P2 ×U : (X 2 +Y 2 +Z 2 )(X 2 −Y 2 −Z 2 )−t(X 4 +Y 4 +Z 4 )}. F-5. Describe Picard–Lefschetz for the families in Exercises B-1 and B-2.

Chapter XI. Elementary deformation theory and some applications

1. Introduction. In this chapter we shall begin our study of moduli of curves by looking at the local picture. To be just a little more specific, we shall fix a stable curve and try to classify the stable curves that are “small perturbations” of it. This will be achieved, in a precise sense, in Section 6, via the construction of the so-called Kuranishi family. The notion of Kuranishi family is central to this book and is the main building block in the construction of all the moduli spaces we will consider. In turn, our construction of the Kuranishi family will rely on the Hilbert scheme and its universal property. A deformation of a complete scheme (or of a compact analytic space) X, parameterized by a pointed scheme (or by a pointed analytic space) (Y, y0 ), consists of a proper flat morphism ϕ : X → Y together with an identification of X with the fiber Xy0 of ϕ over y0 . An infinitesimal deformation of X is just a deformation parameterized by the spectrum of the ring of dual numbers. In the first section we start by describing infinitesimal deformations for a compact complex manifold X of arbitrary dimension. We show that the isomorphism classes of infinitesimal deformations of X are in one-to-one correspondence with the elements of the first cohomology group of X with coefficients in the tangent sheaf TX . One of the central points of this infinitesimal study will be the notion of Kodaira–Spencer map. Given a deformation ϕ : X → (Y, y0 ), there is a homomorphism TY,y0 → H 1 (X; TX ) assigning to each tangent vector v : Spec C[ε] → (Y, y0 ) the class in H 1 (X; TX ) corresponding to the infinitesimal deformation of X obtained pulling back via v the family ϕ to a family over Spec C[ε].1 1

One of the authors, PG, was a graduate student working under the supervision of Don Spencer at Princeton University in the late 1950s and early 1960s. Much of Spencer’s early work in the 1940s had been on the moduli of (not necessarily compact) Riemann surfaces, especially Teichm¨ uller theory. He remarked that, for many years, he had wanted to do deformation theory for higher-dimensional compact complex manifolds but was stuck because he did not know what in higher dimensions should replace quadratic differentials on Riemann surfaces. Spencer said that it was when Kodaira, who was in E. Arbarello et al., Geometry of Algebraic Curves, Grundlehren der mathematischen Wissenschaften 268, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-540-69392-5 3,

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11. Elementary deformation theory and some applications

We then turn our attention to smooth curves. Let (C, p1 , . . . , pn ) be a smooth, n-pointed curve of genus g and set D = p1 + · · · + pn . The infinitesimal deformations of (C, p1 , . . . , pn ) are parameterized by H 1 (C, TC (−D)). Using Schiffer variations, we construct an explicit deformation (1.1)

π : C → (B, 0)

of (C, p1 , . . . , pn ), parameterized by a polydisc of dimension 3g − 3 + n + h0 (C, TC (−D)), and we verify that the Kodaira–Spencer map of this deformation is an isomorphism. Schiffer variations are a useful tool in deformation theory; they consist in changing the complex structure of a Riemann surface around a single point leaving the structure unchanged outside a neighborhood of the chosen point. The explicit deformation we construct is obtained by choosing h1 (C, TC (−D)) general points on C and simultaneously performing independent Schiffer variations around them. Essentially by construction, the resulting deformation has the property that its Kodaira–Spencer map T0 (B) → H 1 (C, TC (−D)) is an isomorphism. In the second section we extend the previous considerations to the case of nodal pointed curves. As a first step, we show that the infinitesimal deformations of an n-pointed nodal curve (C, p1 , . . . , pn ) are parameterized by Ext1 (Ω1C , OC (−D)). To realize why an infinitesimal deformation ϕ : Y → S = Spec C[ε] of an n-pointed nodal curve yields an element of Ext1 (Ω1C , OC (−D)), let us look, for simplicity, at the unpointed case. A natural extension of Ω1C by OC is given by the exact sequence α → Ω1Y ⊗ OC → Ω1C → 0 . 0 → OC ∼ = ϕ∗ Ω1S −

This is the extension associated to the given infinitesimal deformation. As in the smoth case, given a deformation ϕ : C → (Y, y0 ) of (C, p1 , . . . , pn ), one can define a Kodaira–Spencer map TY,y0 → Ext1 (Ω1C , OC (−D)). In order to construct an explicit deformation of (C, p1 , . . . , pn ) parameterized by a polydisc and whose Kodaira-Spencer map is an isomorphism, one would like to mimick what was done in the smooth case using Schiffer variations. To do this, it is necessary to better understand the vector space Ext1 (Ω1C , OC (−D)). Again, we look, for simplicity, at the unpointed Princeton at that time, proved what is now called Kodaira–Serre duality giving the isomorphism ⊗2 ∼ ) = H 1 (C, TC )∨ H 0 (C, ωC

for a smooth curve, that he and Kodaira realized what first-order deformations of a complex structure should be in general. With this insight, Kodaira and Spencer were off and running to lay the foundations for modern deformation theory.

§1 Introduction

169

case. The interpretation of the elements in Ext1 (Ω1C , OC ) is provided by the “local-to-global” spectral sequence of Ext’s which gives the short exact sequence (1.2) 0 → H 1 (C, Hom(Ω1C , OC )) → Ext1 (Ω1C , OC ) → H 0 (C, Ext1 (Ω1C , OC )) → 0 . We first look at the right-hand term of this sequence. The sheaf Ext1 (Ω1C , OC ) is concentrated at the nodes of C, so that H 0 (C, Ext1 (Ω1C , OC )) ∼ = ⊕p∈Sing(C) Cp . Look at a node p. In suitable analytic coordinates, C is of the form xy = 0 near p. It turns out that the infinitesimal deformation corresponding to a generator of Cp is obtained by pasting together the deformation xy = around p with the trivial deformation outside a neighborhood of p. This is the “Schiffer variation” centered at p. It consists in smoothing the node p. Moreover, this construction shows at the same time that this infinitesimal deformation can be easily integrated. We now turn to the left-hand term of (1.2) If α : N → C is the normalization map, and r1 , q1 , . . . , rδ , qδ are the points of N mapping to the nodes α(r1 ) = α(q1 ), . . . , α(rδ ) = α(qδ ) of C, then we will check that (1.3)

Hom(Ω1C , OC ) = α∗ TN (− (ri + qi )) .

The space H 1 (C, Hom(Ω1C , OC )) = H 1 (N, TN (− (ri + qi ))) , as we have seen, parameterizes first-order deformations of N together with the 2δ marked points r1 , q1 , . . . , rδ , qδ . Thus, the elements of H 1 (C, Hom(Ω1C , OC )) parameterize infinitesimal deformations which are locally trivial at the nodes of C or, equivalently, infinitesimal deformations of the pointed curve (N ; r1 , q1 , . . . , rδ , qδ ). The same analysis can be carried out starting from a pointed nodal curve (C, p1 , . . . , pn ). We now proceed as in the smooth case. We can simultaneously integrate Schiffer variations at nodes and at smooth points of C to get a deformation π : C → B, parameterized by a polydisc of dimension 3g − 3 + n + h0 (C, TC (−D)) and check that the Kodaira–Spencer map of this deformation is an isomorphism. In the third section we introduce the concept of Kuranishi family for a nodal pointed curve (C; p1 , . . . , pn ). This is a local universal deformation of (C; p1 , . . . , pn ), π : C → (B, b0 ) ,

σi : B → C , i = 1, . . . , n, χ : (C; p1 , . . . , pn ) ∼ = (π −1 (0); σ1 (0), . . . , σn (0)) .

It turns out that a Kuranishi family for (C; p1 , . . . , pn ) exists if and only if (C; p1 , . . . , pn ) is stable. This existence theorem is proved in the fourth

170


and fifth sections of the chapter. In the third one we prove a number of formal consequences of the basic properties characterizing Kuranishi families. The first important remark is that any deformation of a stable pointed curve whose associated Kodaira–Spencer map is an isomorphism is automatically a Kuranishi family. As a consequence, the explicit deformations constructed in the previous sections via Schiffer variations are indeed Kuranishi families. Another important remark is the following. If one changes the identification χ between (C; p1 , . . . , pn ) and the central fiber of one of its Kuranishi families, the result is passing from the given Kuranishi family to another one. By the universal property, the two Kuranishi families are induced, one from the other, by an automorphism of the base. Thus, in the final analysis, the automorphism group G of the central fiber (C; p1 , . . . , pn ) acts on (B, b0 ) and on the total space C. We then proceed to analyze the action of G on the tangent space to B at b0 and prove that this action is faithful, except when g = n = 1 or g = 2, n = 0. Let (C; p1 , . . . , pn ) be a stable n-pointed curve of genus g. Set D = p1 + · · · + pn . Fix ν ≥ 3. The linear series |(ωC (D))ν | embeds C in Pr , where r = (2ν − 1)(g − 1) + νn − 1. This embedding is the so-called ν-fold log-canonical embedding of C. We consider C as a point of the Hilbert scheme Hilbp(t) r , where p(t) = (2νt−1)(g −1)+νnt, and we look at the locus Hν,g,n whose points represent ν-fold log-canonically embedded stable n-pointed curves of genus g. In Section 5 we prove that Hν,g,n is a smooth (3g − 3 + n + (r + 1)2 − 1)-dimensional subvariety of the Hilbert scheme. In Section 6 we show that Kuranishi families exist in the strongest sense. We start from the Hilbert scheme Hν,g,n . This is a smooth (3g − 3 + n + (r + 1)2 − 1)-dimensional variety acted on by the group G = P GL(r + 1). The set-theoretical quotient Hν,g,n /G is the set M g,n of isomorphism classes of stable n-pointed curves of genus g. Given a point x0 ∈ Hν,g,n corresponding to a ν-fold log-canonically embedded stable n-pointed curve (C; p1 , . . . , pn ) of genus g, the stabilizer Gx0 of x0 can be identified with the automorphism group of (C; p1 , . . . , pn ). Now we take in Hν,g,n a “slice” X through x0 which is transversal to the orbits of G. We can do it in such a way that the following conditions are satisfied: a) b) c) d)

X is affine; X is Gx0 -invariant; for every y ∈ X, the stabilizer Gy of y is contained in Gx0 ; for every y ∈ X, there is a Gy -invariant neighborhood U of y in X, for the analytic topology, such that {γ ∈ G : γU ∩ U = ∅} = Gy .

Now, the restriction π : C → (X, x0 ) has the property of being locally, in the analytic topology, a Kuranishi family for each one of its fibers and in particular for its central fiber (C; p1 , . . . , pn ). The Kuranishi

§1 Introduction

171

universal property is a direct consequence of the universal property of Hν,g,n . We close the section by studying the subloci of the bases of Kuranishi families parameterizing curves with specified automorphisms; in particular, we analyze the loci parameterizing hyperelliptic stable curves. In Section 7, building on our discussion of continuous families of Riemann surfaces begun in Section 9 of Chapter IX, we prove that Kuranishi families satisfy a universal property also with respect to continuous deformations of smooth pointed curves. This property will be essential in our treatement of Teichm¨ uller space in Chapter XV. As Riemann understood, the local perturbation of the complex structure on a compact Riemann surface can be read in the variation of its period matrix. In Section 8 we discuss these ideas from the point of view of deformations. Starting from a family of Riemann surfaces π : C → B, which we may assume to be a Kuranishi family, the period map Z from B to the Siegel upper half-space Hg assigns to each point b in B the period matrix ⎛ ⎜ Ω(b) = ⎝

γi,b

⎞ ⎟ ωα,b ⎠

, α=1,...,g ; i=1,...,2g

where {ωα,b } is a continuously varying basis for H 0 (Cb , C), and {γi,b } is a continuously varying symplectic system of generators for H 1 (Cb , Z). We carry out the local study of the period map introducing the so-called Gauss–Manin connection. We prove that the period map is holomorphic, and we also prove the so-called local Torelli theorem which amounts to the injectivity of the differential of the period map (at nonhyperelliptic curves). The dual of the differential dZ is identified with Max Noether’s map 2 ), S 2 H 0 (C, ωC ) → H 0 (C, ωC whose surjectivity, for a nonhyperelliptic curve C, was proved in the first volume. The introduction of the Gauss–Manin connection leads us, in Section 9, to a digression on the curvature properties of the Hodge bundles. This, although not strictly necessary in the sequel of the book, is quite interesting in its own right and could lead, if pursued further, to an alternative proof of the projectivity of the moduli space of stable curves which is quite different from the one that will be given in Chapter XIV. In Section 10, the last one of the chapter, we present a theorem of Kempf on infinitesimal deformations of symmetric products of smooth curves which we already used in the first volume of this work as an essential step in the proof of Mark Green’s theorem on quadrics through the canonical curve (cf. Chapter VI, Theorem (4.1)).

172


2. Deformations of manifolds. We shall begin by discussing the rudiments of the Kodaira–Spencer deformation theory. Let X be a complete scheme (or a compact analytic space). A deformation of X parameterized by a pointed scheme (resp., a pointed analytic space) (Y, y0 ), y0 ∈ Y , is a proper flat morphism ϕ : X −→ (Y, y0 )

(2.1)

plus a given isomorphism between X and the central fiber ϕ−1 (y0 ). It should be kept in mind that it is absolutely crucial to consider deformations with the same ϕ, but with different identifications between X and ϕ−1 (y0 ), as distinct. A morphism of deformations between (2.1) and another deformation ϕ : X → (Y , y0 ) of X is a cartesian diagram α

X (2.2)

ϕ u (Y, y0 )

β

w X ϕ u w (Y , y0 )

where α and β are morphisms inducing the identity on X. The notion of isomorphism of deformations is the obvious one. An equivalence of deformations is an isomorphism (2.2) such that Y = Y and β is the identity. A first-order deformation of X is simply a deformation of X parameterized by Spec C[ε], where C[ε] is the ring of dual numbers: we shall begin by classifying these in the simplest case, the one in which X is a complex manifold. We shall work in the analytic category. By the GAGA comparison theorems of Serre the results will also apply without change to the algebraic case. For brevity, we shall denote Spec C[ε] by the letter S. We may imagine X as being given by transition data {Uα , zα , fαβ (zβ )}, where ⎧ ⎪ ⎨ U = {Uα } is a finite cover of X , zα = t (zα1 , . . . , zαn ) are holomorphic coordinates in Uα , ⎪ ⎩ zα = fαβ (zβ ) in Uα ∩ Uβ . In triple intersections Uα ∩ Uβ ∩ Uγ the cocycle rule fαβ (fβγ (zγ )) = fαγ (zγ ) holds. Similarly, the total space X of a first-order deformation ϕ : X −→ S

§2 Deformations of manifolds

173

of X may be thought of as being given by gluing the Uα × S via the identifications (2.3)

zα = f˜αβ (zβ , ε) = fαβ (zβ ) + εbαβ (zβ ) ,

while ϕ is given, locally, by ϕ(zα , ε) = ε. The f˜αβ also have to satisfy the cocycle rule f˜αβ (f˜βγ (zγ , ε), ε) = f˜αγ (zγ , ε) , which reduces to the cocycle rule for the fαβ plus (2.4) If

∂ ∂zα

bαβ +

∂fαβ bβγ = bαγ . ∂zβ

stands for the row vector ∂ = ∂zα

∂ ∂ ,..., n ∂zα1 ∂zα

,

then (2.4) just says that the 1-cochain ϑ = {ϑαβ } ∈ C 1 (U, TX ) given by t

ϑαβ = bαβ

t

∂ ∂zα

is a cocycle. The class [ϑ] ∈ H 1 (X, TX ) it defines is called the Kodaira–Spencer class of the first-order deformation ϕ. There is an exact sequence of OX -modules 0 → TX → TX → ϕ∗ TS → 0 . Passing to the cohomology sequence, it is clear from the definitions that the Kodaira–Spencer class is just the coboundary of ϕ

∗

∂ ∂ε

∈ H 0 (X, ϕ∗ TS ) .

Therefore, [ϑ] depends only on the equivalence class of the first-order deformation X → S. Conversely, every 1-cocycle with coefficients in TX defines via (2.3) a first-order deformation of X. Suppose now that

t

ϑ= ϑ =

bαβ

t bαβ

∂ · , ∂zα ∂ · t ∂zα t

174


are cohomologous, i.e., that there are holomorphic vector-valued functions cα ∈ Γ(Uα , OX )n such that

∂fαβ cβ − cα = bαβ − bαβ . ∂zβ

Let ϕ

X − →S,

ϕ

X −→ S

be the first-order deformations of X corresponding to ϑ, ϑ . We may think of ϕ : X → S as being described by (2.3) and of ϕ : X → S as being described by (zβ , ε) = fαβ (zβ ) + εbαβ (zβ ) . zα = f˜αβ

Now set (zα , ε) = Φα (zα , ε) = (zα + εcα , ε) . It is straightforward to check that the Φα define an equivalence between the two first-order deformations. This sets up a 1–1 correspondence between H 1 (X, TX ) and the set of equivalence classes of first-order deformations of X. We may observe, and we shall need this observation, that the notion of first-order deformation makes sense also for an open complex manifold X. One simply has to drop the properness assumption in the definition. More importantly, the proof that equivalence classes of firstorder deformations of X are classified by H 1 (X, TX ) carries over to this case verbatim. Let us next consider an arbitrary deformation ϕ : X −→ (Y, y0 ) of a compact complex manifold X. For any morphism f : (Z, z0 ) → (Y, y0 ), the fiber product

f ∗ ϕ : X ×Y Z → (Z, z0 )

is a deformation of X which is called the pullback of ϕ. Apply this construction to the case where Z is S = Spec C[ε]. Then the tangent space to Y at y0 is the set of all morphisms f , i.e., TY,y0 = Hom(S, (Y, y0 )),


175

and f ∗ ϕ is nothing but the first-order approximation of ϕ in the direction of the tangent vector corresponding to f . We then get a map ρ : TY,y0 −→ H 1 (X, TX ) associating to each f in Hom(S, (Y, y0 )) the Kodaira–Spencer class of the first-order deformation f ∗ ϕ. The map ρ is easily seen to be linear and is called the Kodaira–Spencer homomorphism. We now specialize to the case where X = C is a smooth curve of genus g. The first remark to be made is that the vector space H 1 (C, TC ) parameterizing first-order deformations of C has dimension 3g − 3 unless g = 0, 1, in which case it has dimension 0 and 1, respectively (this, of course, follows from the Riemann–Roch theorem). In this space, there live certain distinguished classes that go under the name of Schiffer variations. Up to multiplicative constants, there is one of these for each point p on C, namely the image δp of a generator of H 0 (C, TC (p) ⊗ Cp ) ∼ =C under the coboundary map of 0 → TC → TC (p) → TC (p) ⊗ Cp → 0 . If U is a small neighborhood of p, z a coordinate on U , and V is the complement of p, then, with respect to the cover {U, V }, a ∂ . Let p1 , . . . , ph be distinct points of cocycle representing δp is 1z ∂z 1 C. Since H (C, TC ( pi )) vanishes for large enough h, the coboundary homomorphism TC (pi ) ⊗ Cpi → H 1 (C, TC ) i

deduced from 0 → TC → TC (

pi ) →

TC (pi ) ⊗ Cpi → 0

i

is onto, showing that H 1 (C, TC ) is generated by Schiffer variations. Thus, the Schiffer variations based at h1 (C, TC ) general points of C form a basis of H 1 (C, TC ). One of the advantages of Schiffer variations is that they can be easily integrated, that is, for each point p of C and for sufficiently small η, one can find a deformation ϕ : D → Δ = {t ∈ C : |t| ≤ η} of C parameterized by a disc which to first-order is δp . Let z : U → {z ∈ C : |z| < b} be a local coordinate centered at p. Choose a positive number a < b, and set A = {w ∈ C : |w| < a} .

176


The total space D can be constructed by pasting together a × Δ and A × Δ C − z : |z| ≤ 2 by means of (2.5)

w=z+

t z

(of course, η depends on a and b). Sometimes we shall refer to this procedure as performing a Schiffer variation at p. Now perform Schiffer variations simultaneously at h = h1 (C, TC ) general points of C, using independent deformation parameters t1 , . . . , th . This yields a deformation (2.6)

ϕ : C → (B, b0 )

of C, parameterized by an h1 (C, TC )-dimensional polydisc, having the remarkable property that the Kodaira–Spencer map (2.7)

ρ : TB,b0 → H 1 (C, TC )

is an isomorphism. This follows at once from the remark that Schiffer variations generate H 1 (C, TC ). All these considerations extend, with minor formal modifications, to deformations of n-pointed smooth curves. A deformation of an n-pointed smooth curve (C; q1 , . . . , qn ) consists of a deformation ψ : X → T of C together with n disjoint section of ψ passing through the points of the central fiber corresponding to q1 , . . . , qn . The notions of morphism, isomorphism, and equivalence of deformations of n-pointed curves are the obvious analogues of the corresponding notions for unpointed curves. When dealing with deformations of n-pointed curves, we shall often omit mention of the sections, when this is unlikely to cause confusion. Equivalence classes of first-order deformations of (C; q1 , . . . , qn ) are n classified by H 1 (C, TC (− i=1 qi )); the easy verification of this fact is left to the reader (see Exercise (2.13)). One can construct a deformation (2.8)

ϕ : C → (B, b0 ) ,

σi : B → C , i = 1, . . . , n,

for (C; q1 , . . . , qn ), parameterized by a polydisc, which is a direct analogue of (2.6). As before, we proceed by integrating Schiffer variations. Some of these are the ordinary Schiffer variations associated to points of C which are distinct from q1 , . . . , qn . The remaining Schiffer variations ηi , i = 1, . . . , n, correspond to first-order deformations moving one marked point and leaving C and the remaining ones fixed. They are defined as follows. One looks at the exact sequence TC,qi → 0 (2.9) 0 → TC (− qi ) → TC →


177

and chooses a coordinate zi centered at qi for each i. Then ηi is the image of ∂/∂zi ∈ TC,qi under the coboundary map

TC,qi → H 1 (C, TC (− qi ))

and can be integrated exactly as one does for an ordinary Schiffer variation, using as gluing, (2.10)

w = zi + t

instead of (2.5). The same argument used in the unpointed case shows (C, TC (− qi )) is given by the Schiffer variations that a basis for H 1 based at h1 (C, TC (− qi )) general points of C. In fact, one can do a little better. Looking at the exact cohomology sequence of (2.9), one easily sees that we can get it by the Schiffer variations based at n − h0 (TC ) + h0 (TC (− qi )) of the marked points, plus h1 (C, TC ) Schiffer variations based at general points of C. In any case, integrating these variations yields a family (2.8) such that the Kodaira–Spencer map TB,b0 → H 1 (C, TC (− qi ))

(2.11)

is an isomorphism. We can summarize what has been achieved in the following statement. Theorem (2.12). Let (C; q1 , . . . , qn ) be a smooth n-pointed genus g curve. There exists a deformation Cu ϕ

σi , i = 1, . . . , n

u (B, b0 )

χ : (C; q1 , . . . , qn ) −→ (ϕ−1 (b0 ); σ1 (b0 ), . . . , σn (b0 ))

of (C; q1 , . . . , qn ) such that the Kodaira–Spencer map ρ : Tb0 (B) → H 1 (C, TC (−

qi ))

is an isomorphism and B is a polydisc of dimension 3g − 3 + n + h0 (C, TC (− qi )). When (C; q1 , . . . , qn ) is stable, i.e., when 2g −2+n > 0, the dimension of B is 3g − 3 + n. Exercise (2.13). Fill in the details of the proof of Theorem (2.12) Exercise (2.14). Let D be an effective divisor, possibly with multiple points, on a smooth curve C. Interpret H 1 (C, TC (−D)) from the point of view of infinitesimal deformations.

178


3. Deformations of nodal curves. We now wish to study deformations of nodal curves. As in the previous section, we will generally work in the analytic category. Needless to say, the definitions of deformations and first-order deformations of nodal curves (and open subsets thereof) are as in the smooth case. We shall begin by showing that the equivalence classes of firstorder deformations of an open subset C of a nodal curve are in 1–1 correspondence with the elements of Ext1OC (Ω1C , OC ). Let ϕ : Y → S = Spec C[ε]

(3.1)

be a first-order deformation of C. In the case at hand the exact sequence (2.17) of Chapter X restricts on C to the following one: (3.2)

α 0 → OC ∼ → Ω1Y ⊗ OC → Ω1C → 0 . = ϕ∗ Ω1S −

We only have to show that α is injective. In fact, ϕ∗ Ω1S is generated by dε, and α(dε) certainly is not zero at smooth points of C, where Y is locally a product, and ϕ a projection. Since C is reduced, α must be injective. We have thus associated to any first-order deformation of C an extension of Ω1C by OC , hence an element of Ext1OC (Ω1C , OC ). Exactly as in the case of smooth curves, we may then conclude that any deformation ψ : X → Y of a nodal curve C = ψ−1 (y0 ) defines a Kodaira–Spencer map ρ : Ty0 (Y ) → Ext1OC (Ω1C , OC ) . We next observe that, when two first-order deformations ϕ:Y →S

and

ϕ : Y → S

define the same extension class, they are equivalent. We must produce a sheaf isomorphism β : OY → O Y inducing the identity on OC and commuting with projections, assuming that there is a commutative diagram

OC

' ) ' ''

Ω1Y ⊗ OC γ u Ω1Y ⊗ OC

' ) ' ''

Ω1C

Indeed, we shall construct a β that induces the equivalence γ of extensions. We shall do so by showing that, for any function h on Y, there is a unique function β(h) on Y such that h|C = β(h)|C

and

˜ ˜ dβ(h) = γ(dh),

§3 Deformations of nodal curves

179

where d˜ stands for the composition of d with the natural map Ω1Y → Ω1Y ⊗ OC . Uniqueness is obvious: if f is a function on Y ˜ = gdε in Ω1 ⊗ OC , such that f |C = 0, then f is of the form εg, and df Y ˜ so that, if df = 0, then g, and hence f , vanish. Having proved the uniqueness, the existence is a local problem. For any h, the restriction ˜ on Y . But the difference between h|C locally extends to a function h ˜ ˜ ˜ dh and γ(dh) restricts to zero on C, so that ˜ − γ(dh) ˜ = gdε , d˜h ˜ − εg. It remains to show that β is a ring and we may set β(h) = h homomorphism. By Leibniz’ rule, ˜ ˜ dβ(hg) = γ(d(hg)) ˜ + h|C γ(dg) ˜ = g|C γ(dh) ˜ = d(β(h)β(g)) , so that, by uniqueness, β(hg) indeed equals β(h)β(g). A similar argument shows that β(h + g) is equal to β(h) + β(g). This ends the construction of β. It may be observed that the arguments developed so far apply, without changes, to deformations of a reduced scheme or analytic space, and not only to deformations of a nodal curve. First-order deformations of a reduced scheme or analytic space X are thus parameterized by a subspace of Ext1OX (Ω1X , OX ). Before showing that any element of Ext1 (Ω1C , OC ) comes from a firstorder deformation of C, we need a few preliminary observations. The “local-to-global” spectral sequence of Ext’s (cf. [318], for example) gives a short exact sequence (3.3)

0 → H 1 (C, HomOC (Ω1C , OC )) → Ext1OC (Ω1C , OC ) → H 0 (C, Ext1OC (Ω1C , OC )) → 0 .

The sheaf Ext1 (Ω1C , OC ) is concentrated at the nodes of C, and hence the sequence (3.3) can also be rewritten as

(3.4)

0 → H 1 (C, HomOC (Ω1C , OC )) → Ext1OC (Ω1C , OC ) → Ext1OC,p (Ω1C,p , OC,p ) → 0 . p∈Sing(C)

To compute it at a node p, we embed a small neighborhood of p as a closed subspace of an open subset V of C2 , so that in suitable coordinates, C is of the form xy = 0. From the conormal sequence d

0 → OV,p (−C) ⊗ OC,p − → Ω1V,p ⊗ OC,p → Ω1C,p → 0

180


we get an exact sequence η

(3.5)

→ Hom(OV,p (−C) ⊗ OC,p , OC,p ) → Hom(Ω1V,p ⊗ OC,p , OC,p ) − Ext1 (Ω1C,p , OC,p ) → 0

since Ω1V ⊗ OC is locally free, and hence Ext1 (Ω1V,p ⊗ OC,p , OC,p ) vanishes. Let ϕ be a homomorphism from Ω1V,p ⊗ OC,p to OC,p . Then η(ϕ)(xy) = ϕ(d(xy)) = xϕ(dy) + yϕ(dx). Thus, the image of η is mp Hom(OV,p (−C) ⊗ OC,p , OC,p ), where mp is the maximal ideal in OC,p , and hence (3.5) gives an isomorphism (3.6)

Ext1 (Ω1C,p , OC,p ) ∼ = (OV,p (−C) ⊗ C)∨ .

It follows in particular that Ext1 (Ω1C , OC ) is (noncanonically) isomorphic to p∈Csing Cp . It is useful to give an alternate intrinsic interpretation of Ext1 (Ω1C,p , OC,p ). Let τ be the set consisting of the two possible orderings of the branches of C at p, and let μ2 = {±1} act nontrivially on it. We claim that there is a canonical isomorphism 2 (mp /m2p )∨ ⊗μ2 τ . (3.7) Ext1OC,p (Ω1C,p , OC,p ) ∼ = This is obtained by composing (3.6) with the transpose of an isomorphism ϑ between 2 (mp /m2p ) ⊗μ2 τ and OV,p (−C) ⊗ C which we now describe. Let I be the maximal ideal of OV,p . For any element ξ of OV,p (resp., of OV,p (−C)), we denote by [ξ] its class in OC,p /mp (resp., in OV,p (−C)/IOV,p (−C)). Let x and y be local equations for the two branches of C at p. Denote by bx (resp., by ) the branch defined by the vanishing of x (resp., of y). One then sets ϑ [x] ∧ [y] ⊗ (bx , by ) = [xy] . It is immediate to see that this is well defined and linear. At first sight the isomorphism (3.7) seems to depend on the choice of an embedding of a neighborhood of p ∈ C in an open subset of C2 , but it is easy to show that this is not the case. In fact, given another embedding in an open subset V ⊂ C2 , possibly after shrinking V and V , there is an isomorphism f : V → V inducing the identity on C. This yields a commutative diagram (OV,p (−C) ⊗ C)∨ [[ t ϑ ∼ [ = ] [ 2 1 1 f∗ Ext (ΩC,p , OC,p ) [[ (mp /m2p )∨ ⊗μ2 τ [[ ] u ∼ = t ϑ (OV ,p (−C) ⊗ C)∨ showing that (3.7) is independent of the embedding. Another version of (3.7) can be obtained as follows. Let N be the normalization of C at


181

p, and denote by p1 , p2 the points of N mapping to p. Then (mp /m2p )∨ = TC,p = TN,p1 ⊕ TN,p2 , whence an identification (3.8)

Ext1OC (Ω1C,p , OC,p ) ∼ = TN,p1 ⊗ TN,p2 .

In fact, the identification between 2 (TN,p1 ⊕ TN,p2 ) and TN,p1 ⊗ TN,p2 depends on the choice of an ordering of the two summands, i.e., of an ordering of the branches of the node, and changes sign if the ordering is reversed; tensoring with τ exactly compensates for this ambiguity. We are now in a position to construct a first-order deformation of C corresponding to a given class in Ext1 (Ω1C , OC ). We shall first deal with the case where C is a neighborhood of the origin in xy = 0. Since C is Stein, in this case Ext1 (Ω1C , OC ) = H 0 (C, Ext1 (Ω1C , OC )) ∼ = C. The family xy = aε, where a is a complex number, is a first-order deformation Y of C which can be regarded as the reduction modulo t2 of the surface X with equation xy = at. As this surface is smooth for a = 0, the sheaf Ω1Y ⊗ OC = Ω1X|C is, in this case, locally free and hence a nontrivial extension of Ω1C by OC . If we denote by ζa the class of this extension, a straightforward computation, based, for instance, on (3.6), shows that ζa = aζ1 . It follows that, in the special case at hand, any class in Ext1 (Ω1C , OC ) can be realized as a first-order deformation of C. In treating the general case we shall assume, for simplicity of notation, that C has a single node at p. Let ξ be an element of Ext1 (Ω1C , OC ). Let U be a small neighborhood of p. We have shown that there exists a first-order deformation U of U inducing ξ|U . On the other hand, since C {p} is smooth and affine, the exact sequence (3.3) shows that ξ|C{p} is trivial. Thus, ξ can be obtained by gluing the trivial extension on C {p} with ξ|U along U {p} by means of an equivalence of extensions of Ω1U {p} by OU {p} . As we have shown, over U {p} there is, between the first-order deformation U of U and the trivial deformation D of C {p}, an equivalence β that induces γ. Gluing U and D by means of β yields a deformation Y of C whose class in Ext1 (Ω1C , OC ) is precisely ξ. The construction we have performed provides an interpretation of the two extreme terms in the sequence (3.3). The term on the right corresponds to deformations that smooth one or more of the nodes of C, while the term on the left corresponds to deformations that are locally

182


trivial, that is, locally products. Another way of explaining this last assertion is to notice that, if α : N → C is the normalization map, and r1 , q1 , . . . , rδ , qδ are the points of N mapping to the nodes α(r1 ) = α(q1 ), . . . , α(rδ ) = α(qδ ) of C, then, as we shall see in a moment, one has (3.9) Hom(Ω1C , OC ) = α∗ TN (− (ri + qi )) . The space H 1 (C, Hom(Ω1C , OC )) = H 1 (N, TN (− (ri + qi ))) , as we have seen, parameterizes first-order deformations of N together with the 2δ marked points r1 , q1 , . . . , rδ , qδ . It remains to justify (3.9). Observe first that the equality is trivially true away from the nodes of C. Let then w be a node, and z1 , z2 the points of N mapping to it. We will be done if we can show that Hom(Ω1C,w , OC,w ) = Hom(IωC,w , OC,w ) = Hom(ωN,zi , ON,zi (−zi )) , i=1,2

where I is the ideal of w. The equality on the left follows from the fact that the natural homomorphism Ω1C,w → ωC,w sends Ω1C,w onto IωC,w and has a one-dimensional vector space as kernel. As for the other equality, let ϕ be a homomorphism from IωC,w = ωN,z1 ⊕ ωN,z2 to OC,w . Clearly, ϕ(ωN,zi ) vanishes on one of the branches of the node, and hence is contained in I = ON,z1 (−z1 )⊕ON,z2 (−z2 ). Thus Hom(IωC,w , OC,w ) equals HomOC,w ωN,z1 ⊕ ωN,z2 , ON,z1 (−z1 ) ⊕ ON,z2 (−z2 ) = Hom(ωN,z1 , ON,z1 (−z1 )) ⊕ Hom(ωN,z2 , ON,z2 (−z2 )) since HomOC,w (ωN,zi , ON,zj (−zj )) vanishes when i = j. It goes almost without saying that one can develop a deformation theory for n-pointed nodal curves paralleling the deformation theory for nodal curves. Recall that an n-pointed nodal curve is the datum of a nodal curve C plus distinct smooth points p1 , . . . , pn of C. Look at a first-order deformation of (C; p1 , . . . , pn ) consisting of a family (3.1) plus sections ρ1 , . . . , ρn . Set D = p1 + · · · + pn and R = ρ1 + · · · + ρn . The extension (3.2) can be enriched to the commutative diagram 0

w OC

0

u w OD

(3.10)

w Ω1Y ⊗ OC

w Ω1C

w0

u w Ω1R ⊗ OD

u w0

w0

This diagram can be regarded as an extension of the complex A• = (A0 → A1 ) = (Ω1C → 0) by the complex D• = (D 0 → D 1 ) = (OC → OD ),


183

that is, as an element of Ext1OC (A• , D• ). We leave it to the reader to show that this sets up a one-to-one correspondence between isomorphism classes of first-order deformations of (C; p1 , . . . , pn ) and Ext1OC (A• , D• ), extending the proof we have given in the unpointed case. Notice that Ext1OC (A• , D• ) ∼ = Ext1OC (Ω1C , OC (−D)) .

(3.11)

In fact, the elements of Ext1OC (A• , D• ) are (isomorphism classes of) commutative diagrams of extensions 0

w D0

w E0

w A0

w0

u u u 0 w D1 w E1 w A1 w0 one such being trivial if and only if the two rows admit compatible splittings. Taking kernels of the vertical arrows and keeping in mind that A1 vanishes and D0 → D 1 is onto, we get a commutative diagram with exact rows and columns,

(3.12)

0

0

0

0

u wK

u wE

u w A0

w0

0

u w D0

u w E0

u w A0

w0

0

u w D1

u w E1

u w0

u 0

u 0

The top row is an extension of A0 = Ω1C by K = OC (−D), and it is clear that if this extension is trivial, that is, if E → A0 has a right inverse, then E 0 → A0 has a right inverse whose composition with E 0 → E 1 is zero, and hence the original diagram of extensions is trivial. This shows that Ext1OC (A• , D• ) ⊂ Ext1OC (Ω1C , OC (−D)). Conversely, any extension 0 → K → E → A0 → 0 can be completed to a diagram (3.12) by setting D0 ⊕ E E0 , E1 = , ∼ E where (d, e) ∼ 0 if and only if d and −e come from the same element of K. In conclusion, we have a natural identification (3.13) first-order deformations isomorphism ∼ = Ext1OC (Ω1C , OC (− pi )) . of (C; p1 , . . . , pn ) E0 =

184


Clearly, there is an exact sequence (3.14)

0 → H 1 (C, Hom(Ω1C , OC (−D))) → Ext1 (Ω1C , OC (−D)) → H 0 (C, Ext1 (Ω1C , OC )) → 0

which generalizes (3.3) to the n-pointed case and admits a similar interpretation. The term on the left classifies first-order deformations which are locally trivial, and the one on the right classifies first-order smoothings of the nodes. Alternatively, one can view the term on the left as classifying first-order deformations of the normalization of C, pointed by the counterimages of D and of the nodes. Finally, one can extend to nodal curves the construction that led to Theorem (2.12). We begin with the following observation. The realization of Ext1 (Ω1C , OC (− pi )) as the space of first-order deformations of the n-pointed nodal curve (C; p1 , . . . , pn ) was largely based on the possibility of infinitesimally smoothing a nodal curve. We want to show that the same can be done in finite terms, namely that, given a node p on C, one can find a deformation ϕ : C → Δ,

ρi : Δ → C

of (C; p1 , . . . , pn ), where Δ is the disk {t ∈ C : |t| < neighborhood of p, the fibers ϕ−1 (t) are smooth for t near p, C will look like xy = t. The construction is as are neighborhoods U of p, not containing any of the pi , thought of as being obtained from two discs

1}, and, in a

= 0. In fact, follows. There which may be

V = {z ∈ C : |z| < 1}, W = {w ∈ C : |w| < 1} via identification of the origins. Consider the regions A = {x, y, t) ∈ C3 : |x| < 1, |y| < 1, |t| < 1, xy = t} , B = ((C − U ) × Δ) ∪ B , where B = {(z, t) ∈ V × Δ : |z| >

|t|} ∪ {(w, t) ∈ W × Δ : |w| > |t|} ⊂ C × Δ .

Then C is the complex manifold obtained from A and B by identifying B to A = A − {(x, y, t) : |x| = |y|} by means of t (z, t) → z, , t , z (3.15) t (w, t) → , w, t , w


185

and ϕ is the projection to the last factor. As for the sections, we simply set ρi (t) = (pi , t) in (C − U ) × Δ ⊂ C.

Figure 1. To construct the analogue of the family in (2.12), we shall think of (C; p1 , . . . , pn ) as being obtained from its pointed normalization (N ; r1 , q1 , . . . , rδ , qδ , p1 , . . . , pn ) by identifying points r1 , q1 , . . . , rδ , qδ in pairs. The family ψ:N →W obtained by applying Theorem (2.12) to the pointed normalization comes equipped with disjoint sections Ri : W → N ; Qi : W → N , i = 1, . . . , δ , Pi : W → N , i = 1, . . . , n , passing, respectively, through the ri , qi , and pi . Identifying Ri and Qi for each i yields a locally trivial deformation ϕ : C → W ,

Pi : W → C ,

i = 1, . . . , n ,

of (C; p1 , . . . , pn ). The Kodaira–Spencer map provides an identification between the tangent space to W at w and the group H 1 (N, TN (−

(ri + qi ) −

pj )) = H 1 (C, Hom(Ω1C , OC (−

pj ))) .

The smoothing of a node which we performed on a fixed curve can be done “with parameters” on a locally trivial family such as ϕ . Doing this independently at every node of C gives a deformation (3.16)

ϕ : C → B = W × Δ,

σi : B → C ,

i = 1, . . . , n,

of (C; p1 , . . . , pn ), where Δ is a δ-dimensional polydisc. Moreover, we may choose coordinates t1 , . . . , tδ on Δ in such a way that, near the

186


node of C arising from the identification of ri with qi , C is of the form xy = ti . It may be observed that the Kodaira–Spencer map provides an identification of the exact sequence 0 → Tw (W ) → T(w,0) (B) → T0 (Δ) → 0 with (3.14). Furthermore, the locus in B parameterizing singular curves is the union of the hyperplanes ti = 0, i = 1, . . . , δ. More precisely, the restriction of the family (3.16) to ti = 0 parameterizes deformations of C which are locally trivial at the ith node. Summing up, here is what has been shown. Theorem (3.17). Let (C; p1 , . . . , pn ) be an n-pointed nodal curve of genus g. There exists a deformation Cu ϕ

σi , i = 1, . . . , n

u (B, b0 )

χ : (C; p1 , . . . , pn ) −→ (ϕ−1 (b0 ); σ1 (b0 ), . . . , σn (b0 ))

of (C; p1 , . . . , pn ) such that the Kodaira–Spencer map ρ : Tb0 (B) → Ext1OC (Ω1C , OC (− pi )) is an isomorphism, and B is a polydisc of dimension 3g − 3 + n + dim Hom(Ω1C , OC ). In particular, when (C; p1 , . . . , pn ) is stable, dim Ext1 (Ω1C , OC (− pi )) = dimb0 (B) = 3g − 3 + n . Finally, if δ is the number of nodes of C, one can choose coordinates t1 , . . . , tδ , . . . on B, vanishing at b0 , in such a way that the locus parameterizing deformations which are locally trivial at the ith node is ti = 0; in particular, the locus parameterizing singular curves is t1 · · · tδ = 0. We close with a useful generalization of the sequence (3.14). Let (C; p1 , . . . , pn ) be an n-pointed nodal curve, and let W = {w1 , . . . , w } → C be the partial normalization at be a set of nodes of C. Let α : C mapping to wi . these nodes, and denote byri , qi the two points of C We set D = pi and E = (ri + qi ), and write D also to designate the Then (3.9) immediately generalizes to preimage of D in C. − E)) , Hom(Ω1C , OC (−D)) = α∗ Hom(Ω1, OC(−D C

so that, in particular, Hom(Ω1 , O (−D − E))) = H 1 (C, Hom(Ω1 , OC (−D))) . H 1 (C, C C C

§4 The concept of Kuranishi family

187

then fit in a The exact sequence (3.14) and its analogue for C commutative diagram

whence an exact sequence − E)) → Ext1 (Ω1 , OC (−D)) → 0 → Ext1 (Ω1, OC(−D C C 1 1 Ext (ΩC,w , OC,w ) → 0

(3.18)

w∈W

Of course, the term on the left classifies first-order deformations which are locally trivial at the nodes belonging to W , and the one on the right classifies first-order smoothings of these nodes. It is useful to notice that, by (3.8),

(3.19)

Ext1 (Ω1C,w , OC,w ) =

w∈W

i=1

TC,r ⊗ TC,q . i i

When W is the set of all the nodes of C, the sequence (3.18) reduces to (3.14). Its first two terms parameterize first-order deformations of (as an (n + 2)-pointed curve) and of (C; p1 , . . . , pn ), respectively. C The homomorphism connecting them is given by the clutching operation described in Section 7 of Chapter X. Exercise (3.20). Construct an example of family X → S of npointed nodal curves over a connected base such that the dimension of Ext1 (Ω1Xs , OXs (− pi (s))) is not constant as a function of s. 4. The concept of Kuranishi family. One of the central notions in the theory of deformations and moduli is the one of Kuranishi family. We will need it only for curves, and we limit ourselves to this case. We work in the analytic category. Let (C; p1 , . . . , pn ) be an n-pointed connected nodal curve. A deformation (4.1) Cu ϕ

σi , i = 1, . . . , n

u (B, b0 )

χ : (C; p1 , . . . , pn ) −→ (ϕ−1 (b0 ); σ1 (b0 ), . . . , σn (b0 ))

188


of (C; p1 , . . . , pn ) is said to be a Kuranishi family for (C; p1 , . . . , pn ) if it satisfies the following condition: ψ

→ (E, e0 ) of (C; p1 , . . . , pn ) and for any K) For any deformation D − sufficiently small connected neighborhood U of e0 , there is a unique morphism of deformations of n-pointed curves F wC D U

(4.2) u

(U, e0 )

f

ϕ u w (B, b0 )

Kuranishi families need not exist in general. In the next two sections we shall prove the following basic result. Theorem (4.3). A Kuranishi family for (C; p1 , . . . , pn ) exists if and only if (C; p1 , . . . , pn ) is stable. The mere existence of a Kuranishi family has several important formal consequences, which we now detail. Most of them need no proof. Corollary (4.4). Kuranishi families are essentially unique, in the sense that any two Kuranishi families for (C; p1 , . . . , pn ), up to restricting to sufficiently small connected neighborhoods of the base points, are isomorphic via a unique isomorphism. Corollary (4.5). The Kodaira–Spencer map of a Kuranishi family at the base point is an isomorphism. Corollary (4.6). Let there be given a deformation of a stable n-pointed curve (C; p1 , . . . , pn ) over the pointed analytic space (E, e0 ). Suppose that its Kodaira–Spencer map at e0 is an isomorphism and that E is smooth at e0 . Then the deformation is a Kuranishi family for (C; p1 , . . . , pn ). The proof is immediate. In fact, the universal property of the Kuranishi family asserts the existence of a diagram (4.2). The Kodaira–Spencer map in question can be identified with the differential of f , which is thus a local isomorphism at e0 . Corollary (4.7). When (C, p1 , . . . , pn ) is stable, the deformation constructed in (3.17) is a Kuranishi family for (C, p1 , . . . , pn ). Corollary (4.8). The base of the Kuranishi family of a stable n-pointed curve (C, p1 , . . . , pn ) of genus g is smooth of dimension 3g − 3 + n. A family X → S of stable n-pointed curves can be viewed as a deformation of a fiber Xs0 , taking as identification between Xs0 and itself the identity. Corollary (4.9). Let X → S be a family of stable n-pointed curves, and s0 a point of S. If X → S is a Kuranishi family for Xs0 , then it is a Kuranishi family for Xs , for all s in an open neighborhood U of s0 . If X → S is an algebraic family, then we can take U to be Zariski open.


189

It follows from the previous results that X → S is Kuranishi for Xs if and only if s is a smooth point of S and the Kodaira–Spencer map at s is an isomorphism. The first of these conditions is clearly an open one, both in the ordinaryand in the Zariski topologies. Since the dimension of Ext1 (Ω1Xs , OXs (− σi (s))) is independent of s, the second condition translates into a rank condition for a map between vector bundles and hence is open as well. The corollary follows. Corollary (4.10). Let (4.1) be a Kuranishi family for (C; p1 , . . . , pn ). Denote by G the automorphism group of (C; p1 , . . . , pn ). Then there are arbitrarily small neighborhoods V of b0 such that the action of G on (C; p1 , . . . , pn ) extends to compatible actions on V and on CV taking each distinguished section to itself. Proof. As usual, we write Cb to denote the fiber ϕ−1 (b), γ be an element of G. Composing γ −1 with χ : C → Cb0 , identification of (C; p1 , . . . , pn ) with the central fiber of hence a new deformation of (C; p1 , . . . , pn ). The universal Kuranishi family, applied to this deformation, then gives diagram γ idC wC wC C χ χ χ u u u Fγ CVγ w CU γ y wC ϕ ϕ ϕ u u u fγ w Uγ y wB Vγ

and so on. Let we get another the family and property of the a commutative

where Vγ , Uγ are suitable neighborhoods of b0 , and fγ , Fγ are isomorphisms. Replacing each Vγ with ∩{Vγ : γ ∈ G}, we may suppose that all the Vγ are equal to the same neighborhood V of b0 . By uniqueness, we may also suppose that fγ fη = fγη and Fγ Fη = Fγη where defined and that fγ and Fγ are the identities when γ is the identity. Further replacing V with ∩{Uη : η ∈ G}, we may also assume that Uγ = fγ (V ) = V for all γ ∈ G. The required actions of G on V and CV are given, respectively, by γ → fγ and γ → Fγ . Q.E.D. The action of G on CV given by (4.10) is clearly faithful, while the same is not necessarily true of the action on V . This action is faithful (B) is. Notice that, under if and only if the corresponding action on Tb 0 the isomorphism Tb0 (B) ∼ = Ext1OC (Ω1C , OC (− pi )), the action of G on Tb0 (B) corresponds to the natural one on Ext1OC (Ω1C , OC (− pi )). The situations in which this action is not faithful can be completely classified. Proposition (4.11). Let (C; p1 , . . . , pn ) be a stable n-pointed curve of genus g, and let γ be a nontrivial automorphism of (C; p1 , . . . , pn ) which acts trivially on Ext1OC (Ω1C , OC (− pi )). Then one of the following occurs:

190


i) g = 1, n = 1, and γ is the symmetry about the marked point; ii) g = 2, n = 0, and γ is the hyperelliptic involution. Proof. We begin by illustrating the main ingredient of the proof in the case of a smooth C. First of all, a smooth stable n-pointed curve of genus zero has no nontrivial automorphisms, so we may 1. The assume that g ≥ 2 ( pi )). dual of Ext1OC (Ω1C , OC (− pi )) = H 1 (C, TC (− pi )) is H 0 (C, ωC On the other hand, 2 ( pi ) ≥ deg ωC + 3, deg ωC unless one of the following occurs: - g = 1, n ≤ 2; - g = 2, n = 0.

2 Then, outside of these exceptions, |ωC ( pi )| embeds C in PH 1 (C, TC (− pi )), and hence if γ acts trivially on H 1 (C, TC (− pi )), it must act trivially on its projectivization and therefore on C. It remains to examine the exceptional cases. The first of them will be dealt with when we treat the case of a possibly singular C. The second is taken care of by the argument we have just used. In fact, in genus two the bicanonical mapping is the composition of the hyperelliptic double covering and of a Veronese embedding, so that an automorphism of C acting trivially on H 1 (C, TC ) must be the identity or the hyperelliptic involution. We now move on to the general case. Set D = pi . The vector space Ext1 (Ω1C , OC (−D)) sits in an exact sequence 0 → H 1 (C, Hom (Ω1C , OC (−D))) → Ext1 (Ω1C , OC (−D)) → Cp → 0 p∈Csing

whose dual is (4.12)

0→

2 Cp → H 0 (C, Ω1C ⊗ ωC (D)) → H 0 (C, J ωC (D)) → 0,

p∈Csing

where J stands for the ideal of the singular locus of C. Let γ be an automorphism of (C; p1 , . . . , pn ) that acts as the identity on H 0 (C, Ω1C ⊗ ωC (D)). Since γ acts trivially on the leftmost term in (4.12), it leaves every node of C fixed. As a consequence, γ carries every singular component of C into itself. It is also obvious that γ carries every component containing a marked point to itself. Now let C = ∪Ci be the decomposition of C in irreducible components, let νi : Ni → Ci be the normalization map, denote by Ei the divisor of all points of Ni mapping to singular points of C, and by Di the divisor of all points of Ni mapping to one of the pj . Clearly, 2 2 (D)) = H 0 (Ni , ωN (Ei + Di )) . H 0 (C, J ωC i i


191

Each summand is nonzero, except those for which Ni is rational and Ei + Di has degree three. In particular, for any i such that Ni is not 2 of this kind, J ωC (D) has nonzero sections vanishing on all components 2 (D)), it follows from other than Ci . Since γ acts trivially on H 0 (C, J ωC this remark and the previous ones that γ carries every component of C into itself, except possibly those smooth rational components which meet the rest of the curve in three points and do not contain marked points. Since, however, the singular points of C are left fixed, the only case where there can be an interchange of components is where C is the union of two smooth rational curves meeting at three points (hence a curve of genus two), n = 0, and γ is the hyperelliptic involution. We may then assume that each component of C is carried into itself by γ. It follows in particular that γ must be the identity on each smooth rational component of C. If Ci is not one of these components, one immediately sees that 2 (Ei + Di ) ≥ deg ωNi + 3 deg ωN i with the following four exceptions: a) Ci is smooth of genus 1 and contains exactly two points which are either marked or points of contact with the rest of C; b) Ci is a curve of genus 1 with one node and contains exactly two points which are either marked or points of contact with the rest of C; c) Ci = C has genus 1, n = 1; d) Ci has genus 1, does not contain marked points, and meets the rest of C at one point; e) Ci = C has genus two, and n = 0. 2 Except in these cases, we then conclude that |ωN (Ei + Di )| embeds Ni i 2 in projective space, and since γ acts trivially on H 0 (Ni , ωN (Ei + Di )), it i must act trivally on Ci as well. We now examine cases a), b), c), d), e) above, in this order. In cases a) and b), we denote by p and q the points which are either marked or 2 (Ei + Di )| yields points of contact with the rest of C. In case a), |ωN i 1 a degree two map of Ci = Ni to P , so that the restriction of γ to Ci can only be the identity or the sheet interchange. This last possibility, however, cannot occur, since p and q are mapped to the same point of P1 , and we know that singular points and marked points of C are left fixed by γ. In case b), if z is a suitable local parameter centered at p, γ must be, locally, of the form

(4.13) where ζ is a root dimension 1, and a a(dz)2 /z, where a is be equal to 1, i.e., γ

z → ζz , 2 (Ei + Di )) has of unity. The space H 0 (Ni , ωN i nonzero section is, locally near p, of the form a nonvanishing function. By γ-invariance, ζ must must be the identity on Ci .

192


In cases c) and d), we denote by p the unique point which is either marked or a point of contact with the rest of C. As before, γ is locally of the form (4.13) at p. Proceeding as above, one sees that a nonzero 2 (Ei + Di )) is of the form a(dz)2 at p, where a is section of H 0 (Ni , ωN i a nonvanishing function. Such a section can be invariant under γ only if ζ 2 = 1. In case c), this corresponds to the exceptional case i) in the statement of the proposition. In case d), suppose that γ does not act trivally on Ci , i.e., that ζ = −1. Then γ cannot act as the identity on the component of C meeting Ci . Assume in fact that this is the case. Letting zw = 0 be the equation of C at p, if a belongs to Cp and zw = aε is the corresponding first-order deformation, γ clearly sends it to zw = −aε, that is, γ acts on Cp as multiplication by −1, contrary to our assumption. Thus, if Cj is the component of C meeting Ci , by our previous considerations Cj must also fall under case d). Hence, C equals Ci ∪ Cj and has genus two. Moreover, γ is the symmetry about the point of attachment, both in Ci and in Cj , and therefore is the hyperelliptic involution. It remains to examine case e). When C is smooth, we have already seen at the beginning of the proof that the only automorphisms acting trivially on Ext1 (Ω1C , OC ) are the identity and the hyperelliptic 2 involution. Suppose C = Ci is singular. If C has one node, |ωN (Ei )| i 1 maps Ni in two-to-one fashion to P , so that γ must be the identity or the hyperelliptic involution. If C has two nodes, its only nontrivial automorphism fixing the two nodes is the hyperelliptic involution. Q.E.D. While there exist no Kuranishi families for a nonstable nodal curve (C; p1 , . . . , pn ), it is possible to construct what is called a versal family. By this one means a deformation (4.1) which satisfies condition K) except for uniqueness, which is replaced by the weaker property that the Kodaira–Spencer map at the central fiber be an isomorphism. It is a consequence of the definition that, modulo shrinking the base, a versal deformation of an n-pointed nodal curve is unique up to an isomorphism, which however need not be unique. The existence of versal families follows easily from the one of Kuranishi families for stable curves. We add to the marked points of C the minimum number of additional ones needed to get a stable If curve (C; p1 , . . . , pn , pn+1 , . . . ) and look at its Kuranishi family. one ignores the marked sections that go through the added points, one gets a deformation of (C; p1 , . . . , pn ) which is the required versal family. In fact, given a deformation of (C; p1 , . . . , pn ), by suitably adding sections and possibly shrinking the base, one obtains a deformation of (C; p1 , . . . , pn , pn+1 , . . . ), which comes by pullback from the Kuranishi family. As for the second characteristic property of a versal family, it suffices to observe that the exact sequence (3.14) coincides with the corresponding sequence for (C; p1 , . . . , pn , pn+1 , . . . ). Exercise (4.14). Let C be a stable hyperelliptic curve. Use the cut-

§5 The Hilbert scheme of ν -canonical curves

193

and-paste methods employed in the construction of (3.16) to show that there is a family of stable curves over a disk whose central fiber is C and whose remaining fibers are smooth hyperelliptic. Exercise (4.15). Consider a family of stable n-pointed curves C →B,

σi : B → C, i = 1, . . . , n .

Assume that it is Kuranishi for any one of its fibers. Consider the family π2 : C ×B C → C,

τi , i = 1, . . . , n ,

where π2 is the projection to the second factor, and τi (x) = (σi (x), x). Apply to this family and to the diagonal the stabilization procedure described in Section 8 of Chapter X. Show that the resulting family is a Kuranishi family for any one of its fibers. 5. The Hilbert scheme of ν-canonical curves. Consider the subset of the appropriate Hilbert scheme consisting of all stable curves of genus g embedded by the ν-canonical system for sufficiently large ν. The main aim of this section is to show that this subset is in fact a smooth subscheme of the Hilbert scheme and to compute its tangent space and dimension. We will also generalize this result to the case of stable n-pointed curves. We first need to show that for a flat family of curves, the condition of being nodal is an open one in the Zariski topology. Proposition (5.1). Let ϕ : X → S be a flat proper morphism of schemes or of analytic spaces. Then the set of all s ∈ S such that Xs = ϕ−1 (s) is not a connected nodal curve is Zariski-closed in S. If, in addition, n sections σ1 , . . . , σn of ϕ are given, then the set of all s ∈ S such that (Xs ; σ1 (s), . . . , σn (s)) is not a connected n-pointed nodal curve is Zariski-closed in S. We shall prove the proposition for a morphism of schemes, the proof for a morphism of analytic spaces being essentially identical, except for terminological changes, but simpler. Without loss of generality we may assume that S is connected and that at least one fiber of ϕ (and hence any fiber, by flatness) has dimension 1. Observe that, for any (closed) point s of S, the dimension of H 0 (Xs , OXs ) is 1 when Xs is connected and reduced. Furthermore, if α

→ K1 → K2 → · · · K0 − is a complex of free sheaves on an open subset U of S which functorially calculates the higher direct images of OX (cf. Section 3 in Chapter IX), then the locus of the points s ∈ U such that the dimension of

194


H 0 (Xs , OXs ) is strictly greater than 1 is the locus where the rank of α is rank(K 0 ) − 2 or less. As such, it is Zariski-closed. This means that it suffices to prove the proposition under the additional assumption that H 0 (Xs , OXs ) has dimension 1 for all s. In particular, we may assume that all the fibers of ϕ are connected and do not have embedded components. Since ϕ is proper, to prove the proposition, it suffices to show that, for a point of X, the condition of being neither smooth nor a node in its fiber is a closed one. Possibly after shrinking S, we may embed X in a product Pr × S in such a way that ϕ is the restriction to X of the projection to the second factor. Now suppose that a fiber Xs is a nodal curve. Then Xs is a local complete intersection in Pr . In fact, a nodal curve is obviously a local complete intersection from the anaytic point of view; on the other hand, a subscheme of Cr is a local complete intersection if and only if it is such as an analytic subspace of Cr . As we have seen (cf. Lemma (5.25) in Chapter IX), the property of being a local complete intersection is an open one. Therefore we may suppose that all the fibers of ϕ are local complete intersections. Furthermore, a nodal fiber of ϕ is locally the complete intersection of r − 2 smooth hypersurfaces meeting transversely and a further hypersurface. This also is an open property, and we may therefore suppose that it is enjoyed by all fibers of ϕ. More precisely, we may suppose that Pr × S is covered with a finite number of open sets U isomorphic to open subsets of Cr × S and that X ∩ U is defined by equations F, F3 , . . . , Fr , where the intersection of the subscheme F3 = · · · = Fr = 0 with U ∩ (Pr × {s}) is smooth for any s. We denote by x1 , . . . , xr linear coordinates on Cr and regard Us = U ∩ (Pr × {s}) as an open subset of Cr and hence Xs ∩ U as a locally closed subscheme of Cr . We also regard F, F3 , . . . , Fr as functions on (varying) open subsets of Cr depending on parameters in S. Let s0 be a point of S, and p0 a point of Xs0 . After a linear change of coordinates in Cr we may suppose that x1 , x2 , F3 , . . . , Fr give local coordinates on Us0 at p0 , and hence, possibly after shrinking U and S, at every point of Us for every s ∈ S. We set yi = xi for i = 1, 2 and yi = Fi for i > 3. Clearly, p is a smooth point of Xs if and only if one of the two derivatives ∂F/∂yi , i = 1, 2, does not vanish at p. On the other hand, by Lemma (2.3) in Chapter X, p is a node if and only if these same derivatives vanish at p, but the Hessian determinant det

∂2F ∂y12 ∂2F ∂y1 ∂y2

∂2F ∂y1 ∂y2 ∂2F ∂y22

does not. The proof will be complete if we can show that these are algebraic conditions and not merely analytic ones. That this is the case follows from the observation that all partial derivatives with respect to the y variables of a rational function of x1 , . . . , xr are in fact rational


195

functions of the x variables. To prove this, we may clearly restrict to first derivatives, since the statement for higher derivatives then follows immediately by induction. Let then G be a rational function, and write ∂G ∂G ∂xj = . ∂yi ∂xj ∂yi j The functions ∂G/∂xj are rational functions of the x variables. As for the functions ∂xj /∂yi , it suffices to notice that, by Cramer’s rule, the entries of the Jacobian matrix ∂x ∂y are rational functions of the entries ∂y of ∂x , which are themselves rational functions of the x variables. This finishes the proof of the proposition in the unpointed case. The proof of the last statement is simple and left to the reader.

Remark (5.2). The final part of the argument we used to prove Proposition (5.1) also shows that, in an algebraic family X → S of nodal curves, the locus of points which are singular in their fiber is a closed subscheme of X. We now return to the central subject of this section. Consider a pi . It stable n-pointed genus g curve (C; p1 , . . . , pn ), and set D = follows, for instance, from part iii) of Lemma (6.1) in Chapter X that, for ν ≥ 3, the ν-fold log-canonical sheaf (ωC (D))ν is very ample and embeds C in Pr , where r = (2ν − 1)(g − 1) + νn − 1. Its Hilbert polynomial is (5.3)

pν (t) = (2νt − 1)(g − 1) + νnt .

We denote by H the Hilbert scheme of (n + 1)-tuples (Y ; z1 , . . . , zn ), where Y is subscheme of Pr with Hilbert polynomial pν (t), and z1 , . . . , zn are points of Y (cf. the end of Chapter IX, Section 7). We just proved that the (nonempty) subset U of H parameterizing connected n-pointed nodal curves is Zariski open. Let Yu π σi , i = 1, . . . , n, u U be the restriction of the universal family. Of course, a general point of U does not correspond to an n-pointed curve embedded by the νfold log-canonical sheaf. We wish to construct a subscheme Hν,g,n of U parameterizing such curves. To be more precise, we would like Hν,g,n to satisfy the following universal property. Write L for the pullback to Y of the hyperplane bundle on Pr and observe that F = (ωπ ( σi ))ν L−1 has zero relative degree. Consider a map α : X → U and the corresponding cartesian diagram

196


β

Y ×U X η u X

wY π

α

u wU

Suppose that β ∗ F is isomorphic to η ∗ G for some line bundle G on X. Then α factors through Hν,g,n → U . We denote by W the subscheme of U defined by the sum of the gth Fitting ideal of R1 π∗ (F ) and of the gth Fitting ideal of R1 π∗ (F −1 ), and let π : Y → W be the restriction of Y → U . By construction, the rank of π∗ F and π∗ F −1 is everywhere at least equal to 1. The scheme Hν,g,n is defined as the open subset of W where the rank of the multiplication map π∗ F ⊗ π∗ F −1 → π∗ OY = OW is equal to one. We shall sometimes refer to Hν,g,n as the Hilbert scheme of ν-log-canonically embedded, stable, n-pointed, genus g curves. We end this section by showing that Hν,g,n is smooth of dimension 3g − 3 + n + (r + 1)2 − 1, where r = (2ν − 1)(g − 1) + nν − 1 (recall that ν ≥ 3). We shall do this by explicitly computing the tangent space to Hν,g,n . Let then C ⊂ Pm be a nodal curve, and let p1 , . . . , pn be distinct smooth points of C. Set D = pi . Let IC be the ideal sheaf of C in Pm , and ID the one of D. Look at the commutative diagram 0

w IC /IC2

0

u 2 w ID /ID

d

w Ω1Pr ⊗ OC

w Ω1C

u w Ω1Pr ⊗ OD

u w0

w0

Both rows are exact (cf. Exercise (2.21)). Hence we can view the diagram as an exact sequence 0 → C • → B • → A• → 0 involving the complexes, all concentrated in degrees 0 and 1, (5.4) 2 ) , B • = (Ω1Pr ⊗OC → Ω1Pr ⊗OD ) , A• = (Ω1C → 0) . C • = (IC /IC2 → ID /ID We define another complex D• = (D 0 → D1 ) by setting D0 = OC , D1 = OD . From (5.4) we get another exact sequence (5.5) 0 → HomOC (A• , D• ) → HomOC (B• , D• ) → HomOC (C • , D• ) → Ext1OC (A• , D• ), and our next goal is to understand the terms appearing in it. We begin from the left. Since A1 = 0, the homomorphisms from A• to D • are just


197

the homomorphisms from A0 = Ω1C to the kernel of OC → OD , that is, to OC (−D). Passing to the next term, it is clear that a homomorphism from B • to D• is completely determined by its degree zero part Ω1Pr ⊗OC → OC ; hence this term is just HomOC (Ω1Pr ⊗ OC , OC ). The next term is the tangent space at h = (C; p1 , . . . , pn ) to the Hilbert scheme H of (n + 1)tuples (Y ; q1 , . . . , qn ), where Y is a subscheme of Pm and q1 , . . . , qn points on it (cf. Lemma (8.8)). As for Ext1OC (A• , D• ), we know from formula (3.11) that it is isomorphic to Ext1OC (Ω1C , OC (−D)) and from (3.13) that it classifies isomorphism classes of first-order deformations of (C; p1 , . . . , pn ). Summing up, we get from (5.5) the exact sequence (5.6)

0 → HomOC (Ω1C , OC (−D)) → HomOPm (Ω1Pm , OC ) → β

→ Ext1OC (Ω1C , OC (−D)) Th (H) −

Moreover, the homomorphism β is just the Kodaira–Spencer map at h associated to the universal family over H. Now assume that C is connected and not contained in any hyperplane and that the linear system cut out on C by hyperplanes is complete. Combining the exact cohomology sequence of the Euler sequence (5.7)

0 → OC → OC (1)m+1 → HomOPm (Ω1Pm , OC ) → 0

with (5.6), we get the diagram of exact sequences

0

w H 0 (C, OC )

w H 0 (C, OC (1))⊕(m+1)

0 u 1 Hom(ΩC , OC (−D)) u w Hom(Ω1Pm , OC ) δ w H 1 (C, OC ) uγ Hom(C • , D• ) uβ 1 1 Ext (ΩC , OC (−D))

Th (H)

As we have observed, the map β associates to every first-order embedded deformation of (C; p1 , . . . , pn ) the corresponding abstract deformation. The j

elements of Hom(Ω1Pm , OC ) correspond to fiber space maps C×S → Pm ×S, The where S = Spec C[ε], extending the inclusion of C in Pm . map δ associates to any such object the infinitesimal deformation of line bundles on C given by j ∗ (OPm (1) ⊗ OS )(OC (1) ⊗ OS )−1 . Finally, H 0 (C, OC (1))⊕(m+1) /H 0 (C, OC ) is the tangent space to the projective linear group P GL(m+1). All these statements should by now be familiar, except perhaps those concerning Hom(Ω1Pm , OC ) and δ. We shall return to these later. For the moment, we go on with the proof that Hν,g,n is smooth of dimension 3g − 3 + n + (r + 1)2 − 1, specializing the above

198


considerations to the case where (C; p1 , . . . , pn ) is stable and C ⊂ Pm is embedded by the ν-fold log-canonical system (so that m = r). Pick an element v of Th (H) which is tangent to Hν,g,n , and suppose in addition that it comes via γ from an element α of Hom(Ω1Pm , OC ), i.e., from a map of fiber spaces j : C × S → Pm × S, such that OPm (1) ⊗ OS pulls back to (ωC (D))ν ⊗ OS . Then δ(α) = 0 by the definition of δ. This means that α belongs to H 0 (C, OC (1))⊕(r+1) /H 0 (C, OC ). Conversely, it is clear that the image of H 0 (C, OC (1))⊕(r+1) /H 0 (C, OC ) in Th (H) is contained in the tangent space to Hν,g,n at h. At this point we need the following result, to be proved later. Lemma (5.8). Let (C; p1 , . . . , pn ) be a stable n-pointed curve. Then HomOC (Ω1C , OC (−

pi )) = 0 .

Assuming the lemma, we get an exact sequence 0 → H 0 (C, OC (1))⊕(r+1) /H 0 (C, OC ) → Th (Hν,g,n ) → Ext1 (Ω1C , OC (−D)) , λ

where λ stands for the restriction of β to Th (Hν,g,n ). On the other hand, recall that Ext1 (Ω1C , OC (−D)) classifies abstract infinitesimal deformations of (C; p1 , . . . , pn ). Any such can be embedded via the ν-fold log-canonical system, so the map λ is surjective. In conclusion, there is an exact sequence (5.9) 0 → H 0 (C, OC (1))⊕(r+1) /H 0 (C, OC ) → Th (Hν,g,n ) → Ext1 (Ω1C , OC (−D)) → 0 . In particular, dim Th (Hν,g,n ) = 3g − 3 + n + (r + 1)2 − 1 , and hence this quantity is an upper bound for the dimension of Hν,g,n . If we can show that it is also a lower bound, we will have proved that Hν,g,n is smooth of dimension 3g − 3 + n + (r + 1)2 − 1, as announced. To this end, consider the (3g − 3 + n)-dimensional deformation ϕ : C → (B, b0 ) given byTheorem (3.17). We shall write Cb for ϕ−1 (b) and Db for the divisor σi (b). Set G = P GL(r + 1) and consider the principal G-bundle over B defined by B = {(b, F )| b ∈ B, F a basis for H 0 (Cb , (ωCb (Db ))ν ), modulo homotheties}. Let F0 be the basis corresponding to the embedding C ⊂ Pr . On the pulled back family ψ

X = B ×B C → B ,

τi : B → X , i = 1, . . . , n,

§5 The Hilbert scheme of ν -canonical curves there is a canonical projective frame for ψ∗ ((ωX /B ( used to give a fiber space embedding X[ y

[ ] [ B

199

τi ))ν ) which can be

w Pr × B

By universality this family induces a map ξ from B to Hν,g,n . There is a commutative diagram with exact rows

where ρ is the Kodaira–Spencer map. Since ρ is an isomorphism by construction of the family C → (B, b0 ), we get that dξ is an isomorphism. This shows that ξ is a local isomorphism at (b0 , F0 ). Since B has the expected dimension 3g − 3 + n + (r + 1)2 − 1, we get dimh Hν,g,n = dim Th (Hν,g,n ) = 3g − 3 + n + (r + 1)2 − 1 , proving our claim. We now come to the proof of Lemma (5.8). Set D = pi , and consider the normalization map α : N → C. The exact sequence (2.20) 0 → P → Ω1C → ωC → Q → 0 , splits into two short exact sequences 0 → P → Ω1C → α∗ ωN → 0 ,

0 → α∗ ωN → ωC → Q → 0 .

Since P is concentrated at the nodes, the first one of these tells us that Hom(Ω1C , OC (−D)) = Hom(α∗ (ωN ), OC (−D)) . At a node, α∗ (ωN ) is generated by two sections, one vanishing identically on a branch of the node and one vanishing identically on the other. The image of either one under a homomorphism to OC (−D) vanishes identically on one of the branches and hence at the node. Thus, denoting by J the ideal of the singular locus of C and by D the divisor consisting of all the points of N mapping to nodes of C, Hom(α∗ (ωN ), OC (−D)) = Hom(α∗ (ωN ), J OC (−D)) = Hom(α∗ (ωN ), α∗ (ON (−α∗ (D) − D ))) −1 = H 0 (N, ωN (−α∗ (D) − D )).

200


−1 The last group vanishes, since ωN (−α∗ (D) − D ) is the dual of the log-canonical sheaf of N , which has positive degree on every component of N , by stability. Finally, as we promised, we return to the deformation-theoretic interpretation of the group Hom(Ω1Pm , OC ) and of the coboundary map δ : Hom(Ω1Pm , OC ) → H 1 (C, OC ) in the long cohomology sequence of (5.7). We recall that here C ⊂ Pm is a nodal curve. We set S = Spec C[ε]. Suppose we are given a map

(5.10)

j

C × S → Pm × S

of fiber spaces over S extending the inclusion C ⊂ Pm . If f is a function or a form on Pm , or on C, when no confusion is likely, we shall denote by the same symbol its pullback to Pm × S, or to C × S, via the projection to the first factor. Given a section ϕ of Ω1Pm , pull it back to C × S via the composition of j with the projection of Pm × S to Pm . The pulled-back form can be written as ψ + f dε, where ψ is a form on C, and f is a function on C. Associating f to ϕ yields a well-defined element of Hom(Ω1Pm , OC ). We wish to show that any homomorphism α : Ω1Pm → OC comes, via the procedure we have just described, from a unique morphism as in (5.10). What we have to find is a homomorphism η : OPm ×S → OC×S extending the restriction homomorphism OPm → OC . For any such η, given a function f + εh on Pm × S, where f and h are functions on Pm , we must have η(f + εh) = η(f ) + εh C

and

η(f ) = f C + εσ

for a suitable function σ on C. Thus, η ∗ (df ) = dη(f ) = d(f C + εσ) = df C + εdσ + σdε . If α comes from η, we must then have σ = α(df ) . This proves that α comes from at most one η. To conclude the proof that Hom(Ω1Pm , OC ) classifies maps of fiber spaces as in (5.10) extending C ⊂ Pm , it remains to show that (5.11) η(f + εh) = f C + ε(α(df ) + hC ) defines a homomorphism OPm ×S → OC×S extending OPm → OC . The only thing that needs some verification is that such an η is compatible


201

with multiplication of functions. This is a straightforward calculation which is left to the reader. We also leave it to the reader to check that the interpretation of Hom(Ω1Pm , OC ) as the set of fiber space maps C × S → Pm × S extending OPm → OC is compatible, via γ, with the interpretation of Hom(C • , D• ) as the tangent space to the Hilbert scheme. We now come to the homomorphism δ : Hom(Ω1Pm , OC ) → H 1 (C, OC ) . What we claim is that δ associates to any map j as in (5.10) the infinitesimal deformation of line bundles on C given by j ∗ (OPm (1) ⊗ OS )(OC (1) ⊗ OS )−1 . To see this, we begin by choosing homogeneous coordinates x0 , . . . , xn on Pm ; for each i, we also let Ui be the open set in Pm (or in Pm ×S) where xi does not vanish. Next, let α ∈ Hom(Ω1Pm , OC ) be the homomorphism corresponding to j, and recall that the pullback homomorphism η = j ∗ : OPm ×S → OC×S is given by formula (5.11). For brevity, we write L to denote the line bundle OPm (1) ⊗ OS on Pm × S and M to denote OC (1) ⊗ OS . Relative to the cover {Uh }, the line bundle ∗ L is given by the transition functions ϕhk = xh /x k , so j (L) is given by the transition functions ψhk = η(ϕhk ) = (xh /xk ) C + εα(d(xh /xk )). On the other hand, d

xh xk

=

1 xh dxh xk dxk dxh − 2 xh dxk = − ; xk xk xk xh xh xk

In conclusion, the transition functions for j ∗ (L) are ψhk =

! dxh xh dxk 1 + εα . − xk xh xk C

Thus the Kodaira–Spencer class of the infinitesimal deformation of OC given by j ∗ (L) ⊗ M −1 is the class in H 1 (C, OC ) of the cocycle uhk = α

dxk dxh − xh xk

.

We must show that this same cocycle represents the coboundary δ(α). On each open set Uh there are sections ξih of O(1) such that α corresponds to contraction with the vector field i

ξih

∂ . ∂xi

On the overlap between Uh and Uk we have that ξik = ξih + xi fhk , where the fhk are holomorphic; the coboundary δ(α) is the class in H 1 (C, OC )

202


of the cocycle {fhk }. On the other hand,

dxk dxh − xh xk ! ∂ dxh ∂ dxk − ξik = ∂xi xh ∂xi xk i δih δik = ξik − xh xk i

uhk = α

ξhk ξk − k xh xk ξhh ξk = − k + fhk . xh xk =

Thus the two cocycles {uhk } and {fhk } differ by the coboundary of {ξhh /xh }. This completes the deformation-theoretic description of the homomorphism δ. The following statement summarizes what has been proved in this section. Proposition (5.12). Let g and n be nonnegative integers such that 2g −2+n > 0, and let ν ≥ 3 be an integer. Set r = (2ν −1)(g −1)+νn−1. Then the Hilbert scheme Hν,g,n parameterizing stable n-pointed genus g curves embedded in Pr via the ν-fold log-canonical system is smooth, quasi-projective, and of dimension dim(Hν,g,n ) = 3g − 3 + n + (r + 1)2 − 1 . Moreover, the tangent space to Hν,g,n at a point h fits into an exact sequence (5.9), where λ is the Kodaira–Spencer map at h of the universal family on Hν,g,n . Remark (5.13). It is useful to observe that Hν,g,n is naturally a smooth locally closed subscheme of the product Hilbrpν (t) ×(Pr )n , where pν (t) is given by (5.3). The natural action of G = P GL(r + 1) on this product restricts to an action on Hν,g,n , and the set-theoretical quotient Hν,g,n /G is just the set of isomorphism classes of stable npointed genus g curves. Recall that the explicit construction of the Hilbert scheme we have given in Chapter IX exhibits Hilbpr ν (t) as a closed subscheme of a big projective space PM = P ∧k V , where V = H 0 (Pr , OPr (N ))∨

§6 Construction of Kuranishi families

203

for some large N , and where N +r k= − pν (N ) . r Hence, Hν,g,n ⊂ PM × (Pr )n ⊂ PK , where the last inclusion is the Segre embedding. Of course, the action of G on Hν,g,n is given by G = P GL(r + 1) acting on PK via the obvious representation. 6. Construction of Kuranishi families. We shall now use the Hilbert scheme Hν,g,n introduced in preceding section to construct Kuranishi families. Actually, we shall construct Kuranishi families endowed with several additional properties (e.g., they will be algebraic). We will be amply repaid for the additional effort involved when we construct moduli spaces in Chapter XII. Pick a stable n-pointed curve (C; p1 , . . . , pn ), fix an integer ν ≥ 3 and a ν-fold logcanonical embedding ϕ : C → Pr ,

r = (2ν − 1)(g − 1) − 1 + νn .

We identify C with its image via ϕ and denote by x0 the corresponding point in Hν,g,n (which from now on we will simply denote by H). We let (6.1)

π:Y →H,

σi : H → Y , i = 1, . . . , n ,

be the universal family on H, and for any subscheme or analytic subspace W of H, we shall write πW : YW → W to indicate its restriction to W ; we will often omit mentioning the sections, and in any case their restrictions will always be denoted by the same symbols σ1 , . . . , σn . Following Remark (5.13), we regard H as a smooth locally closed subscheme of a large projective space PK , acted on by G = P GL(r +1) → P GL(K + 1). For any point x ∈ H, let Gx ⊂ G be the stabilizer of x. Since Gx can be identified with the automorphism group of the corresponding stable curve, it is a finite group. Consider the orbit O(x0 ) ⊂ H of x0 under G; this is a smooth subvariety of H of dimension (r + 1)2 − 1 passing through x0 . Since the linear subspace T of PM tangent to O(x0 ) at x0 is obviously Gx0 -invariant, there is a Gx0 -invariant linear subspace L of PM of complementary dimension such that L ∩ T = {x0 }. We now intersect H with L and claim that we can find a Zariski-open neighborhood X of x0 in H ∩ L such that i) X is affine

204 ii) iii) iv) v)


X intersects O(y) transversely at y for every y ∈ X, X is Gx0 -invariant, for every y ∈ X, Gy ⊂ Gx0 . for every y ∈ X, there is a Gy -invariant neighborhood U of y in X, for the analytic topology, such that {γ ∈ G : γU ∩ U = ∅} = Gy .

That the first two properties can be satisfied is obvious, as is obvious that, once i), ii), iv, and v) are met, iii) can be satisfied as well. We next prove v), arguing by contradiction. Suppose there are sequences xn and yn of points of X converging to y, and elements γn ∈ G Gy such that γn xn = yn . By Theorem (5.1) of Chapter X, possibly after passing to a subsequence, there is a γ ∈ G such that lim γn = γ. n→∞

Clearly γ ∈ Gy , so that, replacing γn with γn γ −1 and yn with γ −1 yn , we may assume that γ is the identity element e of G. Now consider the multiplication map F : G × X → H given by F (η, x) = ηx. The transversality condition ii) tells us that F is a local biholomorphism at the point (e, y). On the other hand, we just constructed two sequences (e, yn ) and (γn , xn ), with γn = e for every n, both converging to (e, y), and such that F (e, yn ) = F (γn , xn ). This is absurd. In conclusion, one may find a neighborhood U of y in X such that the set of elements γ in G having the property that γU ∩ U = ∅ is equal to Gy . By the finiteness of Gy the neighborhood U can be chosen to be Gy -invariant. To show that iv) can be met, we argue as follows. Set I = {(y, γ)|y ∈ X, γ ∈ Gy } . Notice that I = IsomX (YX , YX ), and hence I is proper over X, by Theorem (5.1) of Chapter X. Therefore, removing from X the projections of those components of I which do not meet {x0 } × Gx0 , we can assume that, given any component I of I, we have I ∩ ({x0 } × Gx0 ) = ∅. Set J = {(y, γ) ∈ I | γ ∈ Gx0 }. Clearly, J is a nonempty Zariski-closed subset of I . We must show that I = J. To do this, it suffices to show that J contains a subset of I which is open in the ordinary topology. This is an immediate consequence of v) applied to y = x0 . This concludes the proof that one can construct an X ⊂ H ∩ L satisfying i), ii), iii), iv), and v). Now consider the restriction of the universal family (6.1) to X. We shall show that YX (6.2) u

π

(X, x0 )

σi : X → YX , i = 1, . . . , n , ∼ =

ϕ : (C; p1 , . . . , pn ) −→ (π −1 (x0 ); σ1 (x0 ), . . . , σn (x0 ))


205

is a Kuranishi family for (C; p1 , . . . , pn ). Let D (6.3)

η u

τi : S → D , i = 1, . . . , n , ∼ =

ψ : (C; p1 , . . . , pn ) −→ (η −1 (s0 ); τ1 (x0 ), . . . , τn (x0 ))

(S, s0 ) be another deformation of (C; p1 , . . . , pn ). As we already observed, the transversality condition ii) shows that there are a neighborhood B of x0 in X and a neighborhood V of e in G such that the multiplication map F :V × B −→ H (γ, b) → γb is a biholomorphism onto an open neighborhood W of x0 in H. The ν (ν pi )). inclusion ϕ : C → Pr gives a canonical basis for H 0 (C, ωC −1 Transplant this basis to η (s0 ) via theisomorphism ψ : C → η −1 (s0 ) ν (ν τi )) over a neighborhood A and extend it to a frame of η∗ (ωD/S of s0 . This frame can be used to realize η : DA → A as a family of subschemes of Pr : DA y w A × Pr [ η [[ [ ^ u [ A By the universal property of the Hilbert scheme this family of n-pointed curves is induced by a unique holomorphic map f : A → H. Shrinking A, if necessary, we may assume that f (A) ⊂ W . We then get a cartesian diagram f˜ DA w YW ⊂ Y (6.4)

η u A

f

π u wW ⊂H

of families of n-pointed curves. The situation is illustrated by the picture below.

206


We now define α : A → B ⊂ X by setting α = pB F −1 f , where pB : V × B → B is the projection. We also set β = pV F −1 f . We wish to show that the deformation YB u

π

σi : B → YB , i = 1, . . . , n ,

ϕ : C → π −1 (x0 )

(B, x0 ) induces, via α, the restriction of the deformation (6.3) to A. Look at the cartesian diagram (6.4). The group G acts equivariantly on Y → H; therefore, if we replace f and f˜ with f and f˜ defined by f (a) = β(a)−1 f (a) , f˜ (q) = β(η(q))−1 f˜(q) , we get the other cartesian diagram f˜

DA η u A

f

w YW π u wW

On the other hand, by definition, f (a) = F (β(a), α(a)) = β(a)α(a) so that f = α, and the cartesian diagram in question reduces to DA η u A

f˜

α

w YB π u wB⊂W

Since β(s0 ) = e ∈ G, the identification ϕ of C with π −1 (x0 ) pulls back via f˜ to the identification ψ : C → η −1 (s0 ). This shows that (6.2) is a Kuranishi family for (C; p1 , . . . , pn ). Notice that, among properties i)–v) of X, the only one we used in proving the universal property of (6.2) is the transversality condition ii). In particular it follows that (6.2) is a Kuranishi family for any one of its fibers. We may summarize what we have done in the following:


207

Theorem (6.5). Let ν ≥ 3 be an integer. Let (C; p1 , . . . , pn ) ⊂ Pr be a stable n-pointed genus g curve, embedded in Pr , r = (2ν −1)(g−1)+νn−1, via the ν-fold log-canonical system. Let x0 ∈ Hν,g,n be the corresponding Hilbert point, and let Aut(C; p1 , . . . , pn ) = Gx0 ⊂ G = P GL(r + 1) be the stabilizer of x0 . Then there is a locally closed (3g − 3 + n)-dimensional smooth subscheme X of Hν,g,n passing through x0 such that the restriction to X of the universal family over Hν,g,n is a Kuranishi family for all of its fibers and hence, in particular, a Kuranishi family for (C; p1 , . . . , pn ). In addition, one can choose an X with the following properties: a) b) c) d)

X is affine; X is Gx0 -invariant; for every y ∈ X, the stabilizer Gy of y is contained in Gx0 ; for every y ∈ X, there is a Gy -invariant neighborhood U of y in X, for the analytic topology, such that {γ ∈ G : γU ∩ U = ∅} = Gy .

Now let C (6.6) u

π

σi : X → C , i = 1, . . . , n ,

C = π −1 (x0 )

(X, x0 ) be the Kuranishi family constructed in the theorem. Since the projective group P GL(r + 1) acts equivariantly on the Hilbert scheme Hν,g,n and on the universal family over it, the Gx0 -invariance of X implies that the finite group Aut(C; p1 , . . . , pn ) = Gx0 acts equivariantly on C → X. Moreover, the action of Aut(C; p1 , . . . , pn ) on the central fiber π −1 (x0 ) = C is the natural one. This allows us to interpret Theorem (6.5) as the assertion that there exists a standard algebraic Kuranishi family for (C; p1 , . . . , pn ) in the sense of the following definition. Definition (6.7) (Standard algebraic Kuranishi family). Let (C; p1 , . . . , pn ) be a stable n-pointed curve and set G = Aut(C; p1 , . . . , pn ). We will say that a Kuranishi family (6.6) for (C; p1 , . . . , pn ) is a standard algebraic Kuranishi family if the following conditions are satisfied. Denote by Cy the fiber of π over y and let Gy be the automorphism group of (Cy ; σ1 (y), . . . , σn (y)). Then a) X is affine; b) the family is Kuranishi at every point of X; c) the action of the group Gx0 on the central fiber extends to compatible actions on C and X; d) for every y ∈ X, the automorphism group Gy is equal to the stabilizer of y in Gx0 . In particular, Gy is a subgroup of Gx0 ; e) for every y ∈ X, there is a Gy -invariant neighborhood U of y in X, for the analytic topology, such that any isomorphism (of n-pointed curves) between fibers over U is induced by an element of Gy .

208


The only property that may need some explanation is e). But this follows at once from part d) of the theorem. In fact, any isomorphism between fibers over points of U is induced by a projectivity, that is, by an element of G. The following is a local counterpart of the notion of standard algebraic Kuranishi family. Definition (6.8) (Standard Kuranishi family). Let (C; p1 , . . . , pn ) be a stable n-pointed curve and set G = Aut(C; p1 , . . . , pn ). We will say that a Kuranishi family X → (B, b0 )

τi : B → X , i = 1, . . . , n ,

ϕ : C −→ Xb0

for (C; p1 , . . . , pn ) is a standard Kuranishi family if the following conditions are satisfied: i) B is a connected complex manifold; ii) the family is Kuranishi at every point of B; iii) the action of G on the central fiber extends to compatible actions on X and B; iv) any isomorphism (of n-pointed curves) between fibers is induced by an element of G. Standard Kuranishi families always exist. In fact, given any Kuranishi family, there is a neighborhood of the base point such that the restriction of the family to this neighborhood is standard. By the uniqueness of the Kuranishi family (cf. (4.4)), it suffices to notice that this is true for a standard algebraic Kuranishi family, as is implicit in the definitions. Remark (6.9). The universal property of the Kuranishi family (6.6), which has been stated in the analytic setup, has an exact counterpart in the algebraic category, provided that one works in the étale topology. What is true is that, given an algebraic deformation (6.3), there is an étale base change (S , s0 ) → (S, s0 ), with S connected, such that there is a unique morphism of deformations from the pulled-back family to (6.6). Remark (6.10). A useful application of the existence of Kuranishi families is that any family of nodal curves can be locally embedded in a family of nodal curves with a reduced, or even smooth, base. To see this, start with an analytic family of nodal curves η : X → S and let s0 be a point of S. Let π : X → (B, b0 ) be a versal deformation for η −1 (s0 ). Then, possibly after shrinking S, there are a closed embedding S → T , where T is smooth, and a cartesian square X

β

η u S

α

wX π u wB


209

with b0 = α(s0 ). Out of this we construct another diagram of cartesian squares (η, β) w S×X w T ×X X η

(idS , π) (idT , π) u u u (idS , α) w S×B w T ×B S Clearly, T × B is smooth, and S → T × B is an embedding. What we have just proved holds also in the algebraic setup, provided that we interpret “locally” as meaning “locally in the étale topology,” and hence “shrinking S” as meaning that we base change η : X → S via a suitable étale morphism S → S where S is affine, and replace S with S . All the rest goes through unchanged. We close this section by describing the local structure of the Hilbert scheme AutB (X ) = IsomB (X , X ) parameterizing automorphisms of the fibers of a standard Kuranishi family α : X → (B, b0 ) for the stable n-pointed genus g curve (C; p1 , . . . , pn ). We already know, from Section 5 of Chapter X, that AutB (X ) is finite and unramified over B. We denote by G the automorphism group of (C; p1 , . . . , pn ). It follows from property iv) of standard Kuranishi families that there is a closed immersion AutB (X ) → G × B . " Aγ , where Aγ stands for the Lemma (6.11). Write AutB (X ) = intersection of AutB (X ) with {γ} × B, γ ∈ G = Aut(C; p1 , . . . , pn ). For each γ, let B γ be the fixed subspace of B under the action of γ. Then B γ is smooth, and Aγ maps isomorphically to B γ under the natural γ projection AutB (X ) → B. The tangent space toγ B at b0 is the space of 1 γ 1 γ-invariants Tb0 (B) = ExtOC (ΩC , OC (− pi )) . Finally, possibly after shrinking B, in suitable coordinates centered at b0 , each of the B γ is defined by linear equations. There is not much here that needs proof. That the B γ are defined by linear equations is a consequence of the following well-known observation of Henri Cartan [106]. Lemma (6.12). Let G be a finite group acting on a complex manifold U and fixing a point u ∈ U . Then, on a suitable neighborhood of u and in suitable coordinates centered at u, the action of G is linear. Proof. Choose a system of coordinates centered at u. For each element γ ∈ G, denote by γ its tangent transformation in the chosen coordinates. Set 1 −1 α= (γ ) γ. |G| γ∈G

210


This is a biholomorphism on a neighborhood of u, since its Jacobian at u is the identity. Moreover, one immediately checks that αγ = γ α for every g ∈ G. It follows that G acts linearly in the system of coordinates obtained from the original ones via α. Q.E.D. Lemma (6.12) proves, in particular, the smoothness of B γ . It also identifies the action of G on B with the one on Tb0 (B), proving the statement about the tangent space of B γ . Finally, notice that Aγ → B γ is a bijection since G acts equivariantly on X and B, and hence any γ fixing b ∈ B restricts to an automorphism of α−1 (b); that Aγ → B γ is an isomorphism then follows from Theorem (5.1) in Chapter X. This finishes the proof of Lemma (6.11). Remark (6.13). A straightforward consequence of Lemma (6.11) is that, if H is a subgroup of G = Aut(C; p1 , . . . , pn ), then the locus B H of those points of B which are fixed under the action of H is smooth and its tangent space at b0 can be identified with the space of invariants Tb0 (B)H = Ext1OC (Ω1C , OC (− pi ))H . Furthermore, H is clearly a subgroup of Aut(π −1 (b)) for any b ∈ B H . We next apply Lemma (6.11) to the case where C is a hyperelliptic stable curve and γ is its hyperelliptic involution. We first need a general remark. Let X → S be a proper flat morphism of analytic spaces and suppose that a finite group G acts on X, compatibly with π. We claim that X/G → S is flat. In fact, if p is a point of X, q its image in W = X/G, and s its image in S, then OW,q is the subring of H-invariants of OX,p , where H ⊂ G is the stabilizer of p. As such, when viewed as an OS,s -module, it is a direct summand of OX,p . The flatness of OW,q over OS,s thus follows from the one of OX,p . Lemma (6.14). Let C be a stable curve of genus g > 1, and let α : X → (B, b0 ) ,

C∼ = α−1 (b0 ) ,

be a standard Kuranishi family for C. Suppose that there are points b ∈ B arbitrarily close to b0 such that α−1 (b) is smooth hyperelliptic. Then C is hyperelliptic. Proof. Possibly after shrinking B, we may choose coordinates on it, centered at b0 , in which the action of Aut(C) is linear. Pick a point b such that α−1 (b) is smooth hyperelliptic. By property iv) of standard Kuranishi families, there is an automorphism γ of C which fixes b and acts on α−1 (b) as the hyperelliptic involution. Clearly, γ has order two. Let L ⊂ B γ be the (piece of) line joining b0 and b, and set Y = α−1 (L). Then Y → L is a family of stable curves, and γ acts on Y fiberwise. The fixed locus of γ does not contain components of fibers. In fact, if all points of a component E of a fiber were fixed, by Lemma (6.12) γ


211

would have to act nontrivially on the normal space to E at a general point and hence would have to act nontrivially on L, a contradiction. In particular, γ acts on C with isolated fixed points. We know that Z = Y /γ is flat over L. It is easily seen that the fibers of β : Z → L are nodal curves. On the other hand, since α−1 (b) is smooth hyperelliptic and γ induces on it the hyperelliptic involution, the general fiber of β is a P1 . Since the genus of β −1 (h) is locally constant as a function of h by flatness, it follows that C/γ is a nodal curve of genus zero. Q.E.D. Lemma (6.15). Let C be a hyperelliptic stable curve of genus g > 1, and let C∼ α : X → (B, b0 ) , = α−1 (b0 ) , be a standard Kuranishi family for C. Set H = {b ∈ B : α−1 (b) is hyperelliptic} and let W be any component of the locus in B parameterizing singular curves. Then: i) H is a smooth complex submanifold of B of dimension 2g − 1; ii) H and W are transverse at b0 ; iii) if b is a general point of H, then α−1 (b) is smooth. Proof. Since X → B is a Kuranishi family at any point of B, in the proof we may shrink B at will. We may thus assume that, in suitable global coordinates on B, the automorphism group of C acts linearly on B. We begin by proving ii) and iii). Look at the exact sequence (3.4) and rewrite the term on the right, that is, H 0 (C, Ext1 (Ω1C , OC )), grouping its summands according to node type, as the direct sum of the three terms T1 =

i>0

T2 =

p a node of type η0

T3 =

Ext1 (Ω1C,p , OC,p ),

p a node of type δi

Ext1 (Ω1C,p , OC,p ),

i>0

{p,q} a pair of nodes of type ηi

Ext1 (Ω1C,p , OC,p ) ⊕ Ext1 (Ω1C,q , OC,q ) .

The hyperelliptic involution γ acts on all terms of (3.4). More in detail, we claim that it acts on the term on the right via the trivial action on T1 and T2 , and by interchanging the summands in each term Ext1 (Ω1C,p , OC,p ) ⊕ Ext1 (Ω1C,q , OC,q ) of T3 . This last assertion is clear. As for T1 and T2 , it is convenient to use the isomorphism (cf. (3.8)) Ext1 (Ω1C,p , OC,p ) ∼ = TN,p1 ⊗ TN,p2 ,

212


where N stands for the partial normalization of C at p, and p1 , p2 are the points of N mapping to p. If p is a node of type δi , γ acts as multiplication by −1 on TN,p1 and TN,p2 , and hence acts trivially on their tensor product. If p is of type η0 , the action of γ is the interchange of the two factors and hence is again trivial. This proves our claim. What we have shown implies that there are γ-invariant elements of H 0 (C, Ext1 (Ω1C , OC )) which map to a nonzero element of Ext1 (Ω1C,p , OC,p ) for every p ∈ Sing(C). Any such element of H 0 (C, Ext1 (Ω1C , OC )) can be lifted to a γ-invariant element v ∈ Ext1OC (Ω1C , OC ) = Tb0 (B). In other words, v is tangent to B γ . On the other hand, Tb0 (W ) is a codimension 1 vector subspace of Tb0 (B) whose image in H 0 (C, Ext1 (Ω1C , OC )) is the direct sum of all the vector spaces Ext1 (Ω1C,p , OC,p ), with p a node of C, except one. This proves that Tb0 (W ) and Tb0 (B γ ) are transverse. By what we have just shown, there are vectors v ∈ Tb0 (B γ ) which are transverse to all components of the locus in B parameterizing singular curves. Thus, any deformation of C with v as Kodaira–Spencer class has the effect of smoothing all the nodes of C. In conclusion, the fiber above a general point of B γ is smooth. On the other hand, every fiber α−1 (b) with b ∈ B γ is hyperelliptic, and γ acts on it as the hyperelliptic involution. To see this, one argues essentially as in the proof of Lemma (6.14). Set Y = α−1 (B γ ), Z = Y /γ, and consider the family of nodal curves β : Z → B γ . Since the fiber β −1 (b0 ) has genus zero by assumption, it follows that every fiber of β has genus zero. This shows that B γ ⊂ H, proving ii), and at the same time proves iii). It remains to prove i). It is implicit in the proof of (6.14) that H is contained in B γ , so that in fact H = B γ . Thus H is smooth. It remains to prove the dimension statement. Since H is smooth and the fiber of α at its general point is smooth, it suffices to prove it under the additional assumption that C is smooth. What we must show is that H 1 (C, TC )γ has dimension 2g − 1; dually, we must see that this is the dimension of the cotangent space to H at b0 , that is, of the space of (co)invariants 2 γ H 0 (C, ωC ) . Denote by f : C → P1 the quotient map modulo the hyperelliptic involution and by D its branch locus. An elementary local calculation shows that any invariant quadratic differential on C is the pullback via f of a quadratic differential on P1 with simple poles along D, 2 γ ) can be identified with H 0 (P1 , ωP21 (D)) ∼ and conversely. Thus, H 0 (C, ωC = 0 1 H (P , O(2g − 2)) and hence has dimension 2g − 1, as claimed. Q.E.D. Exercise (6.16). Give a complete proof of the claim in Remark (6.9), mimicking the proof we have given in the analytic case. 7. The Kuranishi family and continuous deformations. In Section 9 of Chapter IX we have introduced the notion of continuous, or differentiable, family of compact complex manifolds. We

§7 The Kuranishi family and continuous deformations

213

can therefore also speak about continuous, or differentiable, deformations of compact complex manifolds, and in particular of smooth curves. In this short section we wish to show that the universal property of the Kuranishi family also holds in this context. For simplicity, we shall limit ourselves to deformations of unpointed curves, but the arguments go through with virtually no change in the n-pointed case. Here is what we wish to prove. Proposition (7.1). Let α : Y → B be a C m family of compact Riemann surfaces, and let b0 be a point of B. Let π : C → (X, x0 ), π −1 (x0 ) Yb0 , be a Kuranishi family for Yb0 . Then, for any sufficiently small neighborhood A of b0 , there is a cartesian diagram F

α−1 (A) α

u A

f

wC π u wX

where f (b0 ) = x0 , and (f, F ) is a morphism of C m families of complex manifolds such that F induces the identity on Yb0 . We claim that, possibly after shrinking B, we can find a cartesian diagram Y (7.2)

H

α

u B

h

w Y α u w B

where (h, H) is a morphism of C m families of complex manifolds, and Y → B is a family of complex manifolds in the ordinary sense. The result then follows from the standard universal property of the Kuranishi family, applied to Y → B . In proving the existence of (7.2) we distinguish two cases. Suppose first that Y → B is real-analytic. We may assume that B is a ball in R centered at the origin, and that b0 coincides with the origin. The family Y → B is given by real-analytic transition functions which are holomorphic in the vertical coordinates. The power series expansions of these make perfect sense even if the B-coordinates are allowed to take complex values with sufficiently small imaginary part, and define a complex-analytic family α : Y → B , where B is a neighborhood of B in C ⊃ R , whose restriction to B is just α : Y → B. If m is a nonnegative integer or ∞, we have to argue differently. Let β : Y → S be a C m family of compact complex manifolds. Recall from Y , A0,1 , and AY denote, respectively, Section 9 of Chapter IX that O Y m the sheaf on Y of all C functions whose restrictions to the fibers are

214


holomorphic, the sheaf of adapted relative (0,1)-forms on Y, and the Y -module, for sheaf of adapted functions on Y. If G is a locally free O any s ∈ S, we denote by Gs the locally free OYs -module G ⊗O OYs . Y Lemma (7.3). Let β : Y → S be a C m family of compact complex Y -module. Let s0 be a point of manifolds, and let G be a locally free O S, and let σ be a section of Gs0 . Assume that H 1 (Ys , Gs ) = 0 for all s close to s0 . Then σ extends to a section of G over a neighborhood of Y s0 . Before proving the lemma, we show how it can be used to complete the proof of (7.1). We may of course shrink B if necessary; in particular, we may assume that α has a C m section D. For any n > 2g − 2, L = O(nD) Y -module which satisfies the assumptions of (7.3). Fix is an invertible O some n > 2g, let s1 , . . . , sh be a basis of H 0 (Yb0 , Lb0 ), and let s1 , . . . , sh be sections of L which extend them to a neighborhood of Yb0 ; these give a C m embedding of fiber spaces y α−1 (A)

w Ph−1 × A

u A

where A is any sufficiently small neighborhood of b0 in B. By (9.11) in Chapter IX, the family α−1 (A) → A is the pullback, via a C m map, of p(t) the universal family on Hilbh−1 , where p(t) = tn + 1 − g. This shows the existence of a diagram (7.2) and hence concludes the proof. It remains to prove (7.3). When Y → S is real-analytic, we already know that it comes locally by restriction from a complex-analytic family, so the conclusion follows from the theory of base change in cohomology. In the remaining cases, choose a covering W of a neighborhood of Ys0 by means of adapted coordinate sets and choose an adapted partition of unity {χi } subordinated to W; for each i, pick an element Wi of W containing the support of χi . The restriction of σ to Wi ∩Ys0 extends to a section σi of G over Wi . Then τ = χi σi is a section of A(G) = A⊗ G O Y

that extends σ. Let ∂ β be the relative ∂ operator on G along the fibers of β and set η = ∂ β τ . We will show that one can solve the equation η = ∂ β u with an adapted u that vanishes identically on Ys0 . Then τ − u will be the sought-for extension of σ. We set A0,1 (G) = A0,1 ⊗O G. Y Using an appropriate adapted partition of unity, we can put on A0,1 and on G adapted hermitian metrics. We denote by ϑβ the formal adjoint of ∂ β : A(G) → A0,1 (G) with respect to these metrics and form the Laplace–Beltrami operator (7.4)

β = ∂ β ϑβ + ϑβ ∂ β

§7 The Kuranishi family and continuous deformations

215

operating on relative G-valued (0,1)-forms. The equation β v = η has a unique solution since, by the assumption that H 1 (Ys , Gs ) = 0 for all s, there are no nonzero –harmonic G-valued (0,1)-forms on the fibers of β. In particular, v vanishes identically on Ys0 . Furthermore, ∂ β v = 0. We then set u = ϑβ v. It remains to show that v is adapted. The result we need about the Laplace–Beltrami operator follows from a theorem of Kodaira and Spencer [437] concerning the differentiability properties of solutions of differential equations Ev = η, where E is a C m family of self-adjoint, strongly elliptic, differential operators, and η is an adapted section. To set up, let α: Z → T be a C m family of compact differentiable manifolds, and let F be a C m family of differentiable vector bundles on it. Given two C m families F and F of differentiable vector bundles on α: Z → T , a linear differential operator carrying sections of F to sections of F will be said to be a C m family of linear differential operators on α: Z → T if, when written in adapted coordinates and relative to adapted local trivializations of F and F , it involves only differentiation with respect to vertical coordinates, and its coefficients are adapted functions. For brevity, we will also say that such a differential operator is adapted. Thus, an adapted linear differential operator carries adapted ) the vector space of sections to adapted sections. We denote by A(F adapted sections of F . Let E: A(F ) → A(F ) be a C m family of linear differential operators. A metric on F will be said to be adapted if the inner product of any pair of adapted sections is an adapted function. Adapted metrics always exist and can, for instance, be constructed by gluing together flat local metrics by means of a partition of unity made up of adapted functions. Suppose an adapted metric is given on F , and one on the relative tangent bundle to Z → T . We denote by , t the inner product on Ft and by dVt the volume form on Zt coming from the metric. Consider the inner product (7.5) (u, v) = u, vt dVt Zt

on A(Ft ), the vector space of C ∞ sections of Ft . We will say that E is a family of formally self-adjoint, strongly elliptic differential operators if each Et is self-adjoint with respect to the inner product (7.5) and strongly elliptic. Under these circumstances the kernel of Et is finite-dimensional, and there are linear operators Ft , Gt : A(Ft ) → A(Ft ) , where Ft is the orthogonal projection onto the kernel of Et , and (7.6)

u = Ft u + Et Gt u

for any u ∈ A(Ft ). We shall refer to Ft and Gt , respectively, as the harmonic projector and Green operator associated to Lt and to the chosen metrics. The theorem of Kodaira and Spencer reads as follows.

216


Theorem (7.7) (cf. [437], Theorem 5). Let m be a nonnegative integer or ∞. Let α: Z → T be a C m family of compact differentiable manifolds, and let F be a C m family of differentiable vector bundles on α: Z → T . Suppose that F and the relative tangent bundle to Z → T are endowed ) → A(F ) be a C m family of formally with adapted metrics. Let E: A(F self-adjoint, strongly elliptic linear differential operators. Suppose that the dimension of the kernel of Et is independent of t. Then the family of harmonic projectors F = {Ft }t∈T and the family of Green operators G = {Gt }t∈T are of class C m , in the sense that F u and Gu are adapted for any adapted u. To be precise, the statement proved by Kodaira and Spencer is slightly less general than the one we have given, in two respects. First of all, they deal only with families of the form Z0 × T → T , where Z0 is a compact manifold. More importantly, they treat only the case m = ∞. These, however, are not serious difficulties. Since the statement of Theorem (7.7) is local on S, the first is resolved by Lemma (9.8) in Chapter IX. The second difficulty is also nonexistent, since the proof given by Kodaira and Spencer for their theorem, and in particular their crucial Proposition 1, work equally well, and virtually without changes, in our context, provided that we substitute Lemma (9.8) in Chapter IX, and in particular its second part, for their Lemma 1. We now go back to the Laplace–Beltrami operator (7.4). Notice that, according to our definition, β is a family of strongly elliptic, formally self-adjoint linear differential operators. Since, by assumption, there are no nonzero –harmonic G-valued (0,1)-forms on the fibers of β, the harmonic projector F is zero, so that β Gη = η, where G is the Green operator. But then Theorem (7.7) asserts that u = Gη is adapted whenever η is. 8. The period map and the local Torelli theorem. This section is a utilitarian introduction to the period mapping in the case of 1-forms. Roughly speaking, the period map describes how the Hodge structure on the cohomology groups of a variety depends on the complex structure. In the case of compact Riemann surfaces, the period map associates to a compact Riemann surface C its normalized period matrix, up to the action of the integral symplectic group, and this, in turn, determines the Jacobian of C as a principally polarized abelian variety. This assignment is the Torelli map. The main result about it is Torelli’s theorem, stating that it is injective (cf. Vol. I, Chapter VI, Section 3). In this section we shall study the period map from an infinitesimal point of view, introducing the Gauss–Manin connection and proving the so called local Torelli theorem, which states that the differential of the period map is nondegenerate at nonhyperelliptic curves.

§8 The period map and the local Torelli theorem

217

In the next section we shall discuss the curvature properties of the Hodge bundles, proving their semipositivity for families of curves. To begin with, consider a family ϕ:C→B of compact, genus g Riemann surfaces parameterized by a polydisc B, and let C = ϕ−1 (0) be its central fiber. From a C ∞ point of view this family can be trivialized. Fix then a C ∞ trivialization f : C × B −→ C , and, for b ∈ B, set Cb = ϕ−1 (b) and fb = f |C×{b} . As usual, denote by Ω1C/B the sheaf of relative holomorphic differentials. Choose a frame for ϕ∗ Ω1C/B , that is, a basis η1 (b), . . . , ηg (b) for H 0 (Cb , ωCb ), varying holomorphically with b. Also let γ1 , . . . , γ2g be a symplectic basis for H 0 (C, Z). We may assume that our frame is normalized in such a way that the varying period matrix ⎛ ⎞ ⎜ ⎟ Ω(b) = ⎝ ηα (b)⎠ (fb )∗ (γi )

α=1,...,g ; i=1,...,2g

has the form Ω(b) = (I, Z(b)) , where I is the identity g × g matrix, and Z(b) a point in the Siegel upper half-space Hg (cf. Chapter I, Section 3). The map Z : B −→ Hg is called the period map. We shall prove first of all that Z is holomorphic and then that the differential of Z fits into a commutative diagram Tb0 (B) (8.1)

dZ

w TZ(b0 ) (Hg ) ⊂ TZ(b0 ) (G(g, 2g))

ρ u H 1 (C, TC )

ν

u w Hom(H 1,0 (C), H 0,1 (C))

where ρ is the Kodaira–Spencer map, and ν is defined by cup-product. Before proving the statement we have announced, we shall reinterpret Z as a map into a Grassmannian, as is already suggested in the above diagram. Intrinsically, the map Z associates to each point b the point H 1,0 (Cb ) ∈ G(g, H 1 (C, C)) = G(g, 2g) .

218


In fact, the rows of the period matrix Ω(b) are simply the coordinates (relative to the basis of H 1 (C, C) which is dual to γ1 , . . . , γ2g ) of the holomorphic differentials η1 (b), . . . , ηg (b). In the diagram above we used the well-known identification between the tangent space to G(g, 2g) at the point H 1,0 (C) and Hom(H 1,0 (C), H 1 (C, C)/H 1,0 (C)) ∼ = Hom(H 1,0 (C), H 0,1 (C)) . To prove the holomorphicity of Z and the commutativity of the diagram above, it is convenient to put ourselves in a more general setting, since what has to be proved has nothing to do with the fact that we are dealing with a family of curves, but rather with the fact that we are dealing with a family of compact Kähler manifolds. Let ϕ : X −→ Δ be a smooth proper morphism with fibers compact Kähler manifolds and base a polydisc. Set Xt = ϕ−1 (t), X = X0 . As before, we have a period map Z : Δ −→ G(q, H 1 (X, C)) , where q = h1,0 (X). We will show that Z is holomorphic and that the natural diagram T0 (Δ) (8.2)

dZ

ρ u h H 1 (X, TX )

j hh hν

w Hom(H 1,0 (X), H 0,1 (X))

commutes. We shall need the following general fact about mappings into Grassmannians. Let H : Δ −→ G(k, V ) Let t = (t1 , . . . , tn ) be coordinates on Δ. Let be a C ∞ map. w1 (t), . . . , wk (t) be a frame for H(t), varying smoothly with t. Suppose that ∂wi /∂ t¯j ∈ H(t) for all t, i, and j. We claim that H is holomorphic. It suffices to do this when n = 1. By hypothesis, then ∂wi = gi,j (t)wj (t) . ¯ ∂t The system of differential equations in the fi,j ∂ ( fi,j wj ) + gi,j wj = 0 ¯ ∂t


219

can be solved uniquely with initial conditions fi,j = 0, since by our assumption it can be rewritten in the form ∂fi,j + fi,h gh,j + gi,j = 0 , ∂ t¯

i, j = 1, . . . , k .

h

Then the vectors vi (t) =

fi,j (t)wj (t) + wi (t)

provide, at least near zero, a new frame for H(t) varying holomorphically with t. Assuming now that H is holomorphic, we recall how to compute its differential dH0 : T0 (Δ) → Hom(H(0), V /H(0)) = TH(0) (G(k, V )) . Let w be an element of H(0), and w(t) a path in V such that w(t) ∈ H(t) and w(0) = w. Then ∂w ∂ (w) = projection of in V /H . (8.3) dH0 ∂ti ∂ti t=0 Let us return to our original problem. Let η be a C ∞ closed relative 1-form on X , i.e., a section of forms A1X /Δ such that dϕ η = 0, where dϕ stands for exterior differentiation along the fibers. Via the de Rham theorem, the 1-form η determines a section [η] of R1 ϕ∗ C ⊗ AΔ . Now fix a C ∞ trivialization (8.4) ,

f : X × Δ −→ X

and denote by p the projection from X × Δ to X. Pick closed 1-forms ϕ1 , . . . , ϕ2q on X whose de Rham classes constitute a basis [ϕ1 ], . . . , [ϕ2q ] of H 1 (X, C). Via the trivialization f , these elements give rise to relative closed 1-forms Φ1 , . . . , Φ2q on X , whose relative cohomology classes [Φ1 ], . . . , [Φ2q ] give a frame for R1 ϕ∗ C ⊗ AΔ . By this we mean that Φν = rel(f∗ p∗ ϕν ) , where rel is the homomorphism from differential forms on X to relative differential forms A∗X /Δ . Thus we can write (8.5)

[η] =

aν [Φν ] ,

where the aν are C ∞ functions on Δ. Alternatively, using (8.4), one may view [η] as a family [η(t)] of classes in H 1 (X, C) depending on t. From this point of view, the above expression for η translates into (8.6) [η(t)] = aν (t)[ϕν ] .

220


It now makes sense to differentiate [η(t)] by differentiating its coefficients; the result is an element of the fixed vector space H 1 (X, C). In order to show that the period mapping Z is holomorphic, we use the holomorphicity criterion for mappings into Grassmannians we discussed above. Therefore we must show that when η is of type (1, 0), then ∂[η(t)]/∂ t¯i |t=0 is of type (1, 0) for every i as well. Once the holomorphicity has been established, to show the commutativity of diagram (8.2), we shall explicitly calculate the differential of the period mapping by evaluating the projection of ∂[η(t)]/∂ti |t=0 in H 1 (X, C)/H 1,0 (X), as required by formula (8.3). To carry out this program, it is convenient to rephrase everything in terms of the so-called Gauss–Manin connection, which we now introduce. This is a connection ∇ on the bundle R1 ϕ∗ C ⊗ AΔ . For a brief discussion on connections and their curvatures, the reader may want to look at the beginning of the next section. Given a C ∞ section [η] of R1 ϕ∗ C ⊗ AΔ as in (8.5) and a C ∞ vector field v on Δ, the Gauss–Manin connection is defined by ∇v [η] = v(aν )[Φν ] . It is clear that this is a connection. When, as in (8.6), we think of [η] as a family [η(t)] of classes in H 1 (X, C) depending on t, we may write ∂[η(t)] = ∇ ∂ [η] , ∂ti ∂ti

∂[η(t)] = ∇ ∂¯ [η] , ∂ ti ∂ t¯i

where again we think of ∇ ∂ [η] and ∇ ∂¯ [η] as families of cohomology ∂ti ∂ ti classes on X. In order to compute these derivatives in a form which is better suited to our goal of showing the holomorphicity of Z and calculating its differential, we shall express the Gauss–Manin connection in a different way. We shall show that η ))] , ∇v [η] = [rel(ιv˜ (d˜

(8.7)

where v˜ is any lifting of v to a vector field on X , η˜ is any 1-form on X such that rel(˜ η ) = η, and ι stands for interior product. Assuming for a moment that (8.7) has been proved, we next indicate how one can conclude. To show the holomorphicity of Z, we must prove that ∇ ∂¯ [η] is of type (1, 0) for every i whenever η is. Since ϕ ∂ ti

is holomorphic, if η is of type (1, 0), we may choose η˜ to be of type (1, 0), and the lifting v˜ of v = ∂/∂ t¯i to be of type (0, 1). The fact that ∇ ∂¯ [η] is of type (1, 0) follows, by type considerations, by looking at ∂ ti

η ))]. This proves the holomorphicity of Z. the expression [rel(ιv˜ (d˜ We now show the commutativity of diagram (8.2). Explicitly, in view of (8.3), we must show that ∂[η(t)] ∂ ∪ [η] mod H 1,0 (X) ≡ρ ∂ti ∂ti


221

for every i or, equivalently, that #

∇

$(0,1) ∂ ∂ti

[η]

=ρ

∂ ∂ti

∪ [η] .

We may choose a lifting v˜ of ∂/∂ti which is of type (1, 0), and, again by type considerations, we then see that (rel(ιv˜ (d˜ η )))(0,1) = rel(ιv˜ (∂¯η˜)) . On the other hand, writing locally ∂ ∂ + cj , ∂ti ∂zj

v˜ = η˜ =

ah dth +

bj dzj ,

j

h

we get: ¯ j, ¯ i− cj ∂b ιv˜ (∂¯η˜) = −∂a ⎞ ⎛ ⎞ ⎛ ∂ ¯j⊗ ¯v ∧ η˜ = ⎝ ⎠∧⎝ ah dth + bj dzj ⎠ ∂c ∂˜ ∂z j j j h ¯ j, = bj ∂c ¯ v˜ (˜ ¯j+ ¯j ¯ i+ ∂(ι η )) = ∂a cj ∂b bj ∂c ¯v ∧ η˜ − ιv˜ (∂¯η˜) , = ∂˜ so that: # (8.8)

∇

$(0,1) ∂ ∂ti

[η]

= [rel(∂˜ v ∧ η˜)] = [rel(∂˜ v ) ∧ η] ∂ ∪ [η] , =ρ ∂ti

where the equality rel(∂˜ v ) = ρ( ∂t∂ i ) is left as an exercise in the Dolbeault 1 isomorphism for H (X, TX ). This proves the commutativity of diagram (8.2). It remains to prove (8.7). We begin by showing that the right-hand side does not depend on the choice of liftings for v and η. To do this, we must prove two facts. First of all, that (8.9)

η ))] = 0 [rel(ιw (d˜

222


whenever w is a vector field along the fibers of ϕ. Secondly, that (8.10)

[rel(ιv˜ (dψ))] = 0

whenever rel(ψ) = 0. We prove (8.9): [rel(ιw (d˜ η ))] = [ιw (rel(d˜ η ))] η )))] = [ιw (dϕ (rel(˜ = [ιw (dϕ (η))] = 0, We next prove (8.10). Clearly, ψ is of the form sinceη is dϕ -closed. ψ= αi dti + βi dt¯i . We have % # $& [rel(ιv˜ (dψ))] = rel ιv˜ (dαi ∧ dti + dβi ∧ dt¯i ) =− [rel(ιv˜ (dti )dαi + ιv˜ (dt¯i )dβi )] =− [rel d(ιv˜ (dti )αi + ιv˜ (dt¯i )βi )] =− [dϕ rel(ιv˜ (dti )αi + ιv˜ (dt¯i )βi )] = 0, the second equality being a consequence of the fact that both ιv˜ (dti ) and ιv˜ (dt¯i ) are constant along the fibers of ϕ and can hence be moved under the sign of relative differentiation. This concludes the proof that the right-hand side of (8.7) is independent of the choice of liftings. ˜ v [η], that is, let Let us denote temporarily this right-hand side by ∇ us set ˜ v [η] = [rel(ιv˜ (d˜ η ))] . ∇ ˜ is a connection. Indeed, if g is a C ∞ function on Δ, The operator ∇ then ˜ v (g[η]) = [rel(ιv˜ d(g η˜))] ∇ = [rel(ιv˜ (dg ∧ η˜ + gd˜ η ))] η ))] = [rel(˜ v (g)˜ η ) + g rel(ιv˜ (d˜ ˜ = v(g)[η] + g ∇v [η] . ˜ is in fact the Gauss–Manin To prove (8.7), that is, to show that ∇ connection, it then suffices to show that ˜ v [Φν ] ∇v [Φν ] = ∇ for every v and every ν. But now the left-hand side is zero by definition, and so is the right-hand side ˜ ν ))] , [rel(iv˜ (dΦ


223

˜ ν to be f∗ p∗ ϕν , which is obviously closed. since we may choose Φ The computations that we have just carried out show, among other things, that the Gauss–Manin connection is intrinsic and in particular does not depend on the choice of the trivialization (8.4) of X → Δ that we used to define the frame {[Φν ]}. As we have already announced, one of the applications that we have in mind for the machinery of the period map and its differential is to the local Torelli theorem. This states that, when C is a smooth nonhyperelliptic curve of genus g and ϕ : C → (B, b0 ) is a standard Kuranishi family for C, then the differential of the period map dZ : Tb0 (B) → TZ(b0 ) (Hg ) is injective. To show this, look at diagram (8.1). Under our assumption, the Kodaira–Spencer map is an isomorphism, so that it suffices to show that ν is injective. But the transpose of ν is the cup-product map (8.11)

2 ), ν ∗ : H 0 (C, ωC ) ⊗ H 0 (C, ωC ) → H 0 (C, ωC

which is surjective for nonhyperelliptic C by Max Noether’s theorem (cf. Chapter III, Section 2). The local Torelli theorem fails at hyperelliptic curves of genus g > 2; in fact, if C is such a curve, the cup-product map (8.11) is not onto. However, the local Torelli theorem fails in directions which are transverse to the hyperelliptic locus, and not in those directions which are tangent to it. To explain what we mean by this, recall that the subvariety H of B parameterizing those fibers of π which are hyperelliptic has dimension 2g − 1 and that its tangent space at b0 is H 1 (C, TC )γ , where γ is the hyperelliptic involution of C. Dually, the cotangent space 2 γ ) . Recall also from the proof of Lemma to H at b0 is H 0 (C, ωC 2 γ (6.15) that the elements of H 0 (C, ωC ) are precisely the pullbacks via the hyperelliptic double covering f : C → P1 of the differentials in H 0 (P1 , ωP21 (D)) ∼ = H 0 (P1 , O(2g − 2)), where D is the branch locus of f . Next, recall that H 0 (C, ωC ) consists entirely of anti-invariants, since a nonzero invariant would descend to a nonzero abelian differential on 2 γ ) . We C/γ = P1 . Thus, the cup-product (8.11) lands inside H 0 (C, ωC 0 2 γ claim that, in fact, its image is all of H (C, ωC ) . To see this, notice that one may identify H 0 (C, ωC ) with H 0 (P1 , O(g − 1)), and the cup product map 2 γ 2 ) ⊂ H 0 (C, ωC ) H 0 (C, ωC ) × H 0 (C, ωC ) → H 0 (C, ωC

with H 0 (P1 , O(g − 1)) × H 0 (P1 , O(g − 1)) → H 0 (P1 , O(2g − 2)) ,

224


which is onto. This proves our claim. Dually, what all of this shows is that the map Tb0 (H) → Hom(H 0 (C, ωC ), H 1 (C, OC )) , that is, the differential of the restriction to H of the period map, is injective and that its image equals the image of the differential Tb0 (B) → Hom(H 0 (C, ωC ), H 1 (C, OC )) , of the full period map. We shall refer to this result as the local Torelli theorem for hyperelliptic curves. 9. Curvature of the Hodge bundles. As we mentioned at the beginning of this chapter, the introduction of the Gauss–Manin connection leads naturally to the study of the curvature properties of the Hodge bundles. This will be carried out in this section, although it will not be needed in any other part of this book. The reader will find an indication of some of the many applications of this study in the bibliographical notes. Before talking about Hodge bundles, it is convenient to discuss connections and their curvatures on complex vector bundles in general. Let X be a complex manifold and E a complex vector bundle on it. Denote by Ai (E) the sheaf of E -valued C ∞ i -forms. Recall that a connection ∇ on E is a linear differential operator ∇ : A0 (E) −→ A1 (E) satisfying ∇(f s) = s ⊗ df + f ∇s . If v is a tangent vector to X, one writes ∇v s to denote the interior product ιv (∇s). The ∇ operator can be extended to operators ∇ : Ak (E) −→ Ak+1 (E) by setting ∇(s ⊗ ω) = s ⊗ dω + ∇s ∧ ω . The operator RE = ∇2 : A0 (E) −→ A2 (E) turns out to be a tensor and, more precisely, a global section of A2 (E ∗ ⊗ E). This tensor is called the curvature form of the connection ∇. Suppose that E is equipped with a nondegenerate (not necessarily definite) hermitian product , and extend this product to E-valued forms by setting s ⊗ ω, t ⊗ ϕ = s, tω ∧ ϕ¯ .

§9 Curvature of the Hodge bundles

225

Then one says that ∇ is compatible with the hermitian product if, given sections s and t of E, one has (9.1)

ds, t = ∇s, t + s, ∇t ,

so that at the level of E-valued forms one has (9.2)

dσ, τ = ∇σ, τ + (−1)deg σ σ, ∇τ .

If, in addition, E is holomorphic, one says that ∇ is a hermitian connection if it is compatible with the hermitian product and its (0,1) part is ∂ E . We then write ∇ = ∇ + ∂ E , where ∇ is the (1, 0)-part of ∇. Suppose ∇ is hermitian. We observe that its curvature form is of type (1, 1). In fact, since RE is a tensor, it suffices to calculate RE (s) when s is a holomorphic section of E. Thus, RE (s) = (∇ + ∂)(∇ + ∂)s = ∇ ∇ s + ∂∇ s . On the other hand, we claim that ∇ ∇ s = 0. To show this, it suffices to prove that ∇ ∇ s, t = 0 for any local holomorphic section t of E. Breaking up (9.1) into pure-type components, we find that ∇ ∇ s, t = ∂∇ s, t + ∇ s, ∂t = ∂ 2 s, t − ∂s, ∂t + ∇ s, ∂t = 0, since t is holomorphic and ∂ 2 = 0. In conclusion, (9.3)

RE = ∂∇ ,

which is clearly of type (1, 1). It is also useful to observe that √ −1RE (s), s is a real (1, 1) form. In fact, RE (s), s = ∂∇ s, s = ∂∇ s, s + ∇ s, ∇ s = ∂∂s, s − ∂s, ∂s + ∇ s, ∇ s = ∂∂s, s + ∇ s, ∇ s , which is purely imaginary. Now suppose that F ⊂ E is a holomorphic subbundle on which the hermitian metric is nondegenerate. One can then define a connection D

226


on F by setting Ds = pF (∇s) , where pF stands for the orthogonal projection to We leave to the reader the easy verification is compatible with the hermitian product. As for is holomorphic, it equals pF (∂ E ) = ∂ F . Thus D section s of F , set σs = ∇s − Ds .

F. that the connection D its (0, 1) part, since F is hermitian. For any

Since, for any C ∞ function f , σ(f s) = s ⊗ df + f ∇s − s ⊗ df + f Ds = f σ(s) , σ is a tensor, and it clearly has type (1, 0). We now compute the curvature of D. Let s and t be holomorphic sections of F . Then RE (s), t = ∂∇ s, t = ∂D s, t + ∂σs, t = RF (s), t + ∂σs, t . On the other hand, ∂σs, t = ∂σs, t + σs, ∇ t = ∂σs, t + σs, D t + σs, σt = σs, σt , since σs is orthogonal to F . In conclusion, (9.4)

RF (s), t = RE (s), t − σs, σt .

Suppose for a moment √ √ that the hermitian product is positive. Recall that −1RE (s), s and −1RF (s), s are real. Then the above formula expresses the principle that curvature decreases on holomorphic subbundles, where, of course,√we adopt the convention of saying that ∇ has positive curvature when −1RE (s), s is positive. We now come to the central theme of this section, that is, to the positivity properties of the so-called Hodge bundles. Fix a family of compact K¨ ahler manifolds π:X →B parameterized by a complex manifold B. The Hodge bundles are the k direct images π∗ ωX /B , k ≥ 1. We consider the Gauss–Manin connection ∇ acting on sections of the rank 2q complex vector bundle E associated to the locally free sheaf


227

R1 π∗ C ⊗ AB . We first observe that E has a complex structure which comes by writing R1 π∗ C ⊗ AB = (R1 π∗ C ⊗ OB ) ⊗OB AB . Secondly, the horizontal sections of E for the Gauss–Manin connection are exactly the sections of R1 π∗ C. Since locally holomorphic sections of E are sums of terms of the form s = f σ, where f is holomorphic and σ is a horizontal section, it follows that the (0, 1)-part of ∇ is just ∂ E . Since E is locally generated by horizontal sections, the Gauss–Manin connection is clearly flat, that is, its curvature form vanishes. The bundle E has a natural complex rank q subbundle F whose fiber over b ∈ B is H 1,0 (Xb ). As a C ∞ bundle, F coincides with the Hodge bundle π∗ ωX /B : as such, it carries a holomorphic structure. There is another holomorphic structure which F inherits from the holomorphic structure of E. In fact, when proving that the period mapping is holomorphic, we showed that, given local coordinates t1 , . . . , tn on B, for every section [ψ] of F , i.e., for every closed relative (1, 0)-form along the fibers of π, the covariant derivatives ∇ ∂ [ψ] are still of type (1, 0). Thus ∂ti

F is a holomorphic subbundle of E. The ∂ operator of this holomorphic structure is given locally by ∇ ∂ ⊗ dti . i

∂ti

Luckily, the two holomorphic structures on F turn out to be the same. To see this, let ψ be a section of π∗ ωX /B ; in other words, let ψ be a holomorphic relative (1, 0) form. View [ψ] as a C ∞ section of F . We must show that for every i . ∇ ∂ [ψ] = 0 ∂ti

Recall that ∇

∂ ∂ti

˜ , [ψ] = [rel (ιv (dψ))]

where v is a lifting of ∂t∂ , which we may choose to be of type (1, 0), i ˜ In local coordinates, and ψ˜ is a (1, 0) form such that ψ = rel(ψ). ∂ ∂ v= + cj , ∂z j ∂ti aj dzj . ψ˜ = bh dth + To say that ψ is holomorphic amounts to saying that the aj are holomorphic. Thus, ˜ = rel(ιv (∂ ψ)) ˜ rel(ιv (dψ)) ∂bh ∧ dth )) = rel(ιv ( = rel( ιv (∂bh ) ∧ dth ) = 0.

228


From now on we restrict our study to the case where the fibers of π are one-dimensional, i.e., compact genus g Riemann surfaces. There is defined on E a natural, nondegenerate hermitian form. Given a point b ∈ B and cohomology classes [η], [ξ] ∈ Eb = H 1 (Xb , C), we set √ η∧ξ. (9.5) [η], [ξ] = −1 Xb

Let us verify that the Gauss–Manin connection is hermitian. Given sections [ω] and [ψ] of E and writing them in terms of a local horizontal frame [φ1 ], . . . , [φ2g ] around b0 ∈ B as introduced in the previous section, [ω] = [ψ] = we get d[ω], [ψ] =

ai [Φi ] , bi [Φi ] , Φi ∧ Φj

d(ai bj ) Xb0

= ∇[ω], [ψ] + [ω], ∇[ψ] . Clearly, the hermitian product , on E is not definite; more specifically, it is positive definite on F and negative definite on the subbundle F ⊥ , whose fiber over b ∈ B can be canonically identified with H 0,1 (Xb ). We compute the curvature of the hermitian connection on F . For this, we use formula (9.4) and get (9.6)

RF ψ, η = −σψ, ση .

Since σ equals the difference between ∇ and its projection to F , σψ is an F ⊥ -valued (1, 0) form, and since the inner product is negative definite on F ⊥ , it follows that √ −1RF ψ, ψ ≥ 0 (9.7) for any ψ, that is, F has nonnegative curvature everywhere on B. One may be more explicit about the value of σ. In fact, we already computed in the previous section (cf. (8.8)) that for any tangent vector v to B, (9.8)

ιv (σψ) = ρ(v) ∪ ψ,

where ρ is the Kodaira–Spencer map. Formula (9.7) is a first positivity property satisfied by a Hodge bundle. We would like to be able to talk about positivity properties of the Hodge bundles even for families of algebraic varieties containing singular fibers and specifically for families of nodal curves. There is no problem in defining the Hodge bundles in this situation. The trouble


229

is that the natural metrics they carry are singular at degenerate fibers. One is therefore compelled to look for a positivity notion which is not metric in nature, but rather algebro-geometric. This is the notion of semipositivity, on which we now digress. A locally free sheaf F over a complete scheme X is said to be semipositive if for every morphism f : C −→ X , where C is a smooth complete curve, the pullback f ∗ F has no line bundle quotient of negative degree (notice that, when F is a line bundle, this amounts to saying that F is nef). There are a number of standard constructions for locally free sheaves which yield semipositive sheaves when applied to semipositive ones. It is possible to give a general criterion to discriminate between those constructions which preserve semipositivity and those which do not. Here, however, we shall content ourselves with the following proposition, which is all we shall need in the sequel. Proposition (9.9). Let F and G be semipositive locally free sheaves on a complete scheme X. Then: a) any locally free quotient of F is semipositive; b) if 0→F →E →G→0 is an exact sequence of locally free sheaves, then E is semipositive; c) F ⊗ G is semipositive. An immediate consequence of this proposition is the following: Corollary (9.10). Tensor powers of semipositive sheaves and their quotients, such as symmetric or exterior powers of semipositive sheaves, are semipositive. Property a) in Proposition (9.9) is immediate from the definition. As for b), suppose that L is a quotient line bundle of f ∗ E for some morphism f from a smooth complete curve C to X. Then, either the composite map from f ∗ F to L in nonzero, in which case L contains a line bundle quotient of f ∗ F , so deg L ≥ 0, or else L is a quotient of f ∗ G, and hence deg L ≥ 0 anyway. Proving part c) of Proposition (9.9) will not be as easy. Our strategy will be to rely on the notion of ampleness, which is closely related to the one of semipositivity, proving first the analogue of (9.9) with “ample” substituted for “semipositive” throughout and then deducing (9.9) as a corollary. Let then F be a locally free sheaf on a complete scheme X, and consider the tautological line bundle O(1) = OP(F ∨ ) (1) on P(F ∨ ). We recall that F is said to be ample if O(1) is ample.

230


Notice that ampleness implies semipositivity. In fact, any line bundle quotient of an ample locally free sheaf F on a smooth complete curve has a strictly positive degree. To see this, notice that a line bundle quotient L of F corresponds to a section Γ of P(F ∨ ) and that the degree of L is just the intersection number of Γ with OP(F ∨ ) (1), which is positive when F is ample. It is a well-known fact that, loosely speaking, nef line bundles are the limits of ample (fractional) line bundles. We shall see in a moment that essentially the same relation intercurs between semipositive and ample locally free sheaves. The fact that nef line bundles are limits of ample ones derives from the following fundamental ampleness criterion, due to Seshadri (cf. [355]). Proposition (9.11) (Seshadri’s criterion). A line bundle L on a complete scheme X is ample if and only if there is a positive constant k such that, for any irreducible reduced complete curve C in X, deg(L|C ) ≥ kμ(C) , where μ(C) stands for the maximum of the multiplicities of the points of C. Suppose then that L is ample, so that Seshadri’s criterion is satisfied. If M is nef, then for any positive integer n, deg(L ⊗ M n )|C ≥ kμ(C) , so that L ⊗ M n is ample. Conversely, suppose that L ⊗ M n is ample for any positive integer n and some ample line bundle L. Then, clearly, n ) > 0. n deg(M|C ) + deg(L|C ) = deg(L ⊗ M|C

Passing to the limit as n goes to infinity shows that deg M|C is nonnegative, i.e., that M is nef. In particular, M is nef if and only if, for any ample line bundle L, the line bundle L ⊗ M n is ample, so that if one is willing to deal with fractional line bundles, one sees that nef line bundles are limits of ample ones. An analogue of this characterization of nef line bundles is the following characterization of semipositivity. Proposition (9.12). Let F be a locally free sheaf on a complete scheme X. The following conditions are equivalent: i) F is semipositive. ii) OPF ∨ (1) is nef. iii) For any morphism f : C → X, where C is a smooth complete curve, and any ample line bundle L on C, the locally free sheaf L ⊗ f ∗ F is ample.


231

To prove this, suppose first that F is semipositive. Let Γ be any irreducible complete curve in P = P(F ∨ ). If Γ is contained in a fiber of the projection from P to X, then OP (1) has positive degree on Γ. If, on the other hand, Γ is not contained in a fiber, and we denote by ˜ its normalization and by k the natural morphism from Γ ˜ to X, then Γ h ˜ has a tautological section Γ mapping to Γ, and moreover k∗ P → Γ deg OP (1)|Γ = deg Ok∗ P (1)|Γ . On the other hand, one has a surjective restriction morphism k ∗ F = h∗ Ok∗ P (1) −→ h∗ (Ok∗ P (1)|Γ ) . Since F is assumed to be semipositive, it follows that deg OP (1)|Γ ≥ 0 . This shows that OP (1) is nef, proving that i) implies ii). Now suppose that ii) holds and let C, f , and L be as in iii). Consider any irreducible complete curve D in Pf ∗ F and let π : P = Pf ∗ F ∨ → C be the projection. Then deg (π ∗ L ⊗ OP (1))|D ≥ μ(D) . This is obvious if D is contained in a fiber. Otherwise, since OP (1) is nef, the left-hand side is larger than deg(π ∗ L|D ), which is at least (deg(L))·μ(D). Then, by Seshadri’s criterion, π ∗ L⊗OP (1) = OP(L∨ ⊗f ∗ F ∨ ) is ample, i.e., L ⊗ f ∗ F is ample. We now prove that iii) implies i). Let f be a morphism from a complete smooth curve C to X, let M be a line bundle quotient of f ∗ F , and let L be a line bundle of degree one on C. Then M ⊗ L, being a line bundle quotient of the ample locally free sheaf f ∗ F ⊗ L, has positive degree; thus deg M ≥ 0. This shows that F is semipositive. Let F be a locally free sheaf on a complete scheme X, and let ϕ : Y → X be a finite morphism. A simple remark that will be useful later is that, as is the case for line bundles, F is ample if and only if f ∗ F is. In fact, there is an obvious commutative diagram ψ Pϕ∗ F ∨ w PF ∨ η

ξ

u Y

ϕ

u wX

where ψ is finite. Moreover, ψ ∗ OPF ∨ (1) = OPϕ∗ F ∨ (1), so that our claim follows from the line bundle case. We are now ready to prove, in several steps, the analogue of Proposition (9.9) for ample sheaves. Actually, the proof will be complete only for locally free sheaves over smooth curves, but this is all we shall need.

232


Lemma (9.13). Let F and G be ample locally free sheaves on a complete scheme X. Then: a) any locally free quotient of F is ample. b) If 0→F →E →G→0 is exact, then E is ample. We first prove a). Let F → H be a surjective map onto a locally free sheaf. Let j : PH∨ → PF ∨ be the corresponding inclusion. Then OPH∨ (1) = j ∗ OPF ∨ (1), and the ampleness of H follows. We shall prove b) only in case X is a smooth curve. Let H be an ample line bundle on X, and let η : PF ∨ → X, ξ : PG ∨ → X be the natural projections. There is an integer h > 0 such that OPF ∨ (h)⊗η ∗ H −1 and OPG ∨ (h) ⊗ η ∗ H −1 are ample. Pick a degree h covering ϕ : X → X and let ψ : Pϕ∗ F ∨ → PF ∨ be the corresponding morphism. Then on X h there is a line bundle H such that H = ϕ∗ H; clearly, H is ample. ∨ Set F = ϕ∗ F , let η be the projection PF → X , and so on. Then ∗

OP(F ⊗H −1 )∨ (1) = η (H so

−1

) ⊗ OPF ∨ (1) ,

∗

OP(F ⊗H −1 )∨ (h) = η ϕ∗ (H −1 ) ⊗ OPF ∨ (h) = ψ ∗ (η ∗ (H −1 ) ⊗ OPF ∨ (h))

is ample because it is the pullback of an ample line bundle under a finite morphism. Thus, OP(F ⊗H −1 )∨ (1) is ample, and the same is true, by the same argument, for OP(G ⊗H −1 )∨ (1). By part b) of Proposition (9.9), −1 is semipositive. By Proposition (9.12), E is ample. Since, as E ⊗ H we already observed, ampleness is insensitive to finite base changes, E is ample as well. This concludes the proof of Lemma (9.13). Lemma (9.14). Let F be an ample locally free sheaf on a complete scheme X. Then, for any h > 0, S h F is ample. We begin by showing that the conclusion of the lemma holds for large enough h. To prove this, we let η : PF ∨ → X be the natural projection, and let G be any coherent sheaf on X. Then, if h ≥ 0, H p (X, Rq η∗ (OPF ∨ (h) ⊗ η ∗ G)) = H p (X, G ⊗ Rq η∗ (OPF ∨ (h))) vanishes for q > 0 and equals H p (X, G ⊗ S h F ) for q = 0. Thus, by a Leray spectral sequence argument, H p (X, G ⊗ S h F ) = H p (PF ∨ , η ∗ G(h))


233

for any p and h ≥ 0. Since F is ample, the right-hand side vanishes for p > 0 and large enough h. This implies that, for large enough h, S h F is generated by its sections, i.e., that there is a surjective morphism ⊕N OX −→ S h F .

Thus, there is a surjective morphism F ⊕N −→ S h+1 F . By Lemma (9.13), S h+1 F is ample, proving our claim. Now let h be arbitrary. By what we have just proved, S hk F is ample for large k. Consider the kth power morphism σ : PS h F ∨ −→ PS hk F ∨ . This is finite-to-one, and σ ∗ OPS hk F ∨ (1) = OPS h F ∨ (k). Since OPS hk F ∨ (1) is ample, OPS h F ∨ (k) is also ample. This concludes the proof of the lemma. Corollary (9.15). Let F and G be ample locally free sheaves on a complete scheme X. Then F ⊗ G is ample. To see this, it suffices to notice that F ⊗ G is a direct summand of S 2 (F ⊕ G) = S 2 F ⊕ S 2 G ⊕ F ⊗ G , which is ample by lemmas (9.13) and (9.14). The analogue of Proposition (9.9) for ample sheaves is now fully proved, at least when the base is a smooth curve. Notice that the only place where this extra assumption has been used is in the proof of part b) of Lemma (9.13). To conclude the proof of Proposition (9.9), we argue as follows. By part iii) of (9.12) we must show that, for any morphism f : C → X, where C is a smooth complete curve, and any ample line bundle H on C, H ⊗ f ∗ (F ⊗ G) is ample. Up to a finite base change, we may assume that H is the tensor product of two ample line bundles H1 and H2 . But then H ⊗ f ∗ (F ⊗ G) = (H1 ⊗ f ∗ F ) ⊗ (H2 ⊗ f ∗ G) , which is the tensor product of two ample sheaves and hence ample. Proposition (9.9) is then fully proved. We now state the main result of this section (see, for instance, Koll´ ar’s paper [439]). Theorem (9.16). Let f : X → S be a family of nodal curves over a k complete base S. Then the locally free sheaves f∗ ωX /S are semipositive for any k ≥ 1.

234


Of course, it is enough to prove the theorem when S is a smooth complete curve. The crucial case is the one where k = 1 and the general fiber of f is a smooth curve. This is the case that we shall fully prove. At the end of the proof we shall briefly discuss the general case. Until further notice we then consider the situation where k = 1, S is a smooth complete curve, and the generic fiber of f is smooth. By successive blow-ups we may also assume that X is a smooth surface. The proof that f∗ ωX /S is semipositive will rest on curvature considerations. We recall that, away from singular fibers, f∗ ωX /S has an intrinsic metric defined as follows. If η and ξ are C ∞ sections of f∗ ωX /S , the hermitian product between η and ξ is defined by η, ξs =

√ −1

η∧ξ. Xs

We have shown that the curvature of this metric is nonnegative, away from singular fibers. At singular fibers the curvature has singularities. To prove the semipositivity, we are required to prove that quotient line bundles of f∗ ωX /S have nonnegative degree. Therefore the idea of the proof is to use the principle that curvature increases on quotient bundles and to show that the singularities of the metric are mild enough to make it possible to compute degrees as integrals of the curvature. Control over the singularities of the metric is achieved by means of the following general criterion. Lemma (9.17). Let L be a holomorphic line bundle over a smooth complete connected curve S, and let p1 , . . . , pn be distinct points of S. For each i, denote by zi a local coordinate centered at pi and by li a nonvanishing section of L on a neighborhood of pi . Suppose that L is endowed with a hermitian metric on S {p1 , . . . , pn } and denote by H the corresponding Chern form. Let μ(x) be a real-valued positive function defined for large enough x such that lim

x→∞

μ(x) =0 xN

for any positive exponent N . Suppose that H is nonnegative on S {p1 , . . . , pn }, that log li is summable for each i, and that there are constants bi such that, for every i, i)

li −1 = O(|zi |bi μ(|zi |−1 )) .

Then deg L ≥

S{p1 ,...,pn }

H+

bi .


235

In particular, when bi ≥ 0, one has that deg L ≥ 0. If, in addition, there are constants ai such that, for every i, ii)

li = O(|zi |−ai μ(|zi |−1 )) ,

then deg L ≤

S{p1 ,...,pn }

H+

ai .

In practical applications, μ(x) most often equals a power of log(x), and ai = bi for each i. In this situation, then deg L = H+ ai . S{p1 ,...,pn }

It should also be observed that, if we assume both i) and ii), the summability of log li follows. We now prove Lemma (9.17). For simplicity, and without any real loss of generality, we shall suppose that n = 1 and write p = p1 , z = z1 , and so on. Since log l is summable, ∂∂ log l is a well-defined current on a neighborhood of p. Therefore, although the hermitian metric has ˆ whose local expression singularities, it has a well-defined Chern current H near the puncture is 1 √ ∂∂ log l . π −1 Let χ be a nonnegative C ∞ function on S which is equal to 1 on a neighborhood of p and has support in a neighborhood U of p. Then

ˆ+ χH

ˆ= H

deg L = S

S

(1 − χ)H . S

On the other hand, if Uε is a disc of radius ε centered at p, then, for small enough ε, χ is constant on Uε , so that repeated applications of Stokes’ theorem show that √ ˆ= χ∂∂ log l π −1 χH S U = ∂ log l ∧ ∂χ U ∂ log l ∧ ∂χ = U Uε χ∂∂ log l + χ∂ log l = U Uε ∂Uε χ∂∂ log l + ∂ log l . = U Uε

∂Uε

236


Summing up, for small enough ε, 1 H+ √ ∂ log l . deg L = π −1 ∂Uε SUε √

Passing to polar coordinates z = re −1ϑ , one easily sees that, for any function f , √ −1 ∂f 1 ∂f dϑ . r − ∂f = − 2 ∂r 2 ∂ϑ ∂Uε

In particular, it follows that 1 ∂ log l ∂ log l = √ r dϑ . ∂r 2 −1 ∂Uε ∂Uε Hence, we can rewrite the degree of L as 1 ∂ log l dϑ . H− r deg L = 2π ∂Uε ∂r SUε In

remainder of the proof we shall often write F (ε) for ∂ log l r dϑ. The positivity assumption on H implies that F is ∂r ∂Uε nonincreasing. Thus, for any E ≥ ε > 0, E dr F (E) F (E)(log E − log ε) = r ε E dr ≤ F (r) r ε E 2π ∂ log l 1 drdϑ = 2π ε 0 ∂r 2π 1 (log lr=E − log lr=ε )dϑ . = 2π 0 1 2π

'

the

Keeping E fixed, letting ε vary, and using i), we thus find that F (E) log(ε−1 ) ≤ C + log(μ(ε−1 )) + b log ε for a suitable constant C or, exponentiating, that e−C ≤

μ(ε−1 ) ε−(F (E)+b)

for any ε such that 0 < ε ≤ E. If F (E) + b > 0, this contradicts the → 0 as x → ∞ for any positive N . The conclusion assumption that μ(x) xN is that F + b is nonpositive or, in other terms, that H + b ≤ deg L SUε


237

for any small ε. This proves the first part of the lemma. To prove the second part, one uses a similar argument. Write H+K. deg L = S{p}

Then, for any r, −F (r) ≥ K, so that, for any E ≥ ε > 0, one has

E

K(log E − log ε) ≤ −

F (r) ε

=

1 2π

2π

0

dr r

(log lr=ε − log lr=E )dϑ .

Keeping E fixed and letting ε vary, condition ii) then implies that K log(ε−1 ) ≤ C + a log(ε−1 ) + log(μ(ε−1 )) or, which is the same, e−C ≤

μ(ε−1 ) . (ε−1 )K−a

This implies that K − a ≤ 0, that is, that deg L ≤ H + a, S{p}

as desired. This proves Lemma (9.17). We now complete the basic step in the proof of the semipositivity theorem for the Hodge bundles. Theorem (9.18). Let f : X → S be a family of nodal curves parameterized by a smooth complete curve. Suppose furthermore that the generic fiber of f is smooth and that X is smooth as well. Then f∗ ωX /S is semipositive. Proof. By what we have seen, the curvature of f∗ ωX /S is nonnegative away from singular fibers. If L is any line bundle quotient of f∗ ωX /S , the same is true for L by the principle that curvature increases on quotient bundles. In order to prove that the degree of L is nonnegative, we shall use Lemma (9.17), and thus what we need to do is to check conditions i) and ii) for L. We begin by analyzing the singularities of the canonical metric of f∗ ωX /S . The problem being local, we focus our attention on f : X −→ D = {t ∈ C| |t| < 1}, where D is a disc in S, X = f −1 (D), and f −1 (t) is smooth for t = 0 π and singular for t = 0. Denote by C the central fiber and let N → C be

238


its normalization. Also denote by p1 , . . . , ph the nodes of C. Then there is an exact sequence 0 → H 0 (N, ωN ) → H 0 (C, ωC ) → ⊕Cpi . Let ϕ and ψ be sections of f∗ ωX/D that do not restrict to zero in H 0 (C, ωC ). We wish to estimate √ ϕ, ψt = −1 ϕ∧ψ f −1 (t)

as t goes to 0. We shall do this first when C has only one singular point p. We may choose local coordinate x and y centered at p on X in such a way that f is given in local coordinates by t = xy. Set A = {(x, y)| |x| < 1, |y| < |}, B = X A, At = A ∩ f −1 (t), Bt = B ∩ f −1 (t). We have

f −1 (t)

ϕ∧ψ =

ϕ∧ψ+ At

ϕ∧ψ, Bt

and the second summand is a C ∞ function of t. We now evaluate the first summand. We may write At = {(x, y) | |t| < |x| < 1 , xy = t} = {(x, y) | |t| < |y| < 1 , xy = t} and

dx , x dx ψ = (d + b1 x + b2 y) , x where c and d are constants, and a1 , a2 , b1 , b2 are holomorphic functions of x and y. We also set ϕ = (c + a1 x + a2 y)

dx , x dx β = (b1 x + b2 y) . x

α = (a1 x + a2 y)

In dealing with these expressions we should keep in mind that, on any fiber, dx dy =− . x y


239

Of course, on the central fiber, we use the expression on the right for x = 0, and the one on the left for y = 0. We have

α∧β = At

(9.19)

+

|t|< τ3(g−h)−2 > +(−1)h < τ3h−2 >< τ2g−3h τg−1 > . Q.E.D. We now use Exercise A-3, stating that 1 (7.13) < τ3j−2 >= j , 24 j! and we obtain g−1 1 κ %g−2 ξ h ∗ ψ%x2g−2 − ψ%x2g−3 ψ%y + · · · + ψ%y2g−2 2 h (7.14) g−1 (−1)g−h = < τ3h−2 τg−1 > . 24g−h (g − h)! h=1

From (7.8), from the first formula in Lemma (7.9), and from (7.14) we then get g−1 1 κ %g−2 κ ξh ∗ ψ%x2g−2 − ψ%x2g−3 ψ%y + · · · + ψ%y2g−2 %2g−1 + 2 Mg h=0

=< τg−1 τ2g > − < τ3g−2 > +

+

g−1 h=1

2g−2 1 (−1)j < τ2g−2−j τj τg−1 > 2 j=0

(−1)g−h < τ3h−2 τg−1 > . 24g−h (g − h)!

754

20. Intersection theory of tautological classes

Thus, in order to prove (7.7) and therefore Theorem (7.1), it suffices to prove the equalities:

2g−2

(7.15)

(−1)j < τ2g−2−j τj τg−1 >=

j=0

(7.16)

g h=1

g! , 2g−2 (2g)!

(−1)g−h 1 < τ3h−g τg−1 >= g . − h)! 24 g!

24g−h (g

These equalities are the object of Exercises A-4–A-8. 8. A brief review of equivariant cohomology and the virtual Euler–Poincar´ e characteristic. The purpose of this section is to collect a series of results about group homology and cohomology providing the link between the homology of moduli spaces and the cohomology of mapping class groups. The detailed proofs of the result illustrated in this section can be found in [88]. In what follows we will use the following notation. Let Γ be a group. We shall consistently view any (left) Γ-module M (i.e., any left ZΓ-module) as a Γ-bimodule by setting mg = g −1 m for any g ∈ Γ and m ∈ M . Given a Γ-module M , we set MΓ = Z ⊗ZΓ M ∼ = M/R , where Z is considered as a trivial Γ-module, and R is the submodule of M generated by the elements of type gm − m with m ∈ M and g ∈ Γ. Given Γ-modules M and N , sometimes we will write M ⊗Γ N to mean M ⊗ZΓ N . We have (8.1)

M ⊗Γ N = (M ⊗Z N )Γ ,

where the Γ-module structure on M ⊗Z N is given by g(m ⊗ n) = gm ⊗ gn. If the group Γ acts freely on a CW complex Z, then we have (8.2)

C• (Z/Γ) = C• (Z)Γ .

Here and in the sequel, all the complexes are taken with integral coefficients. Now let Y be a K(Γ, 1). Recall that this means that Y is a connected CW complex with π1 (Y ) = Γ whose universal cover Y% is contractible; for instance, we can take Y = BΓ and Y% = EΓ. The homology of Γ with integral coefficients is defined by H• (Γ; Z) = H• (Y ; Z) . Similarly, for the cohomology, we set H • (Γ; Z) = H • (Y ; Z) .

§8 A brief review of equivariant cohomology

755

More generally, suppose that Γ acts on a cell complex X. Then the Γ-equivariant homology of X is defined as follows. Consider the cell complex Y% × X. The free action of Γ on Y% and the action of Γ on X induce a free action of Γ on Y% × X. We set XΓ = (Y% × X)/Γ , and we define the Γ-equivariant homology of X by setting H•Γ (X; Z) = H• (XΓ ; Z) . Similarly, we define the Γ-equivariant cohomology of X by setting HΓ• (X; Z) = H • (XΓ ; Z) . When X reduces to a point {pt}, we have H•Γ ({pt}; Z) = H• (Γ; Z) ,

HΓ• ({pt}; Z) = H • (Γ; Z) .

Since Γ acts freely on Z = Y% × X, using (8.1) and (8.2), we get (8.3) H•Γ (X; Z) = H• C• (Y% ) ⊗Γ C• (X) . In particular, (8.4)

H• (Γ; Z) = H• C• (Y% )Γ .

Equality (8.3) exhibits the Γ-equivariant homology as the homology of the total complex of the double complex C• (Y% ) ⊗Γ C• (X). As such, it is the abutment of two spectral sequences. The first one is nothing but the Leray spectral sequence of the fibration π : XΓ → Y = Y% /Γ having X as fiber. The second one is more subtle and can be interpreted in terms of the projection σ : XΓ → X/Γ. This is not a fibration unless the action of Γ on X is free. The fiber of σ over an orbit Γx ∈ X/Γ is the quotient Y% /Γx , where Γx is the stabilizer of x in Γ. In particular, the fiber of σ over Γx is a K(Γx , 1). We can then expect the homology of these stabilizers to appear in the spectral sequence. We will discuss this momentarily. Looking at (8.3) and (8.4), it is natural to make the following definitions. Suppose that C• is a complex of Γ-modules. Then we define the homology of Γ with coefficient in C• as H• (Γ, C• ) = H• C• (Y% ) ⊗Γ C• . With this notation H•Γ (X; Z) = H• (Γ, C• (X)). If C• consists of a single module M concentrated in degree 0, we have the homology of Γ with coefficients in the Γ-module M : H• (Γ, M ) = H• C• (Y% ) ⊗Γ M .

756


Considering the ring Z as a trivial Γ-module, we get back (8.4). Observe that, according to the previous definition, we have H0 (Γ, M ) = MΓ .

(8.5)

Now let us go back to the double complex C• (Y% ) ⊗Γ C• (X) whose total complex computes H•Γ (X; Z). The E 1 -term of the second spectral sequence is given by 1 Epq = Hq C• (Y% ) ⊗Γ Cp (X) = Hq (Γ, Cp (X)) . We will give an alternative expression for the cohomology of Γ with coefficient in Cp (X). Let Xp be the set of p-cells of X so that ( Cp (X) = Zσ . σ∈Xp

Here Zσ is a copy of Z in which the two generators correspond to the two possible orientations of σ. Let Γσ = {g ∈ Γ : gσ = σ} be the stabilizer of σ. Then Zσ is a Γσ -module, where g ∈ Γσ acts as +1 if g is orientation-preserving and as –1 if it is orientation-reversing. Let Σp be a set of representatives of Xp /Γ. The Γ-module structure of Cp (X) can be described as follows: ( ( ( gZσ = (ZΓ ⊗ZΓσ Zσ ) . Cp (X) = σ∈Σp

σ∈Σp

g∈Γ/Γσ

We claim that Hq (Γ, ZΓ ⊗ZΓσ Zσ ) = Hq (Γσ , Zσ ) . In fact, since Y% /Γσ is a K(Γσ , 1), we have Hq (Γ, ZΓ ⊗ZΓσ Zσ ) = Hq C• (Y% ) ⊗ZΓ (ZΓ ⊗ZΓσ Zσ ) = Hq C• (Y% ) ⊗Γσ Zσ = Hq (Γσ , Zσ ) . We have proved the following proposition. Proposition (8.6). Let Γ be a group acting on a cell complex X. Then there is a spectral sequence abutting to the equivariant homology H•Γ (X, Z) whose E 1 -term is given by ( 1 = Hq (Γ, Cp (X)) = Hq (Γσ , Zσ ) , Epq σ∈Σp

where Σp is a set of representatives for the p-cells of X/Γ.

§8 A brief review of equivariant cohomology

757

A completely similar result holds in cohomology. Moreover, one can substitute the module of coefficients Z, which is a trivial Γ-module, with any Γ-module M . In this case one computes the equivariant homology H•Γ (X, M ) = H• (Γ, C• (X) ⊗ M ), while the E 1 -term of the above spectral sequence is given by 1 Epq = Hq (Γ, Cp (X) ⊗ M ) = ⊕ Hq (Γσ , Mσ ) , σ∈Σp

where Mσ = Zσ ⊗ M . Two cases are particularly interesting. The first is where Γ acts freely on X and M = Z. Then the spectral sequence degenerates at E 2 . Recalling (8.5) and (8.2), we get H•Γ (X; Z) = H• (C• (X)Γ ) = H• (X/Γ; Z) . For the second case, we assume that the action of Γ on X admits only finite stabilizers. This implies that the groups Γσ are finite groups. In this case it is useful to take rational coefficients. This means that we take M = Q with trivial Γ-action. Now, it is a rather elementary result (see [88], III.10) that the homology groups of a finite group are torsion groups. It follows that, in this case too, the above spectral sequence degenerates, and one gets H•Γ (X; Q) = H• (C• (X) ⊗Γ Q) = H• (X/Γ; Q) . If, moreover, X is contractible, one gets H• (Γ; Q) = H• (X/Γ; Q) , and, for the obvious reason, one says that X/Γ is a rational K(Γ, 1). An example of this situation is provided by the mapping class group Γg,P of a P -pointed genus g surface acting on the Teichm¨ uller space Tg,P . Since Tg,P is contractible and the action of ΓS,P has finite stabilizers, the moduli space Mg,P is a rational K(Γg,P , 1), so that H• (Γg,P ; Q) = H• (Mg,P ; Q) . Completely analogous results hold in cohomology. In order to avoid technical difficulties, from now on we will consider only groups Γ such that there exists a contractible complex X, on which Γ acts, having the following properties: 1) For each cell σ, the stabilizer Γσ is a finite group. 2) X/Γ has finitely many cells. 3) There exists a finite index subgroup Γ ⊂ Γ acting freely on X.

758


These properties imply that X/Γ is a rational K(Γ, 1), while X/Γ is a K(Γ , 1). The mapping class group Γg,n satisfies these properties. In fact, we can take as X the Teichm¨ uller space Tg,n and as Γ any level structure Γg,n [m] with m ≥ 3, (cf. Theorem (2.11) of Chapter XVI). Let then X be a contractible complex satisfying conditions 1), 2), and 3). Let Σ be a set of representatives for the cells of X/Γ. We define the virtual Euler–Poincaré characteristic of Γ by setting (8.7) .

χvirt (Γ) =

(−1)dim σ

σ∈Σ

1 . |Γσ |

We also say that χvirt (Γ) is the virtual Euler–Poincaré characteristic of X/Γ, and we write χvirt (X/Γ) = χvirt (Γ) .

(8.8)

Let us show that χvirt (Γ) only depends on Γ. For this, it will suffice to prove the following three facts. The first is that, if Γ acts freely on X, then (−1)i rank(Hi (Γ ; Z)) . (8.9) χvirt (Γ ) = i≥0

The second is that (8.10)

χvirt (Γ) =

χvirt (Γ ) . [Γ : Γ ]

The third is that the expression on the right-hand side of (8.10) does not depend on the choice of the finite index subgroup Γ acting freely on X. The first assertion is a direct consequence of the fact that Γ acts freely on X, so that H• (Γ ; Z) = H• (X/Γ ; Z). For the second assertion, let Σ be a set of representatives for the cells of X/Γ . Then [Γ : Γ ] · χvirt (Γ) =

σ∈Σ

(−1)dim σ

[Γ : Γ ] = (−1)dim τ = χvirt (Γ ) . |Γσ | τ ∈Σ

For the third assertion, let Γ be another finite index subgroup of Γ acting freely on X. Set Γ = Γ ∩Γ . Then Γ is a finite index subgroup of Γ acting freely on X, and χvirt (Γ )/[Γ : Γ ] χvirt (Γ ) χvirt (Γ ) = = , [Γ : Γ ] [Γ : Γ ] [Γ : Γ ] and similarly χvirt (Γ )/[Γ : Γ ] = χvirt (Γ )/[Γ : Γ ]. We may now prove the following lemma.

§9 The virtual Euler–Poincaré characteristic of Mg,n

759

Lemma (8.11). Set χg,n = 2 − 2g − n. Then χvirt (Γg,n+1 ) = χvirt (Γg,n ) χg,n .

(8.12)

Proof. To prove this formula, we look at the natural projection η : Tg,n+1 −→ Tg,n . This induces the projection η : Tg,n+1 /Γg,n+1 [m] = Mg,n+1 [m] −→ Mg,n [m] = Tg,n /Γg,n [m] between moduli spaces with level m structures. The map η is a fibration with fiber S {x1 , . . . , xn }, where S is a genus g Riemann surface, and x1 , . . . , xn are distinct points on it. It follows that (m) (m) χvirt Γg,n+1 = χ (Mg,n+1 [m]) = χ (Mg,n [m]) χg,n = χvirt Γg,n χg,n . Formula (8.12) follows immediately from (8.10) and from the fact that [Γg,n+1 : Γg,n+1 [m]] = [Γg,n : Γg,n [m]]. Indeed, the group Γg,n [m] is the image of Γg,n+1 [m] under the natural surjection Γg,n+1 → Γg,n . 9. The virtual Euler–Poincar´ e characteristic of Mg,n . In this section we compute the virtual Euler–Poincaré characteristic of Mg,n . Let (S, P ) be a fixed P -pointed suface of genus g with |P | = n. Recall, from Section 12 of Chapter XVIII, the Γg,P -equivariant homeomorphism ◦

Ψ : TS,P × ΔP −→ |A0 (S, P )| . Also recall that the equivalence classes of simplices in A0 (S, P )/ΓS,P are in one-to-one correspondence with the set

) isomorphism classes of connected genus g ribbon n Gg = . graphs with n marked boundary components Lemma (9.1). χvirt (Mg,n ) =

(−1)V (G) , | Aut(G)| n

G∈Gg

where V (G) is the number of vertices of G. The appearance of the cardinality of Aut(G) in the above expression stems from the fact that, in the one-to-one corrrespondence between equivalence classes of simplices in A0 (S, P )/ΓS,P and elements of Ggn , the automorphism group of a ribbon graph G coincides with the stabilizer in ΓS,P of the corresponding simplex aG . What, at first sight, looks odd is

760


the sign (−1)V (G) . In fact, the dimension of aG is equal to the number E(G) of edges of G and not to V (G). Indeed, we have V (G) − E(G) = 2 − 2g − n .

(9.2)

This discrepancy depends on the fact that A0 (S, P ) is obtained by subtracting the subcomplex A∞ (S, P ) from the complex A(S, P ). As such, A0 (S, P ) is not a simplicial complex, and one cannot directly use formula (8.7) to compute the virtual Euler–Poincaré characteristics of Mg,n . To understand what is going on, we go back to Lemma (2.3) of Chapter XIX. We have a simplicial complex A, a subcomplex B ⊂ A, and we set C = A B. We assume that A is acted on by a group Γ and that B is preserved by this action. Let A1 and B 1 be the first barycentric subdivision of A and B, respectively. Let D be the subcomplex of A1 whose vertices are barycenters of simplices of C. Then there is a Gequivariant retraction of |C| onto |D|. We are now going to define a Γ-equivariant cell decomposition of |D| from which one may compute the virtual Euler–Poincaré characteristic of |D|/Γ. If b (resp., b0 , b1 , etc.) is a vertex of D, we will denote by a (resp., a0 , a1 , etc.) the corresponding simplex of A. For each vertex b of D, we define the star of b as the subset of simplices of D given by Star(b) = {b, b1 , . . . , bk ∈ Dk : a < a1 < · · · < ak } . We then set

*

|Star(b)| =

◦ |σ|

.

σ∈Star(b)

From the definition it follows that |Star(b)| is a cell, that dim |Star(b)| = dim |A| − dim |a| , and that we have a bona fide Γ-equivariant cell decomposition |D| =

*

|Star(b)| .

a∈C

It is also clear that the stabilizer Γa of a under the action of Γ on C coincides with the stabilizer of |Star(b)| under the action of Γ on |D|. Now suppose that |C| and hence |D| is contractible. Then χvirt (Γ) = χvirt (|D|/Γ) =

[a]∈C/Γ

(−1)codimA (a) . |Γa |


761

Proof of Lemma (9.1). Apply the formula we just wrote to the case A = A(S, P ), B = A∞ (S, P ), C = A0 (S, P ), and Γ = ΓS,P and notice that, by (9.2), codimA (aG ) = 6g − 6 + 3n − E(G) ≡ V (G)

mod 2 . Q.E.D.

Set χg,n = χ(S P ) = 2 − 2g − n . Following Kontsevich, we will compute the formal series Y (t) =

g≥0, n>0 χg,n k. We set E ν• =

1 (ν1 + 2ν2 + 3ν3 + · · · + kνk ) , 2

Vν• = ν1 + · · · + νk .

We also set

G ν• =

isomorphism classes of ribbon graphs with νj vertices of valency j = 1, 2, 3, . . .

) .

We will denote with the same symbol a ribbon graph and its isomorphism class. From the definitions it follows that Eν• and Vν• are, respectively,

762


the number of edges and the number of vertices of a ribbon graph belonging to Gν• . Next we set

) isomorphism classes of ribbon graphs with n boundary = , Gν(n) • components and with νj vertices of valency j = 1, 2, 3, . . . so that

(E

∪ · · · ∪ Gν• ν• Gν• = Gν(1) •

+1)

.

We also set Gνn•

⎫ ⎧ ⎨ isomorphism classes of ribbon graphs with ⎬ = n marked boundary components and with . ⎭ ⎩ νj vertices of valency j = 1, 2, 3, . . .

Notice that, while in the definition of Ggn the ribbon graphs in question are supposed to be connected, no such restriction is made in the definitions (n) of Gν• , Gν• , and Gνn• . Proof of Lemma (9.3). The proof of this lemma is really a repetition of the argument used to prove formula (5.14). Again, we are using 2 dμ = √12π exp(− x2 )dx as probability measure on R. Changing variable √ from x to y/ t, we obtain +

⎛ ⎞ ⎞ ∞ ∞ j j y2 t x ⎠ y 1 1 ⎠ e− 2 dy √ exp⎝−t exp ⎝− dx = √ 2π R j j ( t)j−2 2π R j=2 j=3 ⎛ ⎞, ∞ j 1 y ⎠ √ = exp ⎝− j−2 j ( t) j=3

⎛

k

−1

t

=

→0

ν•

1 (−1)Vν• < y 2Eν• > tVν• −Eν• νi ν !(i) i i=1

k ν•

# # 1 (−1)Vν• #P2Eν• # tVν• −Eν• , νi ν !(i) i=1 i

where PX denotes the set of pairings of a set of cardinality X, and where, to get the last equality, we used Wick’s lemma. The group H=

k

Sνi × Ciνi .

i=1

operates on P2Eν• . The set of orbits of this action coincides with Gν• . Given an (unmarked) ribbon graph G, assign to it a marking in an arbitrary way, and provisionally denote by Autun (G) the automorphism


763

group of the unmarked graph G and by Aut(G) the automorphism group of G considered as a marked graph. Then 1 1 |P2Eν• | = un ν i ν !(i) | Aut (G)| i=1 i k

G∈Gν•

Eν• +1

=

n=1 G∈G (n) ν•

Eν• +1

=

1 | Autun (G)|

1 1 . n! | Aut(G)| n

n=1 G∈Gν•

It follows that ⎛+ ⎛ ⎞ ⎞ ∞ j t x ⎠ dx⎠ log ⎝ exp ⎝−t 2π R j j=2 ⎛ ⎞ 1 ⎝ 1 ⎠ tχg,n , (−1)V (G)

n! | Aut(G)| t−1 →0 n g≥0,n>0, χg,n 0.

When g = n = 1, we get back the result anticipated in formula (6.5) of Chapter XIX. 10. A very quick tour of Gromov–Witten invariants. The theory of Gromov–Witten invariants goes well beyond the scope of this volume and should constitute the subject of another book. Here we would like to give the idea of what this theory is, in a very quick overview, and, for obvious reasons, we will keep the number of bibliographical references to a bare minimum. After Gromov’s work, enumerative geometry takes a sharp detour. The enumerative problems, per se, lose their center stage position, and their solution starts being viewed as a way of producing invariants of the ambient space. For instance, the fact that there are (as Chasles, de Jonquières, and Zeuthen knew) 640 rational nodal plane quartic through 11 general points in P2 is certainly a property regarding quartic curves, but, in a deeper sense, it is a characteristic property of the projective plane P2 . Fix a smooth projective variety V . This will be the ambient space we want to explore. The idea is to use curves as probes. More precisely, one defines the following, seemingly uncomputable, numbers. Fix a homology class β in H2 (V, Z). The latter is a discrete abelian group, and an element of it should be viewed as a discrete invariant as, for instance, the degree of a plane curve. Next look at all genus g curves C and maps f : C → V such that [f (C)] = β (i.e., curves of fixed “degree” and genus). Also fix a certain number of subvarieties V1 , . . . , Vn ⊂ V and suppose that there is some way of saying that there is only a finite number N of curves C and maps f such that (10.1)

[f (C)] = β,

f (C) meets each of the Vi .

This number N is a Gromov–Witten invariant of V . It is denoted with the symbol < [V1 ], . . . , [Vn ] >Vβ,g . In other words,

#⎧ # ⎫ #⎨ pairs ((C, p1 , . . . , pn ); f ) with C of ⎬ . # # # iso## , genus g, pi ∈ C, f : C → V, < [V1 ], . . . , [Vn ] >Vβ,g = ## ⎩ # [f (C)] = β, f (pi ) ∈ Vi , i = 1, . . . , n ⎭ #

where ((C, p1 , . . . , pn ), f ) is isomorphic to ((C , p 1 , . . . , p n ), f ) if there is a bianalytic map ϕ : C → C such that f = f ◦ ϕ and ϕ(pi ) = pi , i = 1, . . . , n.

§10 A very quick tour of Gromov–Witten invariants

767

Figure 4. Kontsevich introduced the appropriate compact moduli space to express these numbers as intersection numbers. It is the moduli space of stable maps: ⎫ ⎧ ⎨ pairs ((C, p1 , . . . , pn ); f ) with (C, p1 , . . . , pn ) ⎬ . iso . an n-pointed nodal curve of genus g, M g,n (V, β) = ⎭ ⎩ f : C → V a stable map, [f (C)] = β. A stable map is simply an analytic map having the following restrictive property. If it happens to contract some of the irreducible components of the pointed curve C to a point, which it may, then that component (as pointed curve) should only have a finite number of automorphisms. There are natural evaluation maps: (10.2)

ev i : M g,n (V, β) −→ V [(C, p1 , . . . , pn ); f ] → f (pi )

and the Gromov–Witten invariants of V are easily expressed as intersection numbers on M g,n (V, β) by pulling back the Poincaré duals ωVi of the classes [Vi ] to M g,n (V, β), via the various evaluation maps, and then intersecting them: V ev1 ∗ (ωV1 ) ∧ · · · ∧ ev∗n (ωVn ) . (10.3) < [V1 ], . . . , [Vn ] >β,g = M g,n (V,β)

(see, for example, [276] and [497]). It is a rather extraordinary fact that the Gromov–Witten invariants define a deformation of the multiplicative structure of H• (V ). To explain this point, consider a basis [U0 ], . . . , [UN ] of H• (M ) and the classical intersection matrix / [Ui ] · [Uj ] = ωUi ∧ ωUi if i + j = dim(V ), (10.4) gij = V 0 otherwise.

768


This matrix captures the classical multiplicative structure of H• (M ). Now one forms the so-called Gromov–Witten potential Φ(x0 , . . . , xN ) =

< [U0 ]n0 · · · [UN ]nN >Vβ,g

g,n0 +···+nN ≥3 β∈H2 (V )

xnN xn0 0 ··· N n0 ! nN !

and defines a new quantic product by setting [Ui ] ∗ [Uj ] =

kl

∂ 3Φ g kl [Ul ] . ∂xi ∂xj ∂xk

It turns out that this is an associative product in H• (V ) ⊗ Q[[x0 , x1 , . . . , xN ]], and it can be easily checked that the constant term of [Ui ] ∗ [Uj ], as a series in the xi , is the classical intersection product [Ui ] · [Uj ]. Now, the associativity equation for the quantum product ∗ gives differential equations for the Gromov–Witten potential Φ, the socalled WDVV equations. These equations, in some cases, may be used to completely determine a certain number of GW-invariants. It came as a surprise when Kontsevich [446], in 1993, was able to solve by this method the classical enumerative problem of determining the number Nd of degree d genus 0 plane nodal curves passing through 3d − 1 points in general position in P2 . Indeed, he showed that these numbers can be computed inductively starting with Euclid’s 1st axiom N1 = 1. The fact that through 5 points in general position there passes a unique conic (N2 = 1) was determined by Apollonius. Chasles proved that through 8 general points there pass 12 nodal cubics (N3 = 12). As we already mentioned, he, de Jonquières, and Zeuthen proved that N4 = 620. Schubert computed N5 = 87304, but there was no hope at that time to see that, say, N8 = 13525751027392. It turns out that, when V = P2 , the quantum part (i.e., the nonconstant part) Φq of the Gromov–Witten potential Φ can be written in the form (10.5)

Φq (x, y) =

∞

Nd

d=1

y 3d−1 dx e . (3d − 1)!

There are only two variables because H• (P2 ) is two-dimensional. Now the associativity equation for ∗ implies that ∂ 3 Φq = ∂y 3

(10.6)

∂ 3 Φq ∂x2 ∂y

2 −

∂ 3 Φq ∂ 3 Φq , ∂x∂y 2 ∂y 3

and this gives the nontrivial recursive formula 3d − 4 3d − 4 − d31 d2 , Nd1 Nd2 d21 d22 Nd = 3d1 − 2 3d1 − 1 d +d =d 1 2 d1 ,d2 >0

d > 1,

§10 A very quick tour of Gromov–Witten invariants

769

computing Nd for all d > 1 from N1 = 1. In general, the computation of the Gromov–Witten invariants involves: a) The solution of a highly nontrivial foundational problem. This is done by Ruan and Tian [609] in the symplectic setting and by Behrend and Fantechi [57] in the algebro-geometric one. b) The idea of using the Atiyah–Bott localization theorem. This was first conceived by Kontsevich and then very efficiently put to work by Graber and Pandharipande [306]. c) The solution of a general conjecture by Eguchi, Hori, and Xiong [193]. This conjecture goes also under the name of Virasoro conjecture (see Getzler [286]) and is solved in some important cases by Liu and Tian [482], Okounkov and Pandharipande [574,575,576] and Givental [299]. We briefly comment on these three points only looking at the algebrogeometric approach. For the approach via symplectic geometry, we refer the reader to the beautiful survey by Li and Tian [473]. a) K. Behrend and B. Fantechi were able to construct a fundamental class for M g,n (V, β), for any projective variety V . They called it the virtual fundamental class. This major achievement allowed the Gromov–Witten theory to start on very firm ground. After their work the symbol M g,n (V,β)

makes perfect sense. The fact that a fundamental class exists is a remarkable fact in view of the nature of M g,n (V, β), which may be as badly behaved as a Hilbert scheme. What comes to help in defining the virtual fundamental class is the infinitesimal deformation theory for curves. b) When a variety M possesses a torus action, then, using equivariant cohomology, Atiyah and Bott gave a way of reducing the computation Φ to a sum of contribution of the type of integrals of the type M ∗ (i (Φ)/E(N )), where F is a connected component of the fixed F F locus of the action, i is the inclusion, NF is the normal bundle, and E(NF ) is its Euler class. Now consider the case of Gromov–Witten invariants. The variety in question is M g,n (V, β). If the target variety V admits a torus action, as happens when V = Pr , then this action lifts to M g,n (V, β). Graber and Pandharipande proved (1999) that the Atiyah–Bott localization formula works also in the context of the virtual fundamental class, so one may use localization when needed. The pleasant aspect of this idea (originally due to Kontsevich) is that, very often, the components of the fixed locus of a torus action on M g,n (V, β) are simply copies of M p,ν , i.e., moduli spaces of curves. This happens because fixed

770


points of the action appear in correspondence to stable maps C → V that happen to collapse to a point an irreducible component Γ of C. If that component is ν-pointed and of genus p, the fixed locus picks up a component isomorphic to M p,ν . In conclusion, in good cases, one may reduce the computation of GW-invariants to integrals over the much better behaved moduli spaces of curves, a substantial improvement. c) To state the Virasoro conjecture, we need one more ingredient. Until now, we ralated the moduli spaces of stable maps M g,n (V, β) with target variety V via the evaluation maps (2), and by means of these we defined the GW-invariants (10.3). But the moduli space M g,n (V, β) is also closely related to the usual moduli space of stable curves via the obvious forgetful map, M g,n (V, β) −→ M g,n [C,p1 , . . . , pn ; f ] → [C, p1 , . . . , pn ]. Via this map, one can pull back to M g,n (V, β) the classes ψ1 , . . . , ψn ∈ H 2 (M g,n ). We call these pullbacks by the same name. We can then enrich the GW-invariants by mixing them with the ψi . We let {γa : a ∈ A} be a basis of H • (V ). Write < τk1 (γa1 ) . . . τkn (γan ) >Vg,β = ψ1k1 ∧ · · · ∧ ψnkn ∧ ev∗1 (γa1 ) ∧ · · · ∧ ev∗n (γan ) . M g,n (V,β)

It is important to notice that, when the target variety V reduces to {pt} a point {pt}, then the invariants < τk1 ([pt]) . . . τkn ([pt]) >g,[pt] reduce to Witten’s invariants (1.2). Following Witten’s procedure, one then introduces the Gromov–Witten potential F (. . . tki ai . . . ) ∞ 1 = n! k ,...,kn n=0

1 a1 ,...,an

q β < τk1 (γa1 ) . . . τkn (γan ) >Vg,β tk1 a1 . . . tkn an .

β∈H2 (V )

%k , k = Eguchi, Hori, and Xiong define differential operators L −1, 0, 1, 2, . . . , of order less than or equal to 2 in the variables tki ai , % n ] = (m−n)L % m+n , and conjecture %m , L satisfying the Virasoro conditions [L that (10.7)

% k (exp(F )) = 0 , L

k = −1, 0, 1, 2, . . . .

% k defined by Eguchi, This is the Virasoro conjecture. The operators L Hori, and Xiong are a direct generalization of the ones introduced by Witten for the case V = {pt}, and the conjecture (10.7) reduces to Witten’s conjecture when the target variety V is a single point.

§11 Bibliographical notes and further reading

771

At the time of writing this book, the Virasoro conjecture is open for a general projective variety. 11. Bibliographical notes and further reading. There are several proofs of Witten’s conjecture. In our text we followed Kontsevich’s original paper [444] (see also Looijenga’s Bourbaki talk [486]), but at the end of the proof we followed the shortcut proposed by Witten [685]. This shortcut has been analyzed in depth by Fiorenza and Murri [263]. We also departed from Kontsevich’s paper as we are using hyperbolic geometry, instead of the theory of Jenkins–Strebel differentials, in order to define the cellular decomposition of moduli spaces, the reason being that the cellular decomposition obtained in this way seems more suitable to be extended to the boundary of moduli. There are many sources for the graph enumeration techniques and for matrix integrals as, for instance, the papers of Bessis, Di Francesco, Itzykson, Zuber, Kontsevich, Mulase, and Zvonkin [388], [69], [390], [389], [168], [445], [545], [543], [693]. The cell decomposition of moduli spaces makes it possible to define a number of combinatorial cycles in rational homology. A comparison between combinatorial cycles and algebro-geometric cycles can be found in the papers of Penner, Mondello, Igusa, [591], [516], [384], [383], and in [27]. The first algebro-geometric proof of Witten’s conjecture (and by this we mean one not using the cellular decomposition of moduli spaces in terms of ribbon graphs) is due to Okounkov and Pandharipande. The starting point of Okounkov–Pandharipande’s proof is the fundamental work [209] by Ekedahl, Lando, Shapiro, and Vainshtein, where the authors provide a remarkable link between Hurwitz numbers and the intersection theory on the moduli space of curves. Let d be a positive integer, and let μ = (μ1 , . . . , μl ) be a partition of d = |μ| into l parts. Let g ≥ 0 be an integer. Denote by Hg,μ the Hurwitz number associated to these data, which may be defined as a weighted count of genus g degree d covers of P1 with ramification profile over ∞ ∈ P1 given by μ and simply branched over (2g − 2 + |μ| + l) points. The formula for Hurwitz numbers proved in [209], the so-called ELSV formula, is (2g − 2 + |μ| + l)! μμi i = Aut(μ) μi ! l

(11.1)

Hg,μ

i=1

$g

M g,1

l

k=0 (−1)

i=1 (1

k

λk

− μ i ψi )

.

A considerable part of [572] is devoted to rederiving the above equality. This involves the localization-of-virtual-class techniques first introduced by Kontsevich and developed in great depth by Pandharipande and his collaborators. In his previous paper [570], while studying the generating function of Hurwitz numbers, Okounkov proposed to use formula (11.1)

772


and the Toda equations to prove Witten’s conjecture. This program, although not literally, was carried out in [572]. More precisely, the idea is that from this formula one can derive Kontsevich’s fundamental combinatorial identity (4.6). Looking at the RHS of (11.1), Okounkov and Pandharipande make their first remarkable observation, namely that, via Laplace transform, the asymptotic behavior of Hurwitz numbers is governed by the LHS of (4.6). To get Kontsevich’s identity, they proceed in two steps. By dissecting the surface C along paths joining ramification points, they first establish that another way to look at the asymptotics of Hurwitz numbers is to relate them with the asymptotics of random trees, and then they relate these to ribbon graphs, showing that the asymptotics of Hurwitz numbers are also governed by the RHS of 6. In this way Okounkov and Pandharipande are able to short-circuit the need for a combinatorial expression of the ψi which was a delicate point of Kontsevich’s proof. A different proof of the ELSV formula is given by Graber and Vakil [309], who refined an idea of Fantechi and Padharipande [245]. A shorter algebro-geometric proof of Witten’s conjecture is given by Kazarian and Lando [407]. Also in this proof the starting point is the ELSV formula (or better its inverse) which expresses Hodge integrals in terms of Hurwitz numbers. Then the authors use a combinatorial trick to express the ψ-integrals in terms of Hodge integrals (and, therefore, in terms of Hurwitz numbers via the ELSV formula). This allows them to write the generating function of ψ-integrals in terms of generating functions of Hurwitz numbers. At this point they use a result of Okounkov [570], which says that the generating function of Hurwitz numbers satisfies the KP hierarchy, and this implies Witten’s conjecture. The key idea of inverting the ELSV formula was already contained in the papers of Shadrin [638] and Zvonkine [694]. A completely different and rather astonishing proof of Witten’s conjecture has been given by Mirzakhani [513]. In her paper she establishes a relationship between the Weil–Petersson symplectic form on the moduli space of hyperbolic Riemann surfaces with geodesic boundary components of given length, and the intersection numbers of the ψ-classes on moduli spaces of curves, and as a biproduct she obtains a result which is equivalent to Witten’s conjecture (cf. also [546]). A treatement of the link between matrix models and the KdV equation can be found in the book by Kaku [402]. This link was found in the late 1980s by a group of theoretical physicists, Kazakov [405], Brézin and Kazakov [86], Douglas [184], Douglas and Shenker [185], Gross and Migdal [323], in connection with the nonperturbative description of 2D gravity. The discrete model used in this description and the corresponding partition function led to the Toda lattice, and passing from a discrete to a continuous parameter the KdV equation made its appearance. In the same period Witten, Dijkgraaf, Verlinde, and Verlinde [684], [182], [181],

§12 Exercises

773

[179] discovered that the same partition function arises in 2D topological gravity, and in this setting the string equation and the KdV equation appear, so to speak, on equal footing by virtue of the Virasoro operators. There are many sources for the general theory of the KdV and KP equations. Among them are the ones by Segal and Wilson [617], Dickey [178], Date, Kashiwara, Jimbo, and Miwa [158], Kac and Raina [401], and Mulase [544] (cf. also [21]). The virtual Euler–Poincaré characteristic of Mg,n was first computed by Harer and Zagier in their fundamental paper [344] and by Penner [589,590]. In [498], Manin gives a formula for the Poincaré polynomial of M0,n . This formula was independently established by Getzler in [284], where the Sn -invariant Poincaré polynomial of M0,n is also computed. The Euler–Poincaré characteristic of Mg,n has been computed by Bini and Harer in [72]. An important generalization of Witten’s conjecture is the so-called r-spin Witten’s conjecture. Building on work by Givental [300], this conjecture has been proved by Faber, Shadrin, and Zvonkine in [240] (see also Polishchuk [597]). 12. Exercises. A. Virasoro equations and intersection numbers We presented two ways of recursively computing intersection numbers. One is to use the Virasoro equations Ln Z = 0, and the other is to use the string equation coupled with the KdV hierarchy. The reader should compare these two methods in explicit cases. We propose, as exercises, the following computations. For this, we go back to the notation of (3.3), (3.4), and (3.5). We also set (12.1)

Fg (t0 , t1 , . . . ) =

d1 ,...,dn di ≥0 d1 +···+dn −n=3g−3

so that F =

$∞ 0

Fg . Furthermore, we set >g =

(12.2)

1 < τd1 · · · τdn > td1 · · · tdn , n!

∂ ∂ ∂ ··· Fg . ∂td1 ∂td2 ∂tdn

A-1. Prove that

>=

1 > > 2k + 1 1 + 2 > > + > 4

774


and 1 >g = 2k + 1 +2

g

g

>h >g−h

h=0

>h >g−h

h=0

1 + >g−1 4

.

A-2. Prove that < τ02 τ3 >=

1 , 24

< τ2 τ3 >=

< τ02 τ22 > − < τ02 τ3 >= 29 , 5760

A-3. Prove that < τ3g−2 >g,1 =

< τ23 >=

1 , 8

7 . 240

1 . (24)g g!

In the next series of exercises the reader will prove the two formulas (7.15) and (7.16). The proof follows Faber [230], Faber and Pandharipande [237,236], and an observation by Dijkgraaf regarding the three-point function (cf. [237]). Set τn z n . τ (z) = n≥0

The general three-point function is defined as (12.3) E(x, y, z) =< τ (x)τ (y)τ (z) >= < τa τb τc > xa y b z c . a,b,≥0

It can be shown that the KdV equation can be written in terms of the three-point function E(x, y, z) as follows: ∂ 2x + 1 (x + y + z)2 E(x, y, z) = E(x, 0, 0)(y + z)2 E(0, y, z) ∂x + xE(x, y, 0)zE(0, 0, z) + xE(x, 0, z)yE(0, y, 0) + x(x + y + x)E(x, y, z) + 2xE(x, 0, 0)(y + z)E(0, y, z) + 2x(x + y)E(x, y, 0)E(0, 0, z) 1 + 2x(x + z)E(x, 0, z)E(0, y, 0) + (x + y + z)4 xE(x, y, z) . 4 In the exercises below we will not use the general three-point function but only two particular cases of it, namely the two-point function E(0, w, z)

§12 Exercises

775

and the special three-point function E(w, z, −z), and we will derive the differential equations they satisfy. A-4. Let Td =

n

dj j=0 τj .

Use Exercise A-1 to prove that ⎞ ⎛ n d j ⎠ ⎝ (< τk−1 τ0 Ta >< τ03 Tb > (2k + 1) < τk τ02 Td > = a j j=0 0≤aj ≤dj

+ 2 < τk−1 τ02 Ta >< τ02 Tb >) +

1 < τk−1 τ04 Td > , 4

where a = {a1 , . . . , an }, and Td = Ta Tb . A-5. Use the preceding exercise, with k = a and Td = τb , to prove that the two-point function < τ0 τa τb > wa z b D(w, z) = E(0, w, z) =< τ0 τ (w)τ (z) >= a,b≥0

satisfies the differential equation ∂ 2w + 1 ((w + z)D(w, z)) = wD(w, z) + D(w, 0)zD(0, z) ∂w (12.4) 1 + 2wD(w, 0)D(0, z) + (w + z)3 wD(w, z) . 4 A-6. Show that the unique solution D(w, z) of the differential equation (12.4) satisfying D(w, 0) = exp(w 3 /24) and D(0, z) = exp(z 3 /24) is given by 3 (w + z 3 ) k! [ 1 wz(w + z)]k . D(w, z) = exp 24 (2k + 1)! 2 k≥0

Show that all the terms of total degree 3g in D(w, z)D(−w, 0) have degree at least g in z, so that, in particular, the coefficient of w 2g+j z g−j in D(w, z)D(−w, 0) is zero. Deduce that, for all j ≥ 1, (12.5)

g h=0

(−1)g−h < τ0 τ3h−g+j τg−j >= 0 . − h)!

24g−h (g

A-7. Use (12.5), the string equation, and recursion to prove formula (7.16). A-8. a) Use Exercise A-4 with k = a and Td = τb τc , together with the definition of D(w, z) in Exercise A-5, to prove that the special three-point function < τa τb τc > (−1)c wa z b+c F (w, z) = E(w, z, −z) = a,b,c≥0

776


satisfies the differential equation (12.6) ∂F (w, z) = w(2w + z)D(w, z)D(0, −z) 4w2 F (w, z) + 2w3 ∂w 1 + w(2w − z)D(w, −z)D(0, z) + w 5 F (w, z) . 4 b) Show that the unique solution of this differential equation is given by (12.7) 3 w (a + b)! a+b+1 F (w, z) = exp . w3a (wz 2 )b a+b−1 24 2 (2a + 2b + 2)! 2a + 1 a,b≥0

c) Looking at the coefficient of w g z 2g in (12.7), prove formula (7.15). A-9. Compute ξirr ∗ (κ1 )

M 1,2

M 1,2

ξ1,∅ ∗ (κ1 × 1) ,

κ21 .

κ2 , M 1,2

M 1,2

B. Asymptotic expansions and graphs B-1. Generalize (5.14) and prove that 0

exp

2k ts x s s=1

s!

1

t→0

Γ∈Fk

1 tvs (Γ) , | Aut(Γ)| s=1 s 2k

where Fk is the set of isomorphism classes of Feynman diagrams with vertices of valency not exceeding 2k, and where vs (Γ) denotes the number of vertices of Γ with valency equal to s. B-2. Recall from (5.16) the measure dμN on the space of N ×N hermitian matrices and recall from (5.17) the meaning of the expectation value. Generalize (5.21) to prove the following asymptotic equality: 0

exp

2k ts Tr X s s=1

s!

1

2k N |X2 (G)| tvs (G) , | Aut(G)| s=3 s

G∈Gk

where Gk is the set of isomorphism classes of ribbon graphs with vertices of valency greater than 2 and not exceeding 2k, and where vs (G) denotes the number of vertices of G with valency equal to s. B-3. Prove equality (5.22). B-4. Prove the second equality in (5.24).

§12 Exercises

777

B-5. Prove the equality in (5.26). B-6. Using the change of variable X → equality (5.27).

√ ΛX, prove the asymptotic

C. Virtual Euler–Poincar´ e characteristic Actually, (8.12) is a consequence of a more general result. C-1. Given an exact sequence of groups 1 −→ Γ −→ Γ −→ Γ −→ 1 , deduce from the Hochschild–Serre spectral sequence (cf. [88], VII.3) that χvirt (Γ) = χvirt (Γ )χvirt (Γ ) . C-2. Prove the existence of an exact sequence 1 −→ π1 (S {x1 , . . . xn }) −→ Γg,n+1 −→ Γg,n −→ 1 .

Chapter XXI. Brill–Noether theory on a moving curve

1. Introduction. As we explained in the preface, when this volume was first conceived, it was planned that a major part of it would be devoted to Brill–Noether theory and its interactions with the moduli theory of curves. Over the years, the focus of the book changed, with the main theme becoming the geometry of the moduli spaces of curves. This left very little space for Brill–Noether theory, which in the meantime had developed at a rapid pace; it was thus impossible to give an up-to-date account of it. On the other hand, the alternative of leaving out Brill–Noether theory entirely seemed impractical, if for no other reason, because various foundational results announced in Volume 1, such as, for instance, Petri’s statement, still had to be further discussed and proved. We thus settled on a compromise solution. When Volume 1 was published, a draft version of parts of the intended second volume already existed. We decided to use this draft, with a few changes and additions, as the basis for the present chapter. As a consequence, the chapter has a bit of an archeological flavor, in content and in style of exposition, and does not completely reflect the present state of Brill–Noether theory. Rather, it gives a snapshot of what the theory looked like some twentyfive years ago. Nevertheless, we believe it may still be useful to the student of the subject. The most fundamental question of the Brill–Noether theory is: For which values of r and d does a general curve of genus g possess a gdr ? In Theorem (2.3) of Chapter V we showed that any smooth curve of genus g possesses a gdr as long as the Brill–Noether number ρ = g − (r + 1)(g − d + r) is nonnegative. If we think of the gdr as being given by an (r + 1)dimensional subspace W of the space of sections H 0 (C, L) of some degree d line bundle L on C, the nonnegativity of the Brill–Noether number says that the dimension of the codomain of the multiplication map μ0,W : W ⊗ H 0 (C, ωC ⊗ L−1 ) → H 0 (C, ωC ) E. Arbarello et al., Geometry of Algebraic Curves, Grundlehren der mathematischen Wissenschaften 268, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-540-69392-5 13,

780

21. Brill–Noether theory on a moving curve

is greater than or equal to the one of its domain; here and in the sequel we write μ0 instead of μ0,W when W = H 0 (C, L). The crudest statement about linear series on general curves states that, if a smooth curve C possesses a gdr with negative Brill–Noether number, then the curve is special in the sense of moduli. A more refined statement was implicitly assumed in a long forgotten paper by K. Petri (see (1.7) and the footnote on page 215 of Chapter V). Theorem (1.1) (Petri’s statement, first version). Let C be a general curve of genus g, and let L be a line bundle on C. Then the cup-product homomorphism μ0 is injective. As we saw in Chapters IV and V, this statement is completely equivalent to the following two statements regarding the so called Brill– Noether varieties of linear series on a smooth curve C (see (1.6), Chapter V). Theorem (1.2) (Petri’s statement, second version). Let C be a general curve of genus g. Let d and r ≥ 0 be integers. Then the variety Grd (C) is smooth of dimension ρ. Theorem (1.3) (Petri’s statement, third version). Let C be a general curve of genus g. Let d and r be integers such that r ≥ 0 and r > d − g. Then the variety Wdr (C) has dimension ρ, and its singular locus is Wdr+1 (C). Of course, in the above statements, having negative dimension means being empty. In this chapter, among other things, we prove the three equivalent statements above. In Sections 2 and 3 we introduce the relative Picard variety Picd (p), parameterizing degree d line bundles on the fibers of a family p : C → S of smooth pointed curves, together with the basic Brill–Noether varieties Cdr (p), Wdr (p), and Gdr (p). In Section 4, using the language just introduced, we digress to prove a theorem by Looijenga stating that the tautological ring of Mg vanishes in degree greater than g − 2. In Section 5 we go back to the Brill–Noether varieties and compute their Zariski tangent spaces in terms of the fundamental homomorphisms μ0 and μ1 . From a closer analysis of the μ1 homomorphism we deduce, in Section 6, the smoothness of Gd1 , and, in preparation for the following section, we present Voisin’s cohomological interpretation of the μ1 homomorphism. In Section 7 we give Lazarsfeld’s elegant proof of Theorem (1.1). In Section 8 we introduce Horikawa’s theory of deformations of mappings, and we use it to give yet another cohomological interpretation of the homomorphisms μ0 and μ1 . We then apply Horikawa’s theory to rigorously justify various naive moduli counts. In passing, we prove the theorem of de Franchis asserting the finiteness of the number of nonconstant morphisms from a fixed smooth curve to a (possibly variable)

§2 The relative Picard variety

781

smooth curve of genus ≥ 2. The starting point of Horikawa’s theory is the elementary observation that infinitesimal deformations of a morphism φ : C → M from a curve to a manifold are classified by H 0 (C, Nφ ), where Nφ is the normal bundle to φ. In Section 9 we then interpret the torsion subsheaf Kφ of Nφ and realize that, along a deformation whose Horikawa class lies in H 0 (C, Kφ ), the complexity of the ramification of φ decreases. In Section 10 we apply this principle to prove that the dimension of the Severi variety Σd,g of plane curves of given degree and genus (whose irreducibility has been proved by Joe Harris) is of dimension 3d + g − 1 and that its general point corresponds to a nodal curve. We then prove that, for any pair of integers (d, r) with d ≥ r ≥ 2, there exist irreducible, nondegenerate nodal curves of degree d and genus g for every value of g between 0 and the Castelnuovo bound π(d, r). In Section 11 we introduce the Hurwitz scheme and prove its irreducibility. In Section 12, after revisiting some classical results by Castelnuovo and Segre on plane curves, we show that the most general curve possessing a gd1 with negative Brill–Noether number does in fact possess a unique such gd1 . Finally, in Section 13 we prove some unirationality results concerning loci of curves with small gonality and, in so doing, we rediscover the classical result asserting the unirationality of Mg for g ≤ 10. 2. The relative Picard variety. In Sections 2 and 3 of Chapter IV we constructed the Poincaré line bundles and the basic varieties of the Brill–Noether theory for a fixed smooth genus g curve. In this section and in the following one we are going to repeat the same construction for moving curves. Some of the steps in these relative constructions are the straightforward generalizations of their counterparts in the case of a fixed curve. We will therefore omit repetitive details. We suggest to the reader to go back and forth between this section and the corresponding section in Chapter IV. All our constructions can be performed both in the analytic and in the algebraic category, but we will be mostly concerned with the algebraic one. From now on, unless otherwise specified, we restrict our attention to smooth curves of genus g > 1. Our first object of study is the relative Picard variety. Here is the basic existence theorem. Theorem (2.1). Let d be an integer. Let p : C → S be a family of smooth curves of genus g > 1 parameterized by a scheme S. Suppose that p admits a section. Then there exist a scheme over S Picd (p) → S and a line bundle Ld = Ld (p) over C ×S Picd (p) which restricts to a degree d line bundle on each fiber of p and which satisfies the following

782


universal property. For every morphism f : T → S and every line bundle L on C ×S T , restricting to a degree d line bundle on each fiber of q : C ×S T → T , there exists a unique lifting ϕ : T → Picd (p) of f such that L = (id ×ϕ)∗ Ld ⊗ q ∗ (Q) for some line bundle Q on T . The line bundle Ld is called a Poincaré bundle of degree d. Before proving this theorem, we make a number of remarks. The first one is that when S is a point and C is a single curve C, then Picd (p) = Picd (C), and the line bundle Ld has the properties of the Poincaré line bundle constructed in Lemma 2.2 of Chapter IV. The second remark is that, if σ is a section of p and Σ ⊂ C is the corresponding divisor, then there is a unique Poincaré line bundle which restricts to the trivial bundle on Σ ×S Picd (p) ∼ = Picd (p). We will say that this particular Poincaré line bundle is normalized with respect to σ. The third remark is that, because of the universal property, given a morphism of families of smooth, 1-pointed, genus g curves X

F

q u T

f

wC p u wS

we have Picd (q) = T ×S Picd (p) . In particular, the fiber over s ∈ S of Picd (p) is Picd (Cs ), where, as usual, we set Cs = p−1 (s). Another consequence of this remark is that, in order to construct the pair (Picd (p), Ld ), with Ld normalized, it suffices to do so locally in S because, by the universal property, there is a canonical way to piece together the local constructions. One futher remark is that one may rephrase Theorem (2.1) by saying that the S-scheme Picd (p) represents the relative degree d Picard functor (2.2)

PicdC/S : Sch/S → Sets

defined by PicdC/S (T ) = Picd (C ×S T )/ Pic(T ). Here the quotient denotes the set Picd (C ×S T ) modulo the equivalence relation that declares [L1 ], [L2 ] ∈ Picd (C ×S T ) equivalent if and only if ∼ L1 ⊗ L−1 = q ∗ (Q), where q : C ×S T → T and Q ∈ Pic(T ). In the 2 literature a scheme representing this functor is often denoted with the symbol PicdC/S , but here we find it more convenient to denote it with the symbol Picd (p). Of course, in the definition of the relative Picard


783

functor, we could have started from a family p : C → S of smooth curves of genus g without requiring the existence of a section. What turns out to be true is that the relative Picard functor is representable only when p : C → S has a section. The theorem proves the positive part of this assertion. The final remark is that it suffices to prove Theorem (2.1) for a single integer d. To see this, we make our first use of the existence of a section of p : C → S. Let σ be a section of p, and let Σ ⊂ C be the corresponding relative Cartier divisor. For every S-scheme T , we let ΣT denote the pullback of Σ to C ×S T . Then, given integers d and e, we can define an isomorphism of functors (2.3)

μ : PiceC/S −→ PicdC/S

by setting μT ([L]) = [L ⊗ O((d − e)ΣT )] . It thus suffices to represent the functor PicdC/S for a single value of d. Proof of Theorem (2.1). Let H be the Hilbert scheme of ν-log-canonical smooth 1-pointed genus g curves (cf. Section 5 in Chapter XI), and let

(2.4)

C ⊂ PN × H ϕ u H

be the universal family. Given a family (2.5)

p:C→S

of 1-pointed genus g curves, we let Σ ⊂ C denote the divisor defined by the section of p. Taking local frames for the locally free sheaf p∗ (ωp (Σ)ν ), we may cover S by open sets U so that the family pU = p|U is the pullback of the universal family (2.4) via a morphism η : U → H. By our third remark above, in order to construct the pair (Picd (p), Ld (p)), it suffices to construct the pairs (Picd (pU ), Ld (pU )), and therefore the pair (Picd (ϕ), Ld (ϕ)), for the universal family (2.4). In conclusion, since H is smooth, this reasoning tells us that we may prove the existence of the relative Picard scheme for a family (2.5) under the additional assumption that this is a family of ν-log canonically embedded, smooth, 1-pointed, genus g curves parameterized by a smooth (connected) variety S:

(2.6)

C ⊂ PN × S p u S

784


Under these assumptions, we consider the relative symmetric product Cd → S , which may be viewed as the Hilbert scheme Cd = HilbdC/S . The relative universal effective divisor of degree d D ⊂ C × Cd (2.7)

π

w Cd

u S

is just the universal family over HilbdC/S . The relative universal divisor of degree d satisfies the universal property which is the relative counterpart of the universal property enjoyed by the universal divisor of degree d over a fixed curve (see Lemma 2.1, Chapter IV). To define the scheme Picd (p), the idea is simply to take the quotient of Cd modulo linear equivalence. We ask the reader to go back to the beginning of Section 3 in Chapter XII, where we introduced the notion of equivalence relation R on a scheme X, and where we explained what an effective quotient X/R is. The existence of such a quotient is a rare occurrence. A beautiful result of Grothendieck describes a situation in which effective quotients exist. Theorem (2.8). (“Scholie”, Section XX of [329]) Let X → S be a f

projective morphism. Let R ⇒ X be a schematic equivalence relation h

such that h is proper and flat. Then there exists an effective quotient Y = X/R. Moreover, Y is projective over S, and the quotient map X → Y is faithfully flat. We are going to assume this result; for a proof, we refer the reader to the lucid, self-contained account by Nitsure that can be found in [243], Section 5.6.3, Chapter 5. Look at the universal divisor (2.7). Suppose that d >> 0 so that π∗ O(D) is locally free. Set h

R = Pπ∗ O(D) → Cd . Clearly, R is a smooth variety, projective over S, whose underlying set may be identified with the set {(D, D ) ∈ Cd ×S Cd | D ∼ D } .


785

We may thus think of R as a smooth S-subvariety of Cd ×S Cd and assume that the morphism h is induced by the projection of Cd ×S Cd on one of its factors. Since the subvarieties S, Cd , and R are all smooth, the fact that R is a set-theoretical equivalence relation makes R, automatically, into a schematic equivalence relation. As the morphism h is proper and flat, Grothendieck’s theorem tells us that the quotient Cd /R exists. We set Picd (p) = Cd /R , and we let u : Cd → Picd (p)

(2.9)

denote the quotient morphism. We now construct the Poincaré line bundle. To construct Ld = Ld (p), we fix a section σ of p and denote by Σ ⊂ C the relative divisor it defines. The morphism σ×id

Cd−1 = S ×S Cd−1 −−−→ C ×S Cd−1 → Cd defines a relative divisor Cσ ⊂ Cd . Exactly as in the case of a single curve (i.e., the case where S is a single point), one verifies that L = (idC ×u)∗ O(D − π∗ Cσ ) is a Poincaré line bundle on C ×S Picd (p) (see Lemma 2.2, Section 2, Chapter IV). Q.E.D. Of course, we could have constructed the relative Picard variety in the analytic category. In this category it is much easier to take quotients. For this reason, in the analytic construction of Picd (p) one can circumvent the use of Grothendieck’s theorem (2.8). The most expedient way to proceed is the following. By the same reduction argument we used in the algebraic case, in the analytic case we can limit ourselves to constructing the relative Picard variety for a local Kuranishi family (2.10)

ϕ:C→B

of smooth, genus g curves. Since the problem is furthermore assume that (2.10) admits an analytic also fix a frame for ϕ∗ ωϕ , where ωϕ is the sheaf of differentials, that is, a basis ω1 (b), . . . , ωg (b) for

local on S, we can section σ. We may relative holomorphic H 0 (Cb , ωCb ) varying

786


holomorphically with b. Using the period map, described in Section 8 of Chapter XI, one then constructs the relative Jacobian J (ϕ) → B . We then have an analytic map u : Cd → J (ϕ)

(2.11)

defined in terms of the familiar abelian sums d pi ub (D) = . . . , ων,b , . . . , i=1

σ(b)

where D = p1 + · · · + pd and ub = u|Cb . When d >> 0, the morphism u identifies J (ϕ) with the quotient of Cd modulo linear equivalence. This is how we define Picd (ϕ) for d >> 0 in the analytic category. Then, using the section σ, and therefore the isomorphism (2.3), we define Picd (ϕ) for every d ∈ Z. The construction of the relative Poincaré line bundle is exactly as in the algebraic case. In the analytic category we also have the exponential sequence 0 → Z → OC → OC× → 0 , and we get the identification Pic0 (ϕ) = R1 ϕ∗ OC /R1 ϕ∗ Z . Let us finally comment on the existence of the relative Picard variety for families p : C → S that do not admit a section. We start with the analytic case. In this setting, using Kuranishi families, we may always assume that S is covered by small open sets where a section exists, and we may fix one such section for each open set U of this cover. One then constructs Picd (pU ) and uses the universal property to patch together these local pieces and define an analytic variety Picd (p). What cannot be constructed is a Poincaré line bundle, since a Poincaré line bundle which is normalized with respect to one section may not be normalized with respect to another one. To see that Picd (p) exists in the algebraic category also when p has no sections , we shall use the following lemma. π

f

Lemma (2.12). Let Z −→ S − → S be morphisms of schemes with f étale , and so on, the underlying analytic spaces and finite. Denote by Zan , πan and morphisms. Suppose that there is a cartesian diagram of analytic spaces F Zan w Y (2.13)

πan

u

San

fan

u

π

w San


787 F

π

→ S Then there are a scheme Y and morphisms of schemes Z −→ Y − such that F wY Z π

u S

f

π u wS

is cartesian and has (2.13) as underlying diagram of analytic spaces. Moreover, Y is unique up to a unique isomorphism. Proof. The morphism F is étale and finite, and Y is a quotient of Zan modulo the equivalence relation = Zan × Zan ⊂ Zan × Zan . R Y ×San Y , we have Since Zan = San = San R ×San Zan = Ran ,

where R = S ×S Z = (S ×S S ) ×S Z is an equivalence relation on Z. On the other hand, p1 : R = S ×S Z → Z is étale and finite since f is. From Theorem (2.8) it follows that there exists an effective quotient Z/R. We then set Y = Z/R and let F : X → Y be the quotient map and π : Y → S the natural projection. The uniqueness of Y follows from the uniqueness of quotients. Q.E.D. We may now prove the existence of Picd (p) in the category of schemes. Start with an algebraic family p : C → S of smooth curves. For any point of S, we may find a Zariski-open neighborhood U and a finite étale base change f : U → U such that the pulled-back family p : C → U has a section. There is a natural projection map of analytic spaces Picd (p ) → Picd (pU ) , where pU : CU → U is the restriction of p : C → S to U . It follows from Lemma (2.12) that Picd (pU ) can be given a scheme structure. The different Picd (pU ) patch together by the uniqueness part of the same lemma. One may wonder what are the functorial properties of the scheme Picd (p) when p : C → S has no sections. Does it represent a functor? The answer is positive but subtle. It represents the functor (PicdC/S )ét defined as follows. Endow the category of S-schemes with the étale topology and consider the contravariant functor PicdC/S as a presheaf in

788


this topology. Then (PicdC/S )ét is the sheafification of PicdC/S . For an exhaustive discussion of these matters, see Kleiman’s Chapter 9 in [243]. 3. Brill–Noether varieties on moving curves. In this section we define the basic varieties of the Brill–Noether theory for moving curves. In simple terms, we will redo, with holomorphic dependence on parameters, the constructions we carried out in Section 3 of Chapter IV. Let (3.1)

p:C→S

be a family of smooth curves of genus g > 1 parameterized by a scheme or analytic space S. As customary, we set Cs = p−1 (s). The first Brill– Noether variety we will consider is denoted Cdr and parameterizes effective divisors D of fixed degree d in the fibers of p such that r(D) ≥ r for some fixed r. Thus, set theoretically, (3.2) supp(Cdr ) = {(s, D) : s ∈ S, D ∈ (Cs )d such that h0 (Cs , OCs (D)) ≥ r + 1} . The second Brill–Noether variety, instead, parameterizes degree d line bundles L on the fibers of p with h0 (L) ≥ r + 1 and is denoted by Wdr (p). Again set-theoretically, (3.3) supp(Wdr (p)) = {(s, D) : s ∈ S, L ∈ Picd (Cs ) such that h0 (Cs , L) ≥ r + 1} . The mechanism which makes it possible to put a scheme structure on these varieties is the same in both cases, and we briefly review it next. Let E be a coherent sheaf on a scheme or analytic space X and α a b suppose that there is an exact sequence OX − → OX → E → 0 (a presentation of E). The hth Fitting ideal of E is the ideal sheaf Ih ⊂ OX generated by the minors of α of size b − h + 1. The important fact about this ideal is that it depends only on E and not on the presentation (cf., for instance, [567]); it is also clear that Ih is functorial under base change. Since any coherent sheaf locally has a presentation, these properties imply that it makes sense to speak of the hth Fitting ideal of a coherent sheaf on X, even in the absence of a global presentation. The subvariety of X defined by the hth Fitting ideal of E is precisely the locus of those x such that dim(E ⊗ k(x)) ≥ h. Now let f : Y → X be a family of smooth curves of genus g, and L a line bundle of relative degree d on Y . Since L is flat over X, the basic theory of base change in cohomology (reviewed, for instance, at the beginning of Section 3 of Chapter IX) implies that there is, locally on X, a complex (3.4)

α : K0 → K1

§3 Brill–Noether varieties on moving curves

789

of free sheaves which calculates, functorially with respect to base change, the direct images of L. In other words, there is an exact sequence α

→ K 1 → R1 f ∗ L → 0 , 0 → f∗ L → K 0 − and the same is true after an arbitrary base change. We will denote the subscheme defined by the ith Fitting ideal of R1 f∗ L with X r . Its support is the locus of all x ∈ X such that R1 f∗ L ⊗ k(x) = H 1 (Yx , Lx ) has dimension at least i or, by Riemann–Roch, the locus of all x such that H 0 (Yx , Lx ) has dimension at least r + 1, where r = d − g + i. Returning to the Brill–Noether varieties, we apply this construction to define Cdr and Wdr (p). For the first one, we look at the projection π : C ×S Cd → Cd and at the line bundle O(D), where D ⊂ C ×S Cd is the universal divisor (cf (2.7)). We then define Cdr ⊂ Cd to be the S-subscheme defined by the (g − d + r)th Fitting ideal of R1 π∗ O(D). By what we have just said, (3.2) holds. To define Wdr (p), we provisionally need to assume that p has a section, so that there exists a Poincaré line bundle Ld on C ×S Picd (p). We then define Wdr (p) ⊂ Picd (p) to be the S-subscheme defined by the (g −d+r)th Fitting ideal of R1 q∗ Ld , where (3.5)

q : C ×S Picd (p) → Picd (p)

is the projection to the second factor. Again, (3.3) clearly holds. The crucial remark to be made here is that Wdr (p) does not depend on the particular Poincaré bundle Ld and therefore, more importantly, does not depend on the choice of a section for p. Indeed, if Q is a line bundle on Picd (p), then, for every i ≥ 0, the ith Fitting ideal of R1 q∗ Ld coincides with the ith Fitting ideal of R1 q∗ (Ld ⊗ q ∗ Q). Recalling that the Fitting rank of a coherent sheaf F is defined to be the largest integer h such that the hth Fitting ideal of F vanishes, a consequence of the functorial properties of Fitting ideals is that the scheme Wdr (p) represents the functor (3.6)

Sch/S −→ Sets T → [L] ∈ PicdC/S (T ) : Fitting-rank(R1 pT ∗ L) ≥ g − d + r} ,

where pT : C ×S T → T is the projection.

790


Exactly as in Proposition (3.4) of Chapter IV, the functorial properties of Fitting ideals, the ones of the universal divisor and the ones of the Poincaré line bundle imply the scheme-theoretical equality u−1 (Wdr (p)) = Cdr ,

(3.7)

where u : Cd → Picd (p) is the Abel–Jacobi map (2.9). We also have a relative counterpart of Lemma (2.3) of Chapter IV. Namely, there exist isomorphisms ϕ and ψ making the following diagram commute: du

TCd /S (3.8)

ψ u

π∗ OD (D)

δ

w u∗ TPicd (p)/S ϕ u w R1 π∗ OC×S Cd

In this diagram, TCd /S and TPicd (p)/S denote the relative tangent sheaves, and the matrix of du is the moving Brill–Noether matrix. In other words, if D = q1 + · · · + qd and η1 , . . . , ηg is a local frame for p∗ ωp , then η1,s (q1 ) ⎜ .. =⎝ .

...

⎞ ηg,s (q1 ) ⎟ .. ⎠, .

η1,s (qd )

...

ηg,s (qd )

⎛

(du)s,D

where, as usual, the above matrix should be replaced by the one on page 159 of Chapter IV when the points qi are not all distinct. On the other hand, the coboundary homomorphism δ is part of a locally free presentation of R1 π∗ OC×S Cd (D): δ

π∗ OD (D) → R1 π∗ OC×S Cd → R1 π∗ OC×S Cd (D) → 0 . Thus, diagram (3.8) expresses the fact that the defining ideal of Cdr in Cd , which is by definition the (g − d + r)th Fitting ideal of R1 π∗ OC×S Cd (D), is generated by the (g −d+r)×(g −d+r) minors of the moving Brill–Noether matrix. We now come to the third, and probably most important, Brill– Noether variety for moving curves. While Wdr (p) parameterizes complete degree d linear series of dimension greater than or equal to r on the fibers of (3.1), the scheme Gdr (p) we are going to construct parameterizes all gdr ’s on the fibers of (3.1). To describe the construction, it is convenient to go back to the setup consisting of a family f : Y → X of smooth curves of genus g and a line bundle L of relative degree d on Y . Suppose first that a complex (3.4) calculating the direct images of L exists on all of X.


791

β

Let G(r + 1, K 0 ) − → X be the Grassmannian bundle of (r + 1)-planes in the fibers of K 0 , and let j : V → β ∗ K 0 be the universal subbundle on G(r + 1, K 0 ). Consider the composite homomorphism j

β ∗ (α)

→ β ∗ K 0 −−−−→ β ∗ K 1 V − and pick a global frame e1 , . . . , eb for β ∗ K 1 . For any local section v of V , we may write β ∗ (α) ◦ j(v) = ai e i for suitable functions ai . We let J be the ideal sheaf generated by all the ai for all possible choices of local section v, and we let r ⊂ G(r + 1, K 0 ) X be the subscheme defined by J. We also denote by W the restriction of r. V to X Now let γ : T → X be a morphism, denote by pY and pT the projections of Y ×X T to its two factors, and suppose we are given a rank r + 1 locally free subsheaf F of pT ∗ (p∗Y L) with the property that the homomorphism (3.9)

∗ 0 F ⊗ k(t) → H 0 (p−1 T (t), pY L ⊗ k(t)) = H (Yγ(t) , Lγ(t) )

is injective for all t ∈ T . For brevity, we shall refer to such a datum as a family of gdr ’s in f∗ L parameterized by T . The prototypical such object is r . In fact, by the definition of X r , W is a subsheaf the bundle W on X r ∗ ∗ 0 of the kernel of the restriction to X of β (α) : β K → β ∗ K 1 . On the other hand, by the characteristic property of K 0 → K 1 , this kernel is r ×X Y . r of the pullback of L to X just the pushforward to X Going back to γ : T → X and F , the latter is a subsheaf of pT ∗ (p∗Y L) = ker(γ ∗ K 0 → γ ∗ K 1 ), and the injectivity of (3.9) says that it is in fact a vector subbundle of K 0 . By the universal property of the Grassmannian, F is the pullback of the universal subbundle V via a unique morphism T → G(r + 1, K 0 ). In fact, this morphism lands in r , since F is a subsheaf of the kernel of γ ∗ K 0 → γ ∗ K 1 . Conversely, X r gives, by pullback of W , a family of g r ’s in f∗ L any morphism T → X d r represents the functor parameterized by T . This means that X (3.10)

Sch/X −→ Sets T → {families of gdr ’s in f∗ L parameterized by T } .

r does not depend on the choice of the complex K • . In particular, X Before we show how to remove the assumption that there exists, globally on X, a complex (3.4) calculating the direct images of L, we

792


pause for a side remark. If we fix trivializations for K 0 and K 1 , the homomorphism α can be regarded as a morphism σ : X → M (a, b) to the variety of a × b matrices, where a and b are the ranks of K 0 and K 1 . Then X r is the pullback via σ of the generic determinantal variety Mk (a, b) consisting of all matrices of rank at most k, with k = b−r +d−g r is the pullback, via the same (cf. Section 2 of Chapter II). Similarly, X k (a, b) of Mk (a, b), also morphism σ, of the canonical desingularization M discussed in Section 2 of Chapter II. To prove that one can do away with the assumption that there exists a complex (3.4) on all of X, we may argue as follows. We know that X can be covered with open subsets on each of which a complex r, (3.4) exists. Thus, for each such open set U , we can construct U which represents the functor (3.10), of course with X replaced by U . On r is clearly compatible with base the other hand, the construction of U change. In particular, if V is another open subset of X on which a r r complex (3.4) exists, U ∩ V is the same as the pullback to U ∩ V of U r r or of V . In other words, the various U patch together canonically by r and represents universality. The resulting scheme is again denoted by X the functor (3.10). What we have proved can be summarized in the following statement. Lemma (3.11). Let f : Y → X be a family of smooth curves of genus g, and L a line bundle of relative degree d on Y . Then there is a scheme r over X which represents the functor (3.10). X We now have at our disposal the tools needed to construct Gdr (p), where p : C → S is the family (3.1). We assume that p has a section. r constructed The scheme Gdr (p) is a special instance of the scheme X above. We take X = Picd (p) , Y = C ×S Picd (p) , f = the projection C ×S Picd (p) → Picd (p) , L = a Poincaré line bundle Ld . r implies that G r (p) represents The universal property (3.11) enjoyed by X d a well-defined geometric functor. To explain what this is, we make a formal definition. Definition (3.12). Let p : C → S be a family of smooth curves of genus g. A family of gdr ’s on p : C → S parameterized by an S-scheme f : T → S is a pair (L, H), where L is a line bundle on CT = C ×S T whose restriction to each fiber of pT : CT → T has degree d, and H is a locally free subsheaf of pT ∗ L of rank r + 1 such that, for each t ∈ T , the fiber homomorphism H ⊗ k(t) → H 0 (p−1 T (t), L ⊗ k(t))


793

is injective. Two families (L, H) and (L , H ) are said to be equivalent if there are a line bundle Q on T and an isomorphism L −→ L ⊗ p∗T Q inducing an isomorphism H −→ H ⊗ Q. The universal property of Gdr (p) may now be expressed by the following result. Theorem (3.13). Let p : C → S be a family of smooth, curves of genus g > 1. Suppose that p admits a section. Then there exist an S-scheme Gdr (p) representing the functor (3.14)

Sch/S −→ Sets equivalence classes of families of gdr ’s T → on p : C → S parameterized by T

and a morphism χ : Gdr (p) → Picd (p) over S, factoring through the inclusion Wdr (p) ⊂ Picd (p). The morphism χ corresponds to the forgetful morphism of functors which associates to a family (L, H) of gdr ’s the line bundle L. An obvious but important onbservation is that, when the family p : C → S consists of a single curve C, i.e., when S is a point, the varieties Wdr (p) and Gdr (p) coincide with the varieties Wdr (C) and Grd (C) defined in Section 3 of Chapter IV. It should also be noticed that the morphism χ : Gdr (p) → Wdr (p) is biregular off Wdr+1 (p) and that, over each point w ∈ Wdr+1 (p) corresponding to a degree d line bundle L on C, the fiber χ−1 (w) is the Grassmannian G(r + 1, H 0 (C, L)). Let us briefly comment on the existence of the varieties Wdr (p) and when the family p : C → S admits no section. Here we may repeat, word by word, the argument at the end of Section 2 where we showed how to define Picd (p) when p has no sections. To this end, we start from a family p : C → S of smooth, genus g curves. Since local analytic section of p exist, we may define Wdr (p) and Gdr (p) as analytic spaces. After shrinking S, if necessary, we pass to a finite étale cover f : S → S such that the pulled-back family p : C → S has a section. Thus, both Wdr (p ) and Gdr (p ) are algebraic. By universality, Wdr (p ) = Wdr (p) ×S S and Gdr (p ) = Gdr (p) ×S S . We may then apply Lemma (2.12). Of course, when p has no sections, neither Wdr (p) represents the functor (3.6), nor Gdr (p) represents the functor (3.14). Gdr (p)

Finally, let us introduce the Brill–Noether subloci of Mg . Let p : C → S be a family of smooth, genus g curves. Look at the induced

794


map pd : Cd → S. We define the subscheme Sdr ⊂ S as the image of Cdr via pd : Sdr = pd (Cdr ) .

(3.15)

Attached to the family p is the moduli morphism m : S → Mg . In the algebraic category we say that (3.1) is a family of genus g curves with general moduli if the moduli map is dominant, while in the analytic category a family of genus g curves with general moduli is one for which m(S) is open in Mg . For example, the Kuranishi family (2.10) is a family with general moduli. Notation (3.16). For a family p : C → S of genus g > 1 curves which is r a Kuranishi family at every point of S, we often write Picdg , Wg,d , and r d r r d Gg,d or, even more simply, Pic , Wd , and Gd instead of Pic (p), Wdr (p), and Gdr (p). Consider a family of curves with general moduli and assume that its r ⊂ Mg by moduli map is surjective. We then define the subvarity Mg,d setting r Mg,d = m(Sdr ) .

(3.17)

r is the locus in Mg described by curves possessing a gdr . The variety Mg,d One of the basic tasks of the Brill–Noether theory for moving curves is to study the local nature of Wdr and Gdr and to compute the dimension r of the irreducible components of Mg,d .

The following diagram summarizes the configuration of Brill–Noether varieties for a family p : C → S: Cdr ⊂ Cd

(3.18)

ud

w Picd (p) ⊃ Wdr (p) u h pd h πd h u h k m r r Sd ⊂ S w Mg ⊃ Mg,d

χ

Gdr (p)

An important sublocus of Mg is the one whose points represent curves satisfying Petri’s condition, in the following sense. Definition (3.19). A smooth genus g curve C is said to satisfy Petri’s condition if for every line bundle L on C, the cup product map μ0 : H 0 (C, L) ⊗ H 0 (C, ωC L−1 ) → H 0 (C, ωC ) is injective.


795

Proposition (3.20). The locus of points in Mg parameterizing curves satisfying Petri’s condition is open. Proof. Look at a Kuranishi family p : C → B and consider the projection qdr : Gdr (p) → B. Then by iii), Proposition (4.1), Chapter IV, we have that {b ∈ B : Cb satisfies Petri’s condition} = r,d b ∈ B : (qdr )−1 (b) is smooth . Q.E.D. Clearly, Petri’s statement (1.1) says that a general curve satisfies Petri’s condition. In view of Proposition (3.20), to prove (1.1), it suffices to exhibit one curve satisfying Petri’s condition. We now make some dimension-theoretic considerations preliminary to the computation of the tangent spaces to Picd , Wdr , and Gdr . Recall the Brill–Noether number ρ = g − (r + 1)(g − d + r) . From Propositions (4.1) and (4.2) in Chapter IV one gets the following result. Proposition (3.21). When g ≥ 2 every component of Gdr has dimension at least 3g − 3 + ρ. Similarly, when r ≥ 0 and r ≥ g − d, every component of Wdr has dimension at least 3g − 3 + ρ, and every component of Cdr has dimension at least 3g − 3 + ρ + r. A number of remarks are in order. Remark (3.22). Note that Proposition (3.21) has meaning even when ρ is negative, since it still may happen that ρ + 3g − 3 is nonnegative. For example, W21 has always dimension 3g − 3 + (2 − g) = 2g − 1. From the existence theorems proved in Chapter VII we already know that the fibers of Wdr (p) → B, where p : C → B is a Kuranishi family, have dimension at least ρ. On the other hand, there are certainly curves, e.g., the Castelnuovo extremal curves C ⊂ Pr of degree d > 2r encountered in Section 2 of Chapter III, for which ρ is negative but Wdr (C) is nonempty, and consequently, dim Wdr (C) > ρ . There still remain the possibility that the whole variety Wdr of special linear series has the expected dimension given by (3.23)

dim Wdr = 3g − 3 + ρ .

For example, look at the case of smooth, degree d plane curves. It is

796


clear that dim Wd2 = h0 (P2 , O(d)) − 1 − dim Aut(P2 ) d(d + 3) −8 = 2 (d − 1)(d − 2) + 3d − 9 = 2 = g + 3d − 9 = 4g − 3 − 3(g − d + 2) = 3g − 3 + ρ , so that (3.23) holds in this case. Lest we become too optimistic, suppose that we go to the next nontrivial family of Castelnuovo extremal curves, which is the case of space curves of degree d = 8 of maximal genus. Thus r = 3 and g = 9. Moreover, by the lemma on page 119 of Chapter III, the general curve of this type is the complete intersection of a nonsingular quadric Q and a quartic. The left-hand side of (3.23) is given by dim W83 = h0 (Q, O(4)) − 1 − dim Aut(Q) = 24 − 6 = 18 , while the right-hand side is 4g − 3 − 4(g − 8 + 3) = 17 , so (3.23) does not hold. Remark (3.24). Proposition (3.21) does not assert that the varieties involved are nonempty when their expected dimension is nonnegative. In fact, this may be quite false. For instance, for r = d = 2, the number 3g − 3 + ρ is nonnegative as soon as g ≥ 3, but Gdr is empty because of Clifford’s theorem. Remark (3.25). Proposition (3.21) covers the genera greater than 1. When g = 0 or g = 1, Gdr is smooth of dimension g + ρ while, for r ≥ d − g, Wdr and Cdr are smooth of dimensions g + ρ and g + ρ + r. 4. Looijenga’s vanishing theorem. As announced in Section 6 of Chapter XVII, to which we refer for the notation, we are now going to prove a vanishing theorem for the tautological ring of Mg , due to Looijenga [489]. The proof of this theorem involves the basic theory of gd1 ’s on moving curves, which is now at our disposal. The statement we want to prove is the following. Theorem (4.1). Ri (Mg ) = 0 for i > g − 2.

§4 Looijenga’s vanishing theorem

797

Before giving the proof of this result, we need to make some preliminary consideration. Instead of working with the stack Mg , we express Mg as the quotient of a smooth variety M by a finite group G. We can also assume that there is a family of smooth genus g curves π:C→M whose moduli map is surjective. We can take as M the moduli space of genus g curves with an appropriate level structure. Points in C will be denoted as pairs (t, x) where t ∈ M and x ∈ Ct = π−1 (t). We also consider the n-fold product C n = C ×M · · · × M C and the n-fold symmetric product Cn = C n /Sn . We denote by ωi the relative dualizing sheaf of the projection πi : C n → C n−1 obtained by i its first Chern class as an omitting the ith component and by K element of A1 (C n ). We will prove the following proposition Proposition (4.2). Let π : C → M be as above. Then any monomial of n ∈ A1 (C n ) vanishes. 1, . . . , K degree d > g + n − 2 in K From this proposition Theorem (4.1) follows easily. In fact we have the cartesian diagrams αi πi Cn Cn wC w C n−1 πn πn−1 π u u u πi β C n−1 wM C n−1 w C n−2 where αi is the projection from C n to the ith factor. With an obvious notation we have =K i , ) = K i . α∗ (K) β ∗ ( κa ) = κ a , π ∗ (K

(4.3)

πi

u

i

n

i

The first and third equalities follow from the cartesianness of diagrams (4.3). For the second, again by cartesianness and by Example 17.4.1 in [275] (cf. the proof of Lemma (4.19) in Chapter XVII), we have, for any a ≥ 0, a+1 a = β ∗ π∗ K β∗κ a+1 ) = (πi )∗ α∗ (K i

a+1

i ) = (πi )∗ (K =κ a . From this, using induction and the push–pull formula, it readily follows that, if ρ = βπi : C n → M is the natural projection, then 1 ρ∗ (K

a1 +1

n ···K

an +1

)=κ a1 · · · κ an .

It is now clear how Theorem (4.1) follows from Proposition (4.2). This proposition is proved by stratifying C n and by using the following elementary lemma.

798


Lemma (4.4). Let X be a scheme, and let X = V0 ⊃ V1 ⊃ · · · ⊃ VN be a stratification by closed subschemes. Let L1 , . . . , LN be line bundles on X such that c1 (Li ) vanishes on Vi−1 Vi , i = 1, . . . , N . Then c1 (L1 ) · · · c1 (LN ) is supported on VN . From now on we fix an integer n >> 0 and an integer d ≥ g + n. We will construct a stratification C n = V0 ⊃ V1 ⊃ · · · ⊃ Vd−1 = ∅

(4.5) of C n such that (4.6)

h|V V = 0 , K i−1 i

i = 1, . . . , d − 1 , h = 1, . . . , n .

Proposition (4.2) will then be a consequence of the elementary lemma above. In order to construct the above stratification, we introduce a number of Brill–Noether varieties. Consider the family of 1-pointed curves (4.7)

q : X = C ×M C → C .

We set Pici = Pici (q), Gd1 = Gd1 (q), and we have natural morphisms η = ηi : Pici → C ,

ζ : Gd1 → C .

We denote by x the image in Pic1 of the section of q, so that x · η1−1 ((t, x)) = [x]. We are going to construct subloci of Gd1 of Gd1 ×M Cd and of Gd1 ×M Cd × C n . The definition of the scheme structure of these subloci will be evident and left to the reader. A point in Gd1 is a triple (t, x, P ), where t is a point in M , x a point on Ct , and P is a pencil (i.e., a gd1 ) on Ct . The first sublocus consists of those pencils P for which dx ∈ P . 1 = {(t, x, P ) ∈ Gd1 : dx ∈ P } . Gd 1 , the morphism ϕP : C → P1 associated This implies that, for (t, x, P ) ∈ Gd to P has degree less than or equal to d and is totally ramified at x. In particular, if d ≤ g, the point x is a Weierstrass point of C. Next, we set 1 V = {(t, x, P, D) ∈ Gd ×M Cd : D ∈ P , | supp(D)| ≤ d − 1} .

In this definition, D is a divisor in P containing a point of multiplicity greater than one, which is then either a base point of P or a ramification point of φP : C → P1 . Set (4.8)

D = mx + D ,

with

x∈ / supp(D ) .

0, then φP is a (d − m)-sheeted ramified cover of P1 and may If D = think that φ−1 φ−1 P (∞) = (d − m)x , P (0) = D .


799

In what follows it will be useful to order the points of the divisor D appearing in (4.8) or at least some of them. For this, we set Z = {(t, x, P, D, x1 , . . . , xn ) ∈ V ×M C n : xi ∈ supp(D) , i = 1, . . . , n} . As we anticipated, all these loci have a natural scheme structure whose definition is left to the reader. For example, 1 Gd = χ−1 ([dx]) ,

where χ : Gd1 → Picd is the natural projection. Since d >> g, it is clear 1 has the structure of a (d − g − 1)-dimensional from its definition that Gd 1 projective bundle over C. In particular, Gd is irreducible of dimension 1 is finite-to-one, so that 2g + d − 3. The natural projection from Z to Gd also Z has dimension equal to 2g + d − 3. We now stratify Z by setting Z k = {(t, x, P, D, x1 , . . . , xn ) ∈ Z : | supp(D) {x}| ≤ d − k − 1} , so that Z = Z 0 ⊃ Z 1 ⊃ · · · ⊃ Z d−1 . The last stratum is somewhat special: for any point (t, x, P, D, x1 , . . . , xn ) ∈ Z d−1 , the two divisors dx and D completely determine the pencil P , while, on the other hand, any point in Z d−1 is of the form (t, x, P, dx, x, . . . , x, ). In particular, under the natural 1 (finite-to-one) projection Z → Gd , the stratum Z d−1 maps isomorphically 1 onto Gd , so that Z d−1 is an irreducible component of Z. Notice that each Z k is closed and that, by Riemann’s existence theorem, Z k−1 Z k is nonempty. The family (4.7) pulls back to a family γ : Y → Z, and we have commutative diagrams F w Cn uY ] [ [ πi τi γ f [ [ u [ u hi Z w C n−1 where F ((y, t, x, P, D, x1 , . . . , xn )) = (x1 , . . . , xn ) , i , . . . , xn ) , hi ((t, x, P, D, x1 , . . . , xn )) = (x1 , . . . , x the section τi : Z → Y of γ is defined by (4.9) τi (t, x, P, D, x1 , . . . , xn ) = (xi , t, x, P, D, x1 , . . . , xn ) , and f : Z → C n by f ((t, x, P, D, x1 , . . . , xn )) = (x1 , . . . , xn ) .

i = 1, . . . , n ,

800


In particular, i = τi∗ c1 (ωγ ) . f ∗K

(4.10) Next, we set

X k = Z k Z d−1 , so that Xk is the union of all the irreducible components of Z k that are distinct from Z d−1 . We have a) X 0 ⊃ · · · ⊃ X d−1 ; b) X 0 = Z; c) X d−1 = ∅ . Finally, we set Vi = f (X i ) ⊂ C n for i = 0, . . . , d − 1. We claim that in this way we obtain a stratification (4.5) satisfying (4.6), thus concluding the proof of Theorem (4.1). The first thing to show is that V0 = C n or, equivalently, that f : X 0 → C n is surjective. Given x1 , . . . , xn in C = Ct , we must find a point x ∈ C, a divisor D in |dx| with a point of multiplicity greater than one and such that xi ∈ supp(D). For this, it suffices to find points y1 , . . . , yd−n−1 such that dx ∼ 2x1 + x2 + . . . xn + y1 + · · · + yd−n−1 . This is possible since, for d − n ≥ g, the morphism C d−n → J(C) (x, y1 , . . . , yd−n−1 ) → [−dx + 2x1 + x2 + . . . xn + y1 + · · · + yd−n−1 ] is surjective by Jacobi’s inversion theorem. Now we have a stratification (4.5). It is also clear that the map f : X 0 → C n is proper. We will next prove the following two lemmas, which are the heart of the entire argument. i |Z k−1 Z k = 0 for k < d and i = 1, . . . , n. Lemma (4.11). f ∗ K Lemma (4.12). Set Uk−1 = X k−1 ∩ f −1 (Vk−1 Vk ) = X k−1 (X k−1 ∩ f −1 (Vk )). Then Uk−1 is contained in Z k−1 Z k . These two lemmas prove that the stratification (4.5) satisfies property (4.6). Indeed, since Uk−1 ⊂ Z k−1 Z k , it follows from Lemma (4.12) i vanishes on Uk−1 . But f : Uk−1 → Vk−1 Vk is proper and that f ∗ K i vanishes on Vk−1 Vk . onto, and thus K Proof of Lemma (4.11). Let W be a connected component of Z k−1 Z k . Look at the restriction to W of the family γ : Y → Z, which, by abuse of notation, we denote by γ : Y → W , and also consider the sections τi defined in (4.9). We have two relative divisors D and D∞ on Y. If D and D∞ are the restrictions of D0 and D∞ to the fiber of γ over a point w = (t, x, P, D, x1 , . . . , xn ) ∈ W , then D∞ = dx ,

D = mx + D

with x ∈ / supp(D ), | supp(D )| = d − k .


801

The relevant remark is that, by the way W is defined, the integer m and the multiplicity νi of xi in D are independent of w ∈ W (of course, if xi = x, then νi = m). For every w ∈ W , there exists a finite morphism πt : Ct → P1 with πt∗ (0) = D and πt∗ (∞) = (d − m)x. Setting yi = πt (xi ), such a morphism determines isomorphisms (4.13) νi +1 C = mP1 ,yi /m2P1 ,yi −→ mνCit ,xi /mC = Tx∨i (Ct )⊗νi = (τi∗ ωγ )⊗νi ⊗ k(w) . t ,xi Were the morphisms πt uniquely defined, the isomorphism (4.13) would give a trivialization of τi∗ (ωγ )νi , thus proving Lemma (4.11), in view of (4.10). But the morphism πt is defined only up to a multiplicative action of C× . Its nonuniqueness can be cured as follows. Let R be ∞}. Write the part of the ramification divisor of πt lying over P1 {0, ri (πt )∗ (R) = i ri zi and normalize πt in such a way that i zi = 1. defined up to multiplication by an rth root At this stage, πt is of unity, where r = ri . We then get a canonical isomorphism C −→ Tx∨i (Ct )⊗rνi = (τi∗ ωγ )⊗rνi ⊗ k(w). Q.E.D. Proof of Lemma (4.12). It suffices to show that f (X k−1 ∩ Z d−1 ) ⊂ f (X k ). Fix a point z ∈ X k−1 ∩ Z d−1 and a one-dimensional analytic arc in X k−1 passing through z and meeting Z d−1 only at z. Restricting the family of curves over Z to this arc, we get an analytic family of pointed curves over Δ = {t ∈ C : |t| < ε}. More precisely, we have an analytic family of triples {Ct , xt , Pt }t∈Δ , where Ct is a smooth curve, xt ∈ Ct , and Pt is a pencil on Ct containing dxt . Moreover, there exists an analytic family of divisors {Dt }t∈Δ , Dt ∈ Pt , such that, for t = 0, we can write Dt = mxt + Dt ,

with xt ∈ / supp(Dt ), | supp(Dt )| = d − k .

Writing Qt = Pt − mxt , ν = d − m, and r = k − m, we are in the assumptions of the following lemma. Lemma (4.14). Let ν be a positive integer, and let {Ct , xt , Qt }t∈Δ be an analytic family of triples, where Ct is a smooth curve, xt ∈ Ct , and Qt is a pencil on Ct containing νxt . Suppose that there exists an analytic family of divisors {Dt }t∈Δ , Dt ∈ Qt , such that, for t = 0, supp(Dt ) is disjoint from xt and has ν − r points, while D0 = νx0 . Then Q0 can be written as Q0 = rx0 + Q for some pencil Q . Assume this lemma. Then the pencil P0 is of the form (m + r)x0 + Q = kx0 + Q and therefore contains a divisor E different from dx0 , with | supp(E) {x}| ≤ d − k − 1, which means that f (z) ∈ f (X k ). This concludes the proof of Lemma (4.12). It now remains to prove Lemma (4.14). For this, observe that if r = 0, there is nothing to prove, while for r > 0, that is, when Dt contains a ramification point of the pencil Qt , Lemma (4.14) is an immediate consequence of the following result.

802


Lemma (4.15). Let ν and {Ct , xt , Qt }t∈Δ be as in Lemma (4.14). Suppose that the pencil Qt is base-point-free for t = 0. For t = 0, let Rt be the intersection with Ct {xt } of the ramification divisor of the morphism associated to Qt . Let R0 be the limit of Rt as t tends to 0. The the multiplicity of x0 in R0 equals the multiplicity of x0 as a fixed point of Q0 . Proof. Let (z, t) be local coordinates around x0 ∈ C0 in the total space of {Ct }t∈Δ , chosen so that the image of the section t → xt is z = 0. We may choose generators f1 and f2 of the pencil Qt which, in terms of these coordinates, have local expressions f2 = ai (t)z i , f1 = z d , i =d

where f2 is divisible neither by t nor by z. The multiplicity of x0 as a base point of Q0 is the least index h for which ah (0) = 0 (h < d). In the ∂ given chart thedivisor Rt is the locus where ∂z (f2 /f1 ) vanishes. This is i (i − d)a (t)z for t = 0. This function is not divisible the divisor of i i =d by either z or t, so that R0 is the divisor of zeroes of i =d (i − d)ai (0)z i , proving that x0 occurs with multiplicity h in R0 . Q.E.D. In the original paper [489] an additional important result is proved. Namely, it is shown that all classes in Rg−2 (Mg ) are proportional to the class [Hg ] of the hyperelliptic locus. This, together with Theorem (7.1) of Chapter XX, shows that [Hg ] = 0 and that Rg−2 (Mg ) = Q · [Hg ]. 5.

The Zariski tangent spaces to the Brill–Noether varieties.

We start with a smooth curve C of genus g > 1 and a Kuranishi family p : C → (B, b0 )

(5.1)

for C ∼ = Cb0 . We assume, in addition, that C → B is a Kuranishi family for every one of its fibers. We set (5.2)

Picd = Picd (p) ,

Wdr = Wdr (p) ,

Gdr = Gdr (p) .

Let ∈ Picd be a point corresponding to a degree d line bundle L on C. We wish to interpret in cohomological terms the exact sequence 0 → T (Picd (C)) → T (Picd ) → Tb0 (B) → 0 . We know how to do this for the two extreme terms. In Section 2 of Chapter IV and in Section 2 of Chapter XI we showed that there are isomorphisms T (Picd (C)) ∼ = H 1 (C, OC ) ,

Tb0 (B) ∼ = H 1 (C, TC ) .

§5 The Zariski tangent spaces to the Brill–Noether varieties

803

To interpret the middle term, we need some preparation. Set S = Spec C[ε] and denote by s0 the closed point of S. A first-order deformation of the pair (C, L) is, by definition, a 4-tuple (ϕ, L, α, β) where (5.3)

ϕ:X →S,

α : C −→ Xs0

is a first-order deformation of C, L is a degree d line bundle on X , and β : L → α∗ L is an isomorphism. An isomorphism between two first-order deformations (ϕ, L, α, β) and (ϕ , L , α , β ) of (C, L) is an isomorphism η : X → X of deformations of C (see Chapter XI, Section 2), plus an isomorphism between L and η ∗ L inducing the identity on L. From the universal property of the Poincaré line bundle described in Theorem (2.1) it follows that T (Picd ) is in one-to-one correspondence with the set of isomorphism classes of first-order deformations of (C, L). To describe these, we proceed as follows. As in (2.3), Chapter XI, we imagine C as being given by transition data {Uα , zα , fαβ (zβ )}, and the first-order deformation (5.3) can be thought of as being given by gluing the Uα × S via the identifications (5.4)

zα = f˜αβ (zβ , ε) = fαβ (zβ ) + εbαβ (zβ ) ,

whereas ϕ is given, on Uα × S, by the projection to S. The Kodaira– Spencer class [ϑ] ∈ H 1 (C, TC ) of the first-order deformation ϕ is given by the cocycle ∂ ϑ = {ϑαβ } , ϑαβ = bαβ . ∂zα Next, we let {gαβ } be the transition functions for L relative to the cover U = {Uα }. We may think of L as being given, relative to the cover {Uα × S}, by transition functions (5.5)

gαβ (zβ , ε) = gαβ (zβ ) + εaαβ (zβ )

satisfying the usual cocycle rule, which, in this case, translates into (5.6) (5.7)

−1 aαγ gαγ

gαβ gβγ = gαγ , ∂gαβ −1 −1 −1 = g bβγ + aαβ gαβ + aβγ gβγ . ∂zβ αβ

Set (5.8)

φ = {φαβ } ∈ C 1 (U, OC ) ,

−1 φαβ = aαβ gαβ ,

η = {ηαβ } ∈ C 1 (U, ωC ) ,

−1 ηαβ = gαβ dgαβ .

Notice that η is actually a cocycle representing the first Chern class c1 (L) ∈ H 1 (C, ωC ). Then (5.7) means that (5.9)

δφ + η · ϑ = 0 ,

804


where η · ϑ ∈ C 2 (U, OC ) is the cup product of η and ϑ. We are now going to construct a rank 2, locally free sheaf ΣL on C whose first cohomology group parameterizes isomorphism classes of first-order deformations of (C, L). In fact, ΣL will fit into the exact sequence (5.10)

τ

0 → OC → ΣL → TC → 0 .

We define ΣL to be the rank two locally free OC -module whose sections are the differential operators of order less than or equal to 1 acting on sections of L. To describe ΣL locally and to construct (5.10), we choose, on each open set Uα , a trivialization χα : L|Uα → OUα so that −1 χα χ−1 β (1) = gαβ and set ξα = χα (1) ∈ Γ(Uα , L), which implies (5.11)

ξβ = gαβ ξα ,

in

Uα ∩ Uβ .

Then, on Uα , the sheaf ΣL is free and generated by the two sections 1α (the constant 1) and Dα , where (5.12)

Dα (f ξα ) =

∂f ξα . ∂zα

The symbol map τ : ΣL → TC in (5.10) sends Dα to Dα (f ξβ ) via (5.11) gives (5.13)

−1 Dα = gαβ

∂ ∂zα .

Computing

∂zβ ∂gαβ 1β + Dβ . ∂zα ∂zα

This means that the extension class of (5.10) is precisely η ∈ H 1 (C, ωC ) ∼ = H 1 (C, Hom(TC , OC )) . Thus, given d, all the ΣL ’s for L ∈ Picd (C) are (noncanonically) isomorphic. Now (5.13), coupled with δϑ = 0 and (5.7), says that the cochain (5.14)

σ = {σαβ } ∈ C 1 (U, ΣL ) ,

σαβ = bαβ Dα − φαβ 1α

is in fact a cocycle. Its cohomology class [σ] will be called the Kodaira– Spencer class of the first-order deformation of (C, L) and maps, via H 1 (C, ΣL ) → H 1 (C, TC ), to the Kodaira-Spencer class of the firstorder deformation of C. In complete analogy with the case of deformation of curves, every 1-cocycle with coeficients in ΣL defines a first-order deformation of (C, L), and cohomologous cycles give rise to isomorphic deformations. In other words, isomorphism classes of firstorder deformations of (C, L) are in one-to-one correspondence with the elements of H 1 (C, ΣL ), that is, we get a bijection

H 1 (C, ΣL ) −→ T (Picd ) which can be easily seen to be linear. In conclusion, we have proved the following result.


805

Proposition (5.15). Let p : C → (B, b0 ) be a Kuranishi family for a smooth curve C. Let be the point of Picd = Picd (p) corresponding to a degree d line bundle L on C. Then the tangent space to Picd at is naturally identified with H 1 (C, ΣL ). Moreover, there is an isomorphism of exact sequences 0

w H 1 (C, OC ) u ∼ =

w H 1 (C, ΣL ) u ∼ =

0

w T (Picd (C))

w T (Picd )

τ

w H 1 (C, TC ) u ∼ =

w0

w Tb0 (B)

w0

where the top exact sequence is the cohomology exact sequence of (5.10), and the vertical arrows are Kodaira–Spencer maps. We are now in a position to identify the Zariski tangent spaces to Wdr and Gdr . Our description will be analogous to that of the Zariski tangent space to Wdr (C) as being given by TL (Wdr (C)) = (Image μ0 )⊥ , where h0 (C, L) = r + 1, and μ0 : H 0 (C, L) ⊗ H 0 (C, ωC L−1 ) → H 0 (C, ωC ) is the multiplication map (see Propositions (4.1) and (4.2), Chapter IV). The main philosophical difference is that derivatives of sections will appear in the formulas in an essential way. Actually, exactly as in Section 4 of Chapter IV, it is better to first determine the Zariski tangent spaces to points of Gdr and then analyze the case of Wdr . Fix a gdr on our curve C. This means that we have a degree d line bundle L on C and an (r + 1)-dimensional subspace W of H 0 (C, L). We let w denote the point of Gdr corresponding to these data. Look at the natural morphism χ : Gdr → Picd and set = χ(w). The fiber of χ over is the Grassmannian of (r + 1)dimensional subspaces of H 0 (C, L), so that we have the exact sequence (5.16)

dχ

0 −→ Hom(W, H 0 (C, L)/W ) −→ Tw (Gdr ) −−→ T (Picd ) .

From the universal properties of Gdr and Picd it follows that the space dχ(Tw (Gdr )) can be described as the set of isomorphism classes of firstorder deformations (5.17)

(X → S = Spec C[ε] , L)

806


of (C, L) for which every section in W extends to a section of L. Let the deformation (5.17) be given by patching data (5.4) and (5.5). Let s be a section in W ⊂ H 0 (C, L) given by holomorphic functions sα (zα ) satifying (5.18)

sα = gαβ sβ

in

Uα ∩ Uβ .

An extension s of s to a section of L corresponds to the datum of functions (5.19)

sα (zα , ε) = sα (zα ) + εtα (zα )

satisfying (5.20)

sα = gαβ sβ ,

which is equivalent to (5.6) plus (5.21)

tα = φαβ sα − bαβ

∂sα + gαβ tα , ∂zα

where bαβ and φαβ are as in (5.4) and (5.8). If we denote by t the cochain in C 0 (U, L) whose local representatives are the tα , then (5.21) reads (5.22)

δt = σ · s ,

where the dot stands for the cup product induced by the C-linear pairing ΣL ⊗C L → OC (notice that this pairing is not OC -linear because differentiation is involved). Therefore, if s extend to a section of L, (5.22) can be solved, i.e., [σ] · s = 0 in H 1 (C, ΣL ). Conversely, if this happens, (5.20) defines an extension of s to a section of L. This discussion shows that the isomorphism classes of the firstorder deformations (X → S, L) of (C, L) such that every section of W ⊂ H 0 (C, L) extends to section of L correspond precisely to the elements of the subspace of H 1 (C, ΣL ) defined by (5.23)

{σ ∈ H 1 (C, ΣL ) : σ · s = 0 for all s ∈ W } .

If we dualize the cup product map H 1 (C, ΣL ) → Hom(W, H 1 (C, L)) ,


807

we get a homomorphism μW : W ⊗ H 0 (C, ωC ⊗ L−1 ) → H 0 (C, ωC ⊗ Σ∨ L)

(5.24)

and a commutative diagram

(5.25)

μ0,W W ⊗ H 0 (C, ωC ⊗ L−1 ) h hh hh μW hh hj h

w H 0 (C, ωC ) u H 0 (C, ωC ⊗ Σ∨ L)

where the vertical arrow is the dual of H 1 (C, OC ) → H 1 (C, ΣL ). The subspace of H 1 (C, ΣL ) given by (5.23) can also be described as (Image μW )⊥ = ker μ∨ W . In conclusion, we have the following result. Proposition (5.26). Let W ⊂ H 0 (C, L) be a gdr on a smooth curve C. Let w be the corresponding point in Gdr . Let χ : Gdr → Picd be the natural projection. Then (5.27)

dχ(Tw (Gdr )) = (Image μW )⊥ = ker μ∨ W ,

and there is an exact sequence dχ

(5.28) 0 −→ Hom(W, H 0 (C, L)/W ) −→ Tw (Gdr ) −−→ (Image μW )⊥ −→ 0 . In particular, dim Tw (Gdr ) = (r + 1)(r − r) + 4g − 3 − (r + 1)(g − d + r) + dim ker(μW ) = 3g − 3 + ρ + dim(ker μW ) , where h0 (C, L) = r + 1, and ρ = g − (r + 1)(g − d + r) is the Brill–Noether number. When W = H 0 (C, L) , we usually write μL , or simply μ, for μW . Since the morphism χ : Gdr → Wdr is biregular off Wdr+1 , we have the following result. Proposition (5.29). Let L be a degree d line bundle on a smooth curve C and set r +1 = h0 (C, L). Let be the point in Wdr corresponding to (C, L). Then T (Wdr ) = (Image μ)⊥ = ker μ∨ . In particular, dim T (Wdr ) = 3g − 3 + ρ + dim(ker μ) , where ρ = g − (r + 1)(g − d + r) is the Brill–Noether number.

808


Keeping the notation introduced above and recalling (3.21), we get the following: Corollary (5.30). The cup product map μW (resp., μ) is injective if and only if Gdr (resp., Wdr ) is smooth and of dimension 3g − 3 + ρ = 4g − 3 − (r + 1)(g − d + r) at w (resp., at ). Recalling Definition (3.19), the preceding corollary, together with the commutativity of diagram (5.25), gives the following result. Corollary (5.31). If the curve C satisfies Petri’s condition, then dim Wdr = dimw Gdr = 3g − 3 + ρ for every r and d and for every ∈ Wdr (C) Wdr+1 (C) and every w ∈ Grd (C). Moreover, Gdr is smooth along Grd (C), and the points of Wdr (C) which are singular in Wdr are precisely those belonging to Wdr+1 . 6. The μ1 homomorphism. We now turn to a typical Kodaira–Spencer question: Given a line bundle L on a smooth curve C, how can one interpret, in cohomological terms, the obstructions to extending L, together with a subspace W of its space of sections, along any deformation of C? In this section we will discuss first-order obstructions. It will turn out that these are completely described in terms of a homomorphism 2 μ1,W : ker μ0,W → H 0 (C, ωC ),

where, as usual, μ0,W : W ⊗ H 0 (C, ωC ⊗ L−1 ) → H 0 (C, ωC ) is the multiplication map. When W = H 0 (C, L), we write μL , μ0,L , μ1,L , or simply μ, μ0 , μ1 when no confusion is likely, instead of μW , μ0,W , μ1,W . The map μ1,W is defined to be the unique map which makes the following diagram commute: μ0,W w H 0 (C, ωC ) W ⊗ H 0 (C, ωC ⊗ L−1 ) u W ⊗ H 0 (C, ωC ⊗ L−1 ) u (6.1) y ker μ0,W

μW

μ1,W

w H 0 (C, ωC ⊗ Σ∨ L) u

2 w H 0 (C, ωC ) u

0

§6 The μ1 homomorphism

809

where the exact sequence on the right is a piece of the cohomology exact sequence of the dual of (5.10) tensored with ωC , 2 0 → ωC → ωC ⊗ Σ∨ L → ωC → 0 .

We consider a Kuranishi family p : C → (B, b0 ) for C ∼ = Cb0 as in (5.1) and adopt the conventions (5.2). Write d for the degree of L, r + 1 for the dimension of W , w for the point of Gdr corresponding to W , and for its image in Wdr . As usual, we also write Bdr for the image of Wdr in B. Dualizing (6.1) and recalling (5.26), we get a commutative diagram

(6.2)

dχ(Tw (Gdr )) ∼ = ker μ∨ W ∩ ∩ T (Picd ) ∼ = H 1 (C, ΣL )

μ∨ W

w Hom(W, H 1 (C, L))

dπ

u u Tb0 (B) ∼ = H 1 (C, TC )

μ∨ 1,W

u w (ker μ0,W )∨

π

where Picd = Picd (p) − → B is the natural morphism. To say that a curve with general moduli possesses a gdr implies that the image of the morphism π ◦ χ : Gdr → B is an open set, so that by Sard’s theorem 1 dπ : ker μ∨ W → H (C, TC )

must be surjective, and therefore μ∨ 1,W must vanish. To summarize, we have proved the following lemma. Lemma (6.3). Let C be a general curve of genus g > 1. Let L be any line bundle on C. Then 2 μ1,L : ker μ0,L → H 0 (C, ωC )

is zero (i.e., ker μ1 = ker μ0 ). It is easy to give an explicit description of μ1,W . Consider an element si ⊗ ri of ker μ0,W . Thus,

si ri = 0 .

810


Think of the si (resp., ri ) as being given by holomorphic functions siα (resp., riα ) satisfying −1 dzβ siα = gαβ siβ , resp., riα = gαβ riβ . dzα Let σ = {σαβ } ,

σαβ = ξαβ Dα + φαβ 1α

be a 1-cocycle with coefficients in ΣL and σ its image in Z 1 (U, TC ): σ αβ = ξαβ

∂ . ∂zα

If we denote by , the duality pairing, we have σ ] = μW [σ]si , ri . si ⊗ ri , [ si ⊗ ri , [σ] = μ1,W The right-hand side of this equality is the integral over C of the cohomology class in H 1 (C, ωC ) represented by the 1-cocycle (6.4)

φαβ

(recall that

(6.5)

siα riα + ξαβ

dsiα

dzα

riα = ξαβ

dsiα dzα

riα

si ri = 0 by assumption). Also notice that 2 dsiα dsiβ dzβ −1 dzβ dgαβ siβ riβ riα = riβ + gαβ dzα dzα dzβ dzα dzα 2 dsiβ dzβ = riβ ; dzα dzβ

2 hence { (dsiα /dzα )riα } is a globally defined section of ωC . It is then clear that the right-hand side of (6.4) is the cup product of σ with this section. Therefore, ds driα iα si ri = (6.6) μ1,W riα = − siα . dzα dzα We end this section by giving the first applications of these considerations and of Corollary (5.30). We will study one-dimensional linear series. Proposition (6.7). Let C be a smooth curve of genus g > 1. Let L be a line bundle on C, and let W be a two-dimensional subspace of H 0 (C, L), that is, an arbitrary gd1 on C. Let Δ be the fixed divisor of the linear series |W |. Then ker μ0,W ∼ = H 0 (C, ωC L−2 (Δ)) . If C is general, then ker μ0,W = 0 .


811

Proof. The isomorphism between ker μ0,W and H 0 (C, ωC L−2 (Δ)) is just the base-point-free pencil trick (cf. Chapter III, page 126). Let s1 , s2 be a basis of W . Suppose that μ0 (s1 ⊗ r1 + s2 ⊗ r2 ) = 0. If C is general, by Lemma (6.3) we must also have μ1 (s1 ⊗ r1 + s2 ⊗ r2 ) = 0. Using the notation of the previous discussion, this means that s1α r1α + s2α r2α = 0 . ds2α ds1α r1α + r2α = 0 . dzα dzα If r1 , r2 are not identically zero, then the determinant s1α

ds2α ds2α − s2α dzα dzα

must vansh identically. But this means that the meromorphic function s2 /s1 is constant, a contradiction. Q.E.D. Despite its deceiving simplicity, the above result is a cornerstone in the study of special divisors. In fact, it has been discovered and rediscovered, from many different points of view, by a large number of authors, over an extended period of time (cf. the bibliographical notes at the end of this chapter). Probably the earliest and certainly one of the clearest formulations of this result was given by Francesco Severi [634]: Sopra una curva di genere p (> 1) a moduli generali, la serie doppia di ogni serie lineare, almeno ∞1 , di ordine ν qualunque esistente sulla curva, è non speciale (e quindi è ν ≥ p2 + 1).8 This is essentially equivalent to Proposition (6.7), which is what we called the first version of Petri’s statement for the case r = 1. It is interesting to notice that, in the case r = 1, Petri’s statement gives the following stronger implication, which is actually false for general r. Proposition (6.8). If g > 1, d ≥ 2, and d ≤ g + 1, then Gd1 is smooth, the singular locus of Wd1 is Wd2 , and dim Gd1 = dim Wd1 = 3g − 3 + ρ = 2g + 2d − 5 . Proof. First of all, notice that Gd1 is nonempty since every hyperelliptic curve has a gd1 . Secondly, observe that, by Lemma (3.5) of Chapter IV, 8

On a curve of genus p (> 1) with general moduli, the double of any linear series of dimension ≥ 1 and arbitrary degree ν is nonspecial (and thus ν ≥ p2 + 1).

812


no component of Wd1 is entirely contained in Wd2 . By Corollary (5.30) it suffices to show that ker μW = 0 for any line bundle L on a smooth genus g curve C and any two-dimensional subspace W ⊂ H 0 (C, L). Now, ker μW is contained in ker μ1,W ⊂ ker μ0,W . Thus, assuming that ker μW = 0, we can use the same argument as in the proof of Proposition (6.7) to reach a contradiction. Q.E.D. For example, for curves of genus 4 the projection W31 → B has one-dimensional fibers over the hyperelliptic locus and, outside this set, is a two-sheeted covering map, branched along the locus of curves lying on singular quadrics. In general, since the fiber of Gdr (resp., Wdr ) over a point b ∈ B is Grd (Cb ) (resp., Wdr (Cb )), the only thing that one can deduce from the second and third version of Petri’s statement is that Gdr (resp., Wdr ) is smooth outside a subvariety that projects to a proper subvariety of B (resp., outside the union of Wdr+1 and a subvariety that projects to a proper subvariety of B). In general the analogue of Proposition (6.8) is false when r ≥ 2. However, in Section 10 we will prove a result about the structure of Gd2 which is more precise than the one implied by Petri’s statement in the case r = 2. At this point we know that, when the dimension of Wd1 (C) exceeds the expected one, the curve C has to be special in the sense of moduli. In Section 5 of Chapter IV, our discussion of Martens’ and Mumford’s theorems showed that it is possible to explicitly describe those curves C for which the dimension of Wd1 (C) is close to its maximum possible value. Along these lines, a natural, more general question is the following. How special (in the sense of moduli) is a curve C carrying a gd1 for which the Brill–Noether number ρ is negative? In terms of diagram (3.18) our 1 ? Or, which is the same, what question is: what is the dimension of Mg,d 1 is the dimension of Bd ? Again the answer is provided by Severi [634]: La pi` u generale curva di genere p > 1, che soddisfa alla condizione di contenere una gν1 (almeno) di ordine ν < p2 + 1 (e perci` o sia a moduli particolari) dipende precisamente da 3p − 3 − i moduli, ove i è l’indice di specialit` a della serie doppia della gν1 .9 This has been rediscovered by Farkas [260]. More precisely: Proposition (6.9) (Severi–Farkas). Let k be the minimum possible 1 , L runs through all value of h1 (C, L2 (−Δ)), where C belongs to Mg,d 9

The most general curve of genus p > 1 having the property of possessing (at least) a gν1 of degree ν < p2 + 1 (which makes the curve one with particular moduli) depends exactly on 3p − 3 − i moduli, where i is the index of speciality of the double of the linear series gν1 .


813

degree d line bundles on C with h0 (C, L) ≥ 2, and Δ stands for the base locus of |L|. If d < g/2 + 1 (i.e., if ρ < 0), then 1 = 3g − 3 − k . dim Mg,d 1 ≤ 3g − 3 + ρ. In particular, dim Mg,d

This follows directly from Proposition (5.26) and Proposition (6.7). It is now natural to ask: when d < g/2 + 1, is it true that 1 dim Mg,d = 3g − 3 + ρ ? To further justify this question, observe that, if ρ < 0, then it is natural to expect that: a) the most general curve C of genus g possessing a gd1 will possess only a finite number of them; b) at least one gd1 on C will have no base points. If this where the case, the preceding proposition would show that 1 dim Mg,d = 3g − 3 + ρ. This agrees with the following naive count of moduli. Let R = 2g + 2d − 2 be the number of branch points of the morphism to P1 attached to a base-point-free gd1 . Then we should have 1 = R − 3 = 3g − 3 + ρ . dim Mg,d

In Section 8 we will present a very nice argument of Beniamino Segre which will enable us to prove the above statement. We will in fact proceed to show that The most general curve C of genus g possessing a gd1 with d < g/2 + 1 has in fact a unique gd1 . In Section 8 we will give a cohomological interpretation of the μ1 homomorphism using the normal bundle sequence (see diagram (8.10)). Following Voisin [672], an equivalent interpretation can be given directly, and we will present it now as a preparation for the next section. Let L be a line bundle on a smooth curve C and assume that it is base-point-free. Consider the evaluation map evC,L : H 0 (C, L) ⊗ OC → L and let ML be its kernel. Since L is base-point-free, ML is locally free of rank r (the reader should prove this as an exercise), and we have an exact sequence of vector bundles on C (6.10)

0 → ML → H 0 (C, L) ⊗ OC → L → 0 .

This sequence is simply the pullback to C, via the morphism φL : C → Pr = PH 0 (C, L)∨ , of the Euler sequence 0 → Ω1Pr → OPr+1 → OPr (1) → 0 . r

814


The bundle ML defined by the sequence (6.10) is called the Lazarsfeld– Mukai bundle [314,464,465,533], an object of paramount importance. Tensoring (6.10) with ωC L−1 , we get (6.11)

0 → ML ⊗ ωC L−1 → H 0 (C, L) ⊗ ωC L−1 → ωC → 0 .

Passing to cohomology, we see that (6.12)

H 0 (C, ML ⊗ ωC L−1 ) = ker μ0 .

Tensoring the evaluation map with the derivation d : OC → ωC , one obtains a homomorphism evC,L ⊗ d : H 0 (C, L) ⊗ OC → ωC L whose restriction to ML is OC -linear. Tensoring with ωC L−1 , this yields a homomorphism of OC modules (6.13)

2 . s : ML ⊗ ωC L−1 → ωC

Using (6.12), one readily sees that the homomorphism induced by s on global sections is nothing but μ1 . There is a completely straightforward generalization of the μ1 map. Given two line bundles L and L on a smooth curve C, let μ0,L,L : H 0 (C, L) ⊗ H 0 (C, L ) → H 0 (C, LL ) be the multiplication map, and define μ1,L,L : ker(μ0,L,L ) → H 0 (C, ωC LL ) We will call this by setting μ1,L,L ( i si ⊗ ti ) = i (si dti − ti dsi ). the generalized μ1 map. For more information on this map, see the bibliographical notes. (6.14)

7. Lazarsfeld’s proof of Petri’s conjecture. In this section we present an elegant proof of Petri’s conjecture, first conceived by Lazarsfeld; we will follow a simplified argument, which we owe to Pareschi. We recall, from Proposition (3.20), that the locus in Mg described by curves satisfying Petri’s condition (3.19) is open, so that, in order to prove Petri’s statement (1.1), it suffices to produce a single genus g curves that satisfies Petri’s condition. It is also clear that, in Petri’s statement (1.1), we can limit ourselves to complete, base-point-free linear series. Lazarsfeld’s idea is to find this curve as a hyperplane section of a K3 surface. Of course, such a curve is by no means a general curve of genus g, at least when g > 11. Indeed, a simple moduli count shows that genus g hyperplane sections of K3 surfaces depend on at most g + 19 moduli. Lazarsfeld’s result is the following:

§7 Lazarsfeld’s proof of Petri’s conjecture

815

Theorem (7.1). Let X be a K3 surface with Pic(X) = Z · [C0 ], where C0 is a smooth irreducible curve of genus g. Then the general element in |C0 | satisfies Petri’s condition. Since, as is well known, for any g ≥ 2, there are K3 surfaces X with Pic(X) = Z · [C0 ] and C0 smooth, irreducible and of genus g, the theorem above implies Petri’s statement (1.1). As we shall see from its proof, Theorem (7.1) holds under a weaker assumption. In fact, it is not necessary to assume that Pic(X) = Z · [C0 ]. It is enough to assume that the linear series |C0 | contains no reducible or nonreduced curve. Proof of Theorem (7.1). Let C be a general element of |C0 |. We proceed by contradiction, assuming that on C there exists a complete, base-pointfree linear series for which the μ0 map is not injective. Let L be the line bundle on C corresponding to this linear series, and set d = deg L and r + 1 = h0 (C, L). By assumption, the kernel of μ0 : H 0 (C, L) ⊗ H 0 (C, ωC L−1 ) → H 0 (C, ωC ) is not zero. In particular, h0 (C, ωC L−1 ) = 0. The linear system |C| cuts out on C the canonical series. Let U ⊂ |C| ∼ = Pg be the nonempty Zariski-open subset consisting of the smooth divisors in |C|. Consider the tautological family C ⊂ X ×U f

(7.2)

u U

We assume that C is the fiber over the point u ∈ U . The characteristic map Tu (U ) → H 0 (C, NC/X ) can be canonically identified with the map H 0 (X, OX (C))/H 0 (X, OX ) → H 0 (C, OC (C)) coming from the cohomology exact sequence of 0 → OX → OX (C) → OC (C) → 0 . Since X is a regular surface, it is an isomorphism, and the Kodaira– Spencer map of the family f can be identified with the coboundary map ρ : Tu (U ) ∼ = H 0 (C, NC/X ) → H 1 (C, TC ) . The natural projection p : Wdr (f ) → U is onto by assumption. Let w ∈ Wdr (C) be the point corresponding to the linear series |L|. Since C is a general element of U , we may assume

816


that dp is surjective at w. From Proposition (5.29) and diagrams (6.1), (6.2) it follows that Image(ρ ◦ dp) ⊂ Image(ρ) ∩ (Image μ1 )⊥ , where, as usual, 2 μ1 : ker μ0 → H 0 (C, ωC ).

In other words, we have ⊥ (7.3) Image(dp) ⊂ ρ−1 Image(ρ) ∩ (Image μ1 )⊥ = (Image(ρ∨ ◦ μ1 )) . Set (7.4)

∨ ). μ1,X = ρ∨ ◦ μ1 : ker μ0 → H 1 (C, ωC NC/X

Since dp is onto, it follows from (7.3) that (Image μ1,X )⊥ = H 0 (C, NC/X ), implying that μ1,X = 0. To prove Theorem (7.1), it then suffices to prove the following result. Proposition (7.5). Let X be a K3 surface with Pic(X) = Z · [C0 ], where C0 is a smooth irreducible curve of genus g. Then for any smooth curve C ∈ |C0 | and any base-point-free line bundle L on C, the map μ1,X is injective. Proof. The first step is to give a cohomological interpretation of the homomorphism μ1,X analogous to the one given for μ1 at the end of the preceding section. View the line bundle L as a coherent sheaf on X and denote by FL the kernel of the evaluation map (7.6)

evX,L : H 0 (C, L) ⊗ OX → L .

Since L is base-point-free, evX,L is onto, hence FL is locally free of rank r + 1, and we have an exact sequence of sheaves on X (7.7)

0 → FL → H 0 (C, L) ⊗ OX → L → 0 .

By Porteous’ formula (formula (4.2) in Chapter II) or, more simply, by an elementary local calculation, it follows that det FL = OX (−C). Now go back to the end of the preceding section, consider the vector bundle ML on C, and notice that det ML = L−1 . Look at the natural surjection FL |C → ML → 0. Since X is a K3 surface, OC (C) = ωC , and hence −1 . det FL |C ⊗ det ML−1 = L ⊗ OC (−C) = LωC

Thus, we get an exact sequence −1 → FL |C → ML → 0 , 0 → LωC

which, after tensoring with ωC L−1 , becomes (7.8)

0 → OC → FL ⊗ ωC L−1 → ML ⊗ ωC L−1 → 0 .

Passing to cohomology, we get an exact sequence 0 → H 0 (C, OC ) → H 0 (C, FL ⊗ωC L−1 ) → H 0 (C, ML ⊗ωC L−1 ) → H 1 (C, OC ) . δ

Now the proposition follows from the next two lemmas.

§7 Lazarsfeld’s proof of Petri’s conjecture

817

Lemma (7.9). The coboundary map δ coincides, up to multiplication by a nonzero scalar, with the homomorphism μ1,X . Lemma (7.10). Under the assumptions of Proposition (7.5), h0 (C, FL ⊗ ωC L−1 ) = 1. Proof of Lemma (7.9). Exactly as we did at the end of the preceding section, to give a cohomological interpretation of μ1,X , we tensor the evaluation map (7.6) with the derivation d : OX → Ω1X , and we obtain a homomorphism evX,L ⊗ d : H 0 (C, L) ⊗ OX → L ⊗ Ω1X whose restriction to FL is OX -linear. Tensoring with ωC L−1 , we obtain a homomorphism of OC -modules t : FL ⊗ ωC L−1 → Ω1X ⊗ ωC

(7.11)

fitting in a commutative diagram 0

0

w OC

w OC

w FL ⊗ ωC L−1

w ML ⊗ ωC L−1

t

s

u w Ω1X ⊗ ωC

u 2 w ωC

w0

w0

where the top row is (7.8), s is as in (6.13), and the bottom row is obtained by tensoring with ωC the dual of 0 → TC → TX |C → NC/X → 0 . 2 ) → H 1 (C, OC ), up to In particular, the coboundary map H 0 (C, ωC multiplication by a nonzero scalar, coincides with ρ∨ . Recalling from the end of the preceding section that the homomorphism induced by s on global sections is μ1 , the lemma follows at once. Q.E.D.

Proof of Lemma (7.10). First of all, we show that FL has the following properties: i) h0 (X, FL ) = h1 (X, FL ) = 0; ii) FL∨ is generated by its global sections away from a finite set of points; iii) h0 (X, FL ⊗ FL∨ ) = h0 (C, FL ⊗ ωC L−1 ). To prove i), it suffices to look at the cohomology sequence of (7.7) and to recall that h1 (X, OX ) = 0. To prove ii), we look at the sequence (7.7) and

818


apply Hom(−, OX ). Since Ext1 (L, OX ) = Ext1 (OC , OX ) ⊗ L−1 = ωC L−1 , we get an exact sequence (7.12)

0 → H 0 (C, L)∨ ⊗ OX → FL∨ → ωC L−1 → 0 .

Since H 1 (X, OX ) = 0, the sequence (7.12) remains exact if we pass to global sections. Since the first term of (7.12) is free and, as h0 (C, ωC L−1 ) = 0, the last one is generated by its global sections away from a finite set of points, ii) follows. To prove iii), it suffices to tensor (7.12) with FL and use i). Lemma (7.10) is now reduced to proving that h0 (X, FL ⊗ FL∨ ) = 1, in other words, that FL∨ has no nontrivial endomorphisms. Suppose that there is a nontrivial endomorphism ϕ : FL∨ → FL∨ . Let λ be an eigenvalue of ϕ(x) for some x ∈ X. The determinant of the endomorphism ψ = ϕ − λ · 1 vanishes at x, but since det ψ ∈ H 0 (X, det(FL ⊗ FL∨ )) = C, it vanishes identically on X. Thus, substituting ϕ with ψ, we can assume that ϕ is everywhere of less than maximal rank. We set A = Image ϕ, B = coker ϕ, and B = B/T (B), where T (B) is the torsion subsheaf of B. By construction, both A and B have positive ranks. Then (7.13)

[C] = c1 (FL∨ ) = c1 (A) + c1 (B ) + c1 (T (B))

in Pic(X). Since, by assumption, Pic(X) ∼ = Z, and since c1 (T (B)) is a nonnegative linear combination of the codimension one irreducible components of the support of T (B), to reach a contradiction, it suffices to show that the remaining two summands in (7.13) are represented by effective divisors. Now A and B are torsion-free sheaves of positive rank and are both quotients of FL∨ . It follows from ii) that they are both generated by their global sections away from a finite set of points. Also, since h0 (X, FL ) = 0, neither A nor B can be trivial. We are then reduced to the following general lemma. Lemma (7.14). Let E be a nontrivial, torsion-free sheaf on a smooth projective surface X. Assume that E is generated by its global sections away from a finite set of points. Then c1 (E) is represented by an effective divisor. Proof. Suppose first that E is locally free. Pick rank(E) sections which generate E at a general point of X. Their wedge product is a nonzero global section of the determinant of E which must vanish somewhere, since E is nontrivial. Hence, its zero locus is an effective divisor representing c1 (E). In case E is not free, the canonical inclusion of E in its double dual E ∨∨ is an isomorphism outside a finite set of points (cf. [429], Chapter V, Section 5). Thus, c1 (E) = c1 (E ∨∨ ), and E ∨∨ is also generated by its global sections away from a finite set of points. As E ∨∨ is locally free, we are done. Q.E.D.

§8 The normal bundle and Horikawa’s theory

819

This concludes the proof of Proposition (7.5) and therefore of Theorem (7.1). 8. The normal bundle and Horikawa’s theory. Our study of special divisors on a curve, and especially of the fundamental map μ0 , has centered around the intrinsic geometry of a moving curve and its moving Picard variety. We now take a different point of view which will enable us to interpret our cohomological approach and our intrinsic constructions in terms of the extrinsic geometry of the curve We begin by recalling some elementary facts concerning the theory of deformations of mappings as developed by Horikawa [367,368,369,370]. Let C be a nonsingular curve, and let M be an n-dimensional complex manifold, which is not assumed to be compact. Let (8.1)

φ:C→M

be a nonconstant analytic map. A deformation of φ parameterized by a pointed analytic space (S, s0 ) is a diagram γ φ˜ w Cs0 = p−1 (s0 ) ⊂ C wM C p (8.2) u (S, s0 ) where (8.3)

p : C → (S, s0 ) ,

γ : C −→ Cs0

is a deformation of C, and φ˜ is a morphism such that φ˜ ◦ γ = φ . The notions of morphism, isomorphism, and equivalence between deformations of maps are the obvious extensions of the analogous notions for deformations of manifolds. A first-order deformation of φ is simply a deformation of φ parameterized by Spec C[ε]. Let us go back to the morphism (8.1). We shall consistently use the symbol T to denote the sheaf φ∗ (TM ). We define the normal sheaf Nφ to the map φ to be the cokernel of the injective sheaf homomorphism dφ : TC → T . We then have an exact sequence of sheaves (8.4)

dφ

0 → TC −−→ T → Nφ → 0 .

Of course, when φ is an embedding, Nφ is just the normal bundle to C in M . Occasionally we shall write N for Nφ when no confusion is likely to occur.

820


Horikawa’s theory sets up a one-to-one correspondence between the vector space H 0 (C, Nφ ) and the set of equivalence classes of first-order deformations of φ. The computations needed to associate to a first-order deformation of φ a section of Nφ closely parallel the various deformationtheoretic arguments that we have encountered in the previous chapters. We ask for a little patience on the part of the reader, while once again we go through the ritual of translating geometric situations into patching data and concocting cohomology classes out of these. As usual, we imagine C as being given by transition data {Uα , zα , fαβ (zβ )} where ⎧ is a finite cover of C by coordinate disks, ⎪ ⎨ U = {Uα } zα is a holomorphic coordinate in Uα , ⎪ ⎩ zα = fαβ (zβ ) in Uα ∩ Uβ . We may assume that, for every α, φ(Uα ) ⊂ Vα ⊂ M , where each Vα is a coordinate patch in M with coordinates wα = (wα1 , . . . , wαn ). Set wα = gαβ (wβ ) in Vα ∩ Vβ and let wα = ψα (zα ) be the expression of the map φ in terms of these local coordinates. The functions fαβ , gαβ , ψα obviously satisfy the compatibility conditions fαβ (fβγ (zγ )) = fαγ (zγ ) (8.5) in Uα ∩ Uβ . gαβ (ψβ (zβ )) = ψα (fαβ (zβ )) Now consider a first-order deformation (8.2) of φ. This is given by: i) transition functions for C, zα = f˜αβ (zβ , ε) = fαβ (zβ ) + εbαβ (zβ ) ; ˜ ii) local expressions for φ, (8.6)

wα = ψ˜α (zα , ε) = ψα (zα ) + εaα (zα ) .

The cocycle bαβ ∂z∂α represents the Kodaira–Spencer class θ ∈ H 1 (C, TC ) of the first-order deformation (8.3). The data i) and ii) must satisfy the compatibility conditions ˜ ˜ fαβ (fβγ (zγ , ε), ε) = f˜αγ (zγ , ε) , gαβ (ψ˜β (zβ , ε)) = ψ˜α (f˜αβ (zβ , ε), ε) .


821

Expanding these in power series in ε, it is easy to check that these conditions are equivalent to (8.5) together with (8.7)

∂gαβ j

∂wβj

ajβ = aα +

∂ψα bαβ . ∂zα

This formula shows that the 0-cochain ∂ ∈ C 0 (U, T ) aiα ∂wαi maps to a section s of Nφ , which is called the Horikawa class of the first-order deformation (8.2). Of course, s depends on the choice of a local parameter ε. Once this is fixed, however, one can easily show that s does not depend on the other choices we have made and that in this way we get a one-to-one correspondence between H 0 (C, Nφ ) and the set of equivalence classes of first-order deformations of φ. Moreover, formula (8.7) also says that the coboundary homomorphism δ : H 0 (C, Nφ ) → H 1 (C, TC ) maps the Horikawa class s to the Kodaira–Spencer class θ. More generally, given a family of smooth curves (8.8)

p:C →S,

a point s0 ∈ S, and a morphism ψ:C→M, we can define a characteristic homomorphism Ts0 (S) → H 0 (Cs0 , Nψs0 ) , where ψs0 stands for the restriction of ψ to Cs0 , as follows. Given v ∈ Ts0 (S) corresponding to a morphism σ : Spec C[ε] → S, we map v to the Horikawa class of the pullback of (8.8) via σ. When ψs0 : Cs0 → M is an embedding, we clearly get back the classical characteristic homomorphism, as defined in Section 5 of Chapter IX. Let C be a smooth curve of genus g. Let L be a line bundle on C of degree d, and W an (r + 1)-dimensional subspace of H 0 (C, L). Assume that the gdr corresponding to W has no base points. Choose a basis X0 , . . . , Xr of W and let φ : C → Pr ∼ = PW ∨

822


be the map corresponding to this choice of basis. As above, we set T = φ∗ (TPr ) . We wish to show that it is possible to interpret the basic homomorphisms μ0,W : W ⊗ H 0 (C, ωC L−1 ) → H 0 (C, ωC ) , 2 ) μ1,W : ker μ0 → H 0 (C, ωC in terms of T and Nφ . In order to do so, we combine the exact sequences (5.10) and (8.4) with the pullback to C of the Euler sequence for Pr 0 → OPr → OPr (1)⊕(r+1) → TPr → 0 . This pulled-back sequence is the middle column of the diagram

(8.9)

0

0

0

0

u OC

u OC σ u

u λ w ΣL W w L⊕(r+1) τ u u dφ w TC wT u 0

w Nφ

w0

w Nφ

w0

u 0

where σ(f ) = (f X0 , . . . , f Xr ) , ∂ i , τ (0 , . . . , r ) = ∂Xi and λW is defined as follows. Let ∇ be a local section of ΣL , that is, a differential operator of order at most one operating on sections of L. Set λW (∇) = (∇X0 , . . . , ∇Xr ) . We claim that diagram (8.9) is commutative. If ∇ is of order zero, i.e., if it is the multiplication by a function f , then clearly λW (∇) = σ(f ) .


823

For the general case, we may find open sets Uα covering C and a local coordinate zα on each Uα such that ∇ = a + bDα

on Uα ,

where Dα is given by (5.12). We then have τ λW (∇) =

∇Xi

∂Xi ∂ ∂ =b ∂Xi ∂zα ∂Xi

by Euler’s theorem. On the other hand, b

∂Xi ∂zα

∂ ∂ = dφ b , ∂Xi ∂zα

proving the commutativity of (8.9). We may also notice that, for each smooth point φ(p) on the curve φ(C), the image λW (ΣL ⊗ k(p)) defines a two-dimensional subspace of Cr+1 containing the one-dimensional space corresponding to σ(OC,p ). The relevant part of the cohomology diagram of (8.9) can be conveniently displayed in the following manner: (8.10)

Here we have used the identifications (8.11)

Hom(W, H 1 (C, L)) ∼ = H 1 C, L⊕(r+1) , H 1 (C, T ) ∼ = coker(μ∨ ) . 0,W

824


Dualizing diagram (8.9), we see that the second identification in (8.11) is nothing but the isomorphism (6.12). A straightforward consequence of diagram (8.10) is the following. Lemma (8.12). There are natural identifications ker μ0,W ker μ1,W

∼ = H 1 (C, φ∗ (TPr ))∨ , ∼ = ker μ ∼ = H 1 (C, N )∨ .

Now notice that the first version (1.1) of Petri’s statement easily follows from the analogous assertion concerning gdr ’s without base points. We are then in a position to give another, quite suggestive formulation of Petri’s statement which brings to light its projective nature. Theorem (8.13) (Petri’s statement - Fourth version). Let C be a general curve of genus g, and let φ : C → Pr ,

r ≥ 1,

be a nondegenerate morphism. Then H 1 (C, Nφ ) = 0 . Observe that, when r = 1, the normal sheaf Nφ is supported on the ramification divisor of φ, so that, obviously, H 1 (C, Nφ ) = 0. This reproves Petri’s statement for gd1 ’s. Notice also that another immediate consequence of diagram (8.10) is that H 1 (C, Nφ ) vanishes when L is nonspecial. As Horikawa [368,369] showed, there is an analogue of the Kuranishi family for deformations of maps. Let φ : C → M be a nonconstant morphism from the smooth curve C to the complex manifold M . Then there exists a universal deformation C (8.14)

w X = q −1 (u ) ⊂ X Φ u0 0 q u (U, u0 )

wM

of φ, in the following sense. For any deformation (8.2) of φ : C → M with S connected and “sufficiently small,” there is a unique morphism C u (S, s0 )

α

wX u w (U, u0 )

˜ Of course, such a universal of deformations of C such that Φ ◦ α = φ. deformation is essentially unique if we allow U to be shrunk as needed. Horikawa proved the following.


825

Theorem (8.15). Any nonconstant morphism φ : C → M from a smooth curve to a complex manifold admits a universal deformation (8.14). Moreover, (8.16)

dim(U ) ≥ h0 (C, Nφ ) − h1 (C, Nφ )

for every component U of U at u0 , and the characteristic map Tu0 (U ) → H 0 (C, Nφ ) is an isomorphism. Instead of proving the theorem in its full generality, we shall limit ourselves to the case where M is a projective manifold. Suppose first that the genus g of C is at least 2. Let ξ : Y → (B, b0 ) be a Kuranishi family for C and consider the Hilbert scheme H = HomB (Y, M × B) parameterizing morphisms from fibers of ξ to M (cf. Section 7 of Chapter IX). We let u0 be the point of H corresponding to the morphism φ and denote by U a small connected neighborhood of u0 in H. We then set X = Y ×B U . The inclusion morphism U → H corresponds to a U -morphism X → M × U ; composing with the projection to M yields a morphism Φ : X → M which, by construction, is a deformation of φ. To see that the deformation thus constructed satisfies the required universal property, we argue as follows. Consider a deformation (8.2) of φ : C → M . After suitably shrinking S there is a unique morphism

(8.17)

C

wY

u

u w (B, b0 )

(S, s0 )

˜ p) : C → M ×S is an S-morphism of deformations of C. The morphism (φ, ∼ ∼ from C = Y ×B S to M × S = (M × B) ×B S and as such is represented by a unique B-morphism S → H which, possibly after further shrinking S, factors through U . The universal property we have just proved shows in particular that the characteristic homomorphism Tu0 (U ) → H 0 (C, Nφ ) is an isomorphism. It remains to prove the dimensionality statement (8.16). For this, we appeal to Theorem (8.20) in Chapter IX. The scheme H = HomB (Y, M × B) is an open subset of the Hilbert scheme Hilb(Y×M )/B , and the point u0 attached to the morphism φ is just the graph Γ of φ, viewed as a subscheme of ξ −1 (b0 ) × M ∼ = C × M . Since M is smooth, Γ is regularly embedded in C × M . Theorem (8.20) of Chapter IX then says that every component of U at u0 has dimension at least (8.18)

h0 (Γ, NΓ/(C×M ) ) − h1 (Γ, NΓ/(C×M ) ) + dim(B) = χ(NΓ/(C×M ) ) + 3g − 3 .

To compute this lower bound, we need to describe the normal bundle NΓ/(C×M ) . If we identify C with Γ via j = (id, φ) : C → C × M , this

826


bundle is nothing but the normal bundle Nj to the morphism j and sits in an exact sequence dj

→ Nj → 0 . 0 → TC −→ j ∗ TC×M − α

Let β : TC → j ∗ TC×M ∼ = TC ⊕ φ∗ TM be the homomorphism x → (x, 0). It is readily seen that the composition α ◦ β : TC → Nj is injective and that its cokernel is just Nφ . The lower bound (8.18) thus becomes (8.19)

χ(Nφ ) + χ(TC ) + 3g − 3 = χ(Nφ ) = h0 (C, Nφ ) − h1 (C, Nφ ) ,

as claimed. This concludes the proof when g ≥ 2. We shall now sketch how the above argument can be modified when g < 2. Suppose g = 1. Pick a point x ∈ C such that φ(x) is neither a singular point of φ(C) nor a critical value for φ. Then pick a smooth divisor D ⊂ M through φ(x) which does not meet the singular locus of φ(C) and is transverse to φ(C). Consider a Kuranishi family for the 1pointed curve (C; x), consisting of a deformation ξ : Y → (B, b0 ) of C plus a section σ : B → Y passing through x. Denote by H ⊂ HomB (Y, M ×B) the Hilbert scheme parameterizing those morphisms from fibers of ξ to M which send the marked point to D. Then H has codimension 1 in (each component of) HomB (Y, M × B). As for g ≥ 2, we let u0 be the point of H corresponding to the morphism φ and denote by U a small connected neighborhood of u0 in H. The family X → (U, u0 ) and the morphism Φ are defined as for g ≥ 2. Now consider the deformation (8.2) of φ : C → M . If S is replaced with a suitably small connected analytic neighborhood of s0 , there is a unique section τ : S → C passing through ˜ (s)) ∈ D for all s ∈ S. This x (or rather through γ(x)) such that φ(τ section makes C → (S, s0 ) into a deformation of the 1-pointed curve (C; x). Hence, possibly after further shrinking S, there is a unique morphism (8.17) of deformations of 1-pointed curves. The remainder of the proof of the universal property of (8.14) is the same as for g ≥ 2. The proof of the lower bound on the dimension of U is also essentially the same. It suffices to notice that, since H has codimension 1 in HomB (Y, M × B), the lower bound (8.18) is replaced by h0 (Γ, NΓ/(C×M ) ) − h1 (Γ, NΓ/(C×M ) ) + dim(B) − 1 = χ(NΓ/(C×M ) ) . When g = 0, the argument is similar to the one for g = 1. We choose three points x1 , x2 , x3 on C and three divisors D1 , D2 , D3 through φ(x1 ), φ(x2 ), φ(x3 ) with the same properties enjoyed by x and D in the genus 1 case, let H be the Hilbert schemes parameterizing morphisms from C to M mapping xi to Di for i = 1, 2, 3, and proceed exactly as above.


827

Remark (8.20). When M = Pr , it is possible to give an alternate proof of Theorem (8.15) without resorting to the Hilbert scheme and to Theorem (8.20) in Chapter IX, but using instead the universal property of Gdr and the dimension estimate (3.21). We shall limit ourselves to a nondegenerate morphism φ : C → Pr , where C is a smooth curve of genus g ≥ 2. Denote by L the pullback to C of the hyperplane bundle on Pr , by d its degree, and set W = H 0 (Pr , O(1)) ⊂ H 0 (C, L). Let ξ : Y → (B, b0 ) be a Kuranishi family for C. Denote by w the point in Gdr (ξ) corresponding to W . As usual, we denote by χ the projection from Gdr (ξ) to Picd (ξ) and by π the projection from Picd (ξ) to B. By Proposition (3.21), every component of Gdr (ξ) at w has dimension at least h1 (C, TC ) + ρ, where ρ = g − (r + 1)(g − d + r) is the Brill–Noether number. Let V be a small neighborhood of w in Gdr (ξ), let L be the pullback to V ×B Y of a degree d Poincaré line bundle, and let η : V ×B Y → V be the natural projection. By definition of Gdr (ξ), there is a locally free rank r + 1 subsheaf H of η∗ L, which is the family of gdr ’s on V . Shrinking V if necessary, we may assume that H is free. Now we let U be the bundle of projective frames of H, so that U → V is a principal P GL(r + 1, C)-bundle, and we denote by u0 the point in U corresponding to the morphism φ. Consider the family of curves X = U ×B Y → (U, u0 ). The frames parameterized by U define a morphism Φ : X → Pr which, by construction, is a deformation of φ. The universal property of this deformation follows immediately from the one of Gdr (ξ). In turn, the universal property shows that the characteristic homomorphism Tu0 (U ) → H 0 (C, Nφ ) is an isomorphism. It remains to prove (8.16). This is a simple dimension count. If G is any component of Gdr (ξ), we denote by U the corresponding component of U . Looking at diagram (8.9) and using Proposition (5.26), we get h0 (C, Nφ ) − h1 (C, Nφ ) = χ(L⊕(r+1) ) − χ(OC ) − χ(TC ) = (r + 1)(d − g + 1) + g − 1 + h1 (C, TC ) = (r + 1)2 − 1 + ρ + h1 (C, TC ) ≤ dim G + (r + 1)2 − 1 = dim U . As a first application of the theory of deformations of mappings, we shall study the following basic question. Consider smooth curves C of genus g which admit a nonconstant morphism φ : C → X of degree d onto some smooth curve X of genus g > 0. The question we ask is: on how many moduli does such a C depend? The answer is best expressed in terms of the ramification divisor R ⊂ C of φ and of its degree w, and goes under the name of Riemann’s moduli count. Of course, w is related to the other invariants of φ by the Riemann–Hurwitz formula (8.21)

2g − 2 = d(2g − 2) + w .

828


Riemann’s moduli count asserts essentially that the moduli of a curve C as above arise from two independent sources. One can vary the moduli of X, or one can freely move the w points of R. This gives a total of (8.22)

3g − 3 + w = 2g − 2 + (3 − 2d)(g − 1)

moduli. The rest of the section will be devoted to justifying this heuristic answer. More precisely, we shall prove the following result. Theorem (8.23) (Riemann’s moduli count). Let d > 1, g > 1, and g > 0 be integers. Denote by Mg (d, g ) ⊂ Mg the locus of those smooth genus g curves which are degree d ramified coverings of smooth genus g curves. Then Mg (d, g ) is nonempty if and only if 2g − 2 ≥ d(2g − 2); in this case it is a closed subscheme of Mg of pure dimension 3g − 3 + w, where w = 2g − 2 − d(2g − 2). If a covering of the type described in the statement of the theorem exists, then w is the degree of its ramification divisor because of the Riemann–Hurwitz formula (8.21) and hence is nonnegative. The converse is also standard and follows, for instance, from Riemann’s existence theorem. Here is an alternate direct proof. Since w is even, we may write w = 2s. There exists an unramified covering f : Y → X of degree d, since H1 (X, Z) ∼ = Z2g , and hence π1 (X) contain (normal) subgroups of any integer index. Now pick distinct points p1 , . . . , ps in X and pairs of distinct points qi , qi ∈ Y , i = 1, . . . , s, such that f (qi ) = f (qi ) = pi for every i. We also choose an open neighborhood Ui of pi for each i; we perform the choice so that the Ui are disjoint and biholomorphic to disks. We also denote by Vi the connected component of f −1 (Ui ) containing qi and by Vi the one containing qi . Now we fix our attention on one specific i. We may assume that there is a local coordinate x centered at pi such that Ui is the disk {|x| < 1}. We denote by y and y the local coordinates on Y at qi and qi obtained by composing x with f . Thus, Vi = {|y| < 1}, Vi = {|y | < 1}, and the map f is given, in these local coordinates, by x = y and by x = y . Choose a small positive ε and set W = {(z, t) ∈ C2 : z 2 + t2 = ε2 , |z| < 1}. The morphism h : W → Ui defined by x = z is a branched covering, with simple ramification precisely at the two points of W lying above x = ±ε, i.e., at the points t = 0, z = ±ε. When δ is a sufficiently small positive number, the region of W where |z| > 1 − δ is the disjoint union of two annuli A and A , both mapped biholomorphically to {x : 1 > |x| > 1 − δ} by h. Now we remove from Y the regions {|y| ≤ 1 − δ} and {|y | ≤ 1 − δ}, and we glue W to the resulting Riemann surface by identifying A to {y : 1 > |y| > 1 − δ} via y = z and A to {y : 1 > |y | > 1 − δ} via y = z. The result is a complete Riemann surface Y together with a branched


829

covering Y → X which agrees with h on W and with f elsewhere; the covering is simply ramified at two points. We may perform this operation simultaneously at all the pi . What we obtain is a degree d branched covering C → X with w = 2s simple ramification points. Thus, the genus of C is g by the Riemann–Hurwitz formula. Remark (8.24). The existence argument outlined above, with a slight modification, works also when g = 0. One takes as Y the disjoint union of d copies of X, and then proceeds as before. The only issue is the connectedness of C. When g = 0, the Riemann–Hurwitz identity immediately implies that s ≥ d − 1, and hence s is just large enough to make it possible, by suitably choosing the qi and qi , to connect all the d sheets of Y . In addition, the condition that g > 1 is clearly unnecessary. In fact, with a little more care, the argument proves the following. Let g ≥ 0 and d > 1 be integers. Let X be a smooth curve of genus g . Suppose that w = 2g − 2 − d(2g − 2) ≥ 0. Let x1 , . . . , xw be distinct points of X. Then there exist a smooth genus g curve C and a degree d covering C → X simply ramified at w points mapping to x1 , . . . , xw . Having disposed of the problem of existence, we next explain why Mg (d, g ) is a subscheme of Mg . First of all, fix an integer n ≥ 3, set S = Mg [n], and let α:C→S be the universal family over S. Similarly, when g > 1, set T = Mg [n], and let β:X →T be the universal family over T . When g = 1, we take as X → T the universal family of 1-pointed genus 1 curves over M1,1 [n]. Then we let Hd ⊂ HomS×T (C × T, S × X ) be the Hilbert schemes parameterizing coverings of degree d from fibers of α to fibers of β. A first observation is that Hd is of finite type over C. To see this, let Γ ⊂ C × X be the graph of a degree d morphism C → X, where C and X are smooth curves of genera g and g . Let p1 and p2 be the projections from C × X to C and X, respectively, and let Fi be a fiber of pi for i = 1, 2. Clearly, (8.25)

(F1 · Γ) = 1 ,

(F2 · Γ) = d ,

(Γ · Γ) = d(2 − 2g ) ,

where the third identity follows from the second and from the fact that ωC×X/C (Γ) is trivial on Γ by adjunction. Now set L = p∗1 ωC ⊗ p∗2 ωX when g > 1 and L = p∗1 ωC ⊗ p∗2 O(x), where x is the marked point on X, for g = 1. Then L3 is a very ample line bundle on C × X, and hence

830


Γ can be realized as a curve of degree n in Pr , where r and n depend only on g, g , and d. This proves our claim. We next show that Hd is proper over S × T . By the properness of the Hilbert scheme, it suffices to show that, given smooth curves C and X of genera g and g , a divisor D in C × X which is the limit of points of Hd , i.e., of graphs of degree d morphisms from curves of genus g to curves of genus g , is in fact the graph of a morphism. Such a D must have the same numerical characters as a true graph. Keeping the notation introduced above, these are expressed by the identities (8.25). It follows from the first of these that D is of the form Γ + E1 + · · · + Ek , where Γ is the graph of a morphism, and the Ei are fibers of p1 , and hence that D is numerically equivalent to Γ + kF1 . Thus (F2 · Γ ) = d − k, and therefore, (Γ · Γ ) = (d − k)(2 − 2g ) . But then d(2 − 2g ) = (D · D) = (d − k)(2 − 2g ) + 2k = d(2 − 2g ) + 2kg , which proves that k = 0 since g > 0 by assumption. The conclusion is that D = Γ is a graph, proving our contention. Since Mg (d, g ) is the image of Hd under the composition of the projection Hd → S × T , of the projection S × T → S = Mg [n], and of the moduli map Mg [n] → Mg , all of which are proper, it follows that Mg (d, g ) is a closed subscheme of Mg . We finally turn to the heart of Theorem (8.23), which is the dimension statement. The dimension of Hd is readily calculated. By Theorem (8.15), the space of degree d morphisms from a variable smooth genus g curve to a fixed smooth curve of genus g is smooth of dimension w. Thus, (8.26)

dim(Hd ) =

3g − 3 + w w+1

if g > 1 , if g = 1 .

This already suffices to conclude that the dimension of Mg (d, g ) does not exceed 3g − 3 + w, since in the g = 1 case one of the dimensions of Hd is accounted for by the translations of the target elliptic curve. The key to the converse inequality is provided by the following classical result. Theorem (8.27) (de Franchis’ theorem). Let g ≥ 1, g ≥ 2, and d ≥ 1 be integers, and let C be a smooth curve of genus g. Then: i) up to isomorphism, there are only finitely many genus g curves X such that there exists a nonconstant morphism from C to X; ii) for any smooth genus g curve X, there are only finitely many nonconstant morphisms from C to X;


831

iii) up to isomorphism, there are only finitely many genus 1 curves X such that there exists a degree d morphism from C to X; iv) for any smooth genus 1 curve X, up to composition with a translation of X, there are only finitely many degree d morphisms from C to X. Proof. The main difference between i) and iii), and between ii) and iv), is that, because of the Riemann–Hurwitz formula, the degree of a nonconstant morphism from a genus g curve to a curve of genus g ≥ 2 is bounded above by the ratio between 2g − 2 and 2g − 2, while no such bound exists for morphisms to a curve of genus one. Thus, in proving statements i) and ii), there is no loss of generality in restricting to morphisms of fixed degree d. This will allow us to deal simultaneously with i) and iii), and with ii) and iv). We begin by proving iii) and iv) when g = 1. Choose an origin e in C; this determines a group law on C. Then any X is isomorphic to the quotient of C modulo a subgroup of order d. Since there are finitely many such subgroups, iii) follows. Now choose an origin e in X and suppose that we are given a degree d morphism φ : C → X. Composing φ with a translation, we may suppose that φ(e) = e and hence that φ is a group homomorphism. Now, two such morphisms sharing the same kernel are obtained from one another by composing with a group automorphism of X. Since there are finitely many group automorphisms of X and there are finitely many possible kernels for φ, we get iv). From now on, we assume that g > 1. As we observed during the second part of the proof of Theorem (8.23), the scheme Hd parameterizing degree d morphisms from smooth genus g curves (with a suitable level structure) to smooth genus g or genus 1 curves (also with a suitable level structure) is of finite type over C. Hence, to prove any one of statements i)–iv), it suffices to show that the morphisms in question are rigid or, in the case of iv), rigid up to translation. Let us start with statements ii) and iv). Let φ : C → X be a nonconstant morphism of degree d. From the normal bundle sequence 0 → TC → φ∗ TX → Nφ → 0 we get an exact sequence (8.28)

0 → H 0 (C, φ∗ TX ) → H 0 (C, Nφ ) → H 1 (C, TC ) .

When g > 1, TX , and hence its pullback via φ, have negative degree, and hence H 0 (C, Nφ ) injects in H 1 (C, TC ). This means that one cannot deform φ without at the same time deforming C, which proves ii). When g = 1, instead, TX is trivial, so H 0 (C, Nφ ) → H 1 (C, TC ) has a onedimensional kernel. This means that the dimension of the Hilbert scheme Homd (C, X) of degree d morphisms from C to X does not exceed 1.

832


On the other hand, composing φ with all possible translations in X yields a one-dimensional complete subscheme of Homd (C, X), and hence a component of Homd (C, X). Since Homd (C, X) is of finite type over C, this proves iv). To prove i) and iii), we follow a different approach. Let φ be as above, and let Q = C ×X C ⊂ C × C be the equivalence relation associated to it. Since the automorphism group of X is finite (which, incidentally, is also a special case of ii)), it will suffice to show that Q is rigid inside C × C. We write Q = Δ + V , where Δ is the diagonal in C × C. For 1 ≤ i < j ≤ 4, let πi,j : C × C × C × C → C × C −1 be the projection to the ith and jth factors, and set Δi,j = πi,j (Δ). Given two correspondences on C, that is, two divisors A and B in C × C, we define their composition A ◦ B to be

A ◦ B = (π1,4 )∗ ((A × B) · Δ2,3 ) . Now, the intersection multiplicity of A and B is equal to the intersection multiplicity of A × B with the diagonal in (C × C) × (C × C), that is, with the degree of the zero-cycle (A × B) · Δ1,3 · Δ2,4 . On the other hand, when B is symmetric (with respect to the interchange of factors), this degree is clearly equal to deg((A × B) · Δ1,4 · Δ2,3 ) , i.e., by the projection formula, to the degree of (A ◦ B) · Δ. In conclusion, when B is symmetric, the intersection multiplicity of A and B is (A · B) = (A ◦ B · Δ) .

(8.29)

We shall apply these considerations to calculate the self-intersection of V . Since φ has degree d, it is clear that Q ◦ Q = dQ = dΔ + dV . On the other hand, Q ◦ Q = Δ ◦ Δ + Δ ◦ V + V ◦ Δ + V ◦ V = Δ + 2V + V ◦ V , whence V ◦ V = (d − 1)Δ + (d − 2)V .


833

Thus, (8.30)

(V · V ) = (V ◦ V · Δ) = (d − 1)(Δ · Δ) + (d − 2)(V · Δ) .

The self-intersection of the diagonal equals 2 − 2g, and we claim that (V · Δ) is equal to w, the degree of the ramification divisor of φ. To see this, let p be a point of C and choose coordinates x centered at p and t centered at φ(p) with respect to which φ is given by t = xm . Choose as coordinates on C × C at (p, p) two copies x1 , x2 of x. In m these coordinates, a local equation for Q is xm 1 = x2 , and one for Δ is x1 = x2 . Hence a local equation for V is m−1

xh1 x2m−h−1 = 0 .

h=0

It follows that the intersection multiplicity of V and Δ at (p, p) is m − 1, which proves our claim. Using the Riemann–Hurwitz formula, (8.30) can thus be rewritten as (V · V ) = (d − 1)(2 − 2g) + (d − 2)r = 2 − 2g + d(d − 2)(2 − 2g ) . This quantity is strictly negative when g ≥ 1 and g > 1, and hence there is a component D of V such that (D · V ) < 0. This component must then be rigid, in the following sense. Let φt : C → Xt , where t varies in a small disk centered at 0 ∈ C, be a family of degree d morphisms to smooth curves of genus g , with X0 = X and φ0 = φ. Let Qt be the equivalence relation on C determined by φt . Then D is a component of Qt for every t. Let Q be the equivalence relation generated by D. If Q = Q, Q is rigid, and we are done. Otherwise, set C = C/Q . Then each φt factors as φt C4 w Xt 4 6 ] [ [ φ t C where the degree of φt is at least 2 but strictly less than d, and we can proceed by induction on d. The initial cases of the induction are those in which g = 1, those we did at the beginning of the proof, and those in which d is prime. In these latter cases, in fact, Q is necessarily equal to Q. Q.E.D. It is now possible to complete the proof of Theorem (8.23). A consequence of de Franchis’ theorem is that the fibers of the composite morphism f : Hd → S × T → S = Mg [n] → Mg have dimension zero when g > 1, and dimension 1 when g = 1. Combining this with (8.26) shows that the dimension of Mg (d, g ) = f (Hd ) is 3g − 3 + w, concluding the proof of Theorem (8.23).

834


Remark (8.31). It is natural to ask what happens to Riemann’s moduli count when g = 0. It is easy to see that (8.23) is false in this case for a number of reasons. First of all, Mg (d, g ) is not closed. It is easy to construct degenerations of degree d morphisms from smooth curves to P1 which are not themselves morphisms of the same kind; in a sense, they may be regarded as morphisms of lower degree from smooth curves to P1 , or alternatively as degree d morphisms from nodal curves to P1 . More importantly, the dimension count fails completely for a very simple reason. When g = 0, for any fixed g the total ramification index w grows linearly with d, so that the quantity −3 + r = 3g − 3 + w can be arbitrarily large while 3g − 3 stays fixed. Some dimensional information can be obtained from the exact sequence (8.28). A degree d morphism φ : C → P1 corresponds to a gd1 on C, that is, to a vector subspace W ⊂ H 0 (C, φ∗ O(1)). Then (8.28), together with (8.12), implies that, at C, the genus g curves which are d-sheeted ramified coverings of P1 depend on at most w − dim(ker(μ0,W )) moduli. Notice that, for d > 1, g > 1, and g ≥ 1, the moduli number 3g − 3 + w given by Riemann’s count is strictly less than 3g − 3. In this case, in fact,

3g − 3 + w = 2g − 2 + (3 − 2d)(g − 1) ≤ 2g − 2 < 3g − 3 . An immediate consequence is the following very well-known and classical result concerning correspondences on curves with general moduli. Corollary (8.32) (Riemann–Hurwitz). Let C be a general curve of genus g > 1. Let φ : C → C be a nonconstant morphism of C onto a curve C . Then either φ is birational, or else C is rational. This result and Petri’s statement for gd1 ’s (cf. Proposition (6.7)) may be combined to derive the following simple but very useful conclusion. Proposition (8.33). Let C be a general curve of genus g. Let φ : C → Pr ,

r ≥ 2,

be a nondegenerate morphism corresponding to a special gdr . Then φ is not composed with an involution (i.e., φ gives a birational map of C onto its image).

§9 Ramification

835

Proof. Set Γ = φ(C). Since C is general, we have just proved that either φ is not composed with an involution or else Γ is rational. We now show that the second case cannot occur. Suppose it does. Let n be the degree of Γ. Obviously, n is at least equal to 2. Let Δ be the inverse image of a point of Γ, and let D be a divisor of the gdr corresponding to φ. Since we are assuming that Γ is rational, we have (8.34)

D ∼ nΔ , h0 (C, O(Δ)) ≥ 2 .

Since, by assumption, |D| is special, we also have h0 (C, ωC (−nΔ)) = 0 ,

n ≥ 2.

This and (8.34) contradict Proposition (6.7). Q.E.D. The conclusion of the preceding proposition is no longer valid if we remove the assumption that φ correspond to a special gdr . However, suppose that the gdr is complete and nonspecial (the case of a special gdr has already been disposed of). Let m be the degree of φ, and let n be the degree of φ(C). Obviously, n ≥ r, that is, (8.35)

m≤

d g = + 1, r r

where the second step follows from the Riemann–Roch theorem. Now assume that φ is composed with an involution. This means that Γ 1 . But then (8.35) contradicts the is rational and that C carries a gm Dimension Theorem of Brill–Noether theory (Theorem (1.5) in Chapter V) unless r = 2 and g is even. In this case d = g + 2. This last case can in fact occur. Suppose that C has even genus. Again by the Dimension Theorem, C can be realized as a branched covering of P1 of degree (g + 2)/2. Let φ be the composition of this covering with the Veronese 2 is then nonspecial by embedding of P1 in P2 . The corresponding gg+2 Proposition (8.33). 9. Ramification. It turns out that, when studying special linear series from the point of view of deformations of mappings, the presence of ramification plays a very special role and has, in fact, unexpected consequences. We shall explore these using the formalism of infinitesimal deformations, as developed in Section 8. There will be no change in notation. Let φ : C → M be a nonconstant analytic map of a smooth curve C into an n-dimensional complex manifold M . The divisor R of zeroes of the

836


differential of φ will be called the ramification divisor of φ. The sheaf homomorphism dφ : TC → T = φ∗ TM extends to a homomorphism TC (R) → T which has maximal rank everywhere. We then get a commutative diagram 0 u Kφ

0 0

u w TC

w T

u w Nφ

w0

0

u w TC (R)

w T

u w Nφ

w0

(9.1)

u Qφ

u 0

u 0 where N = Nφ is a locally free sheaf of rank n − 1, and K = Kφ , Q = Qφ are (noncanonically) isomorphic to the structure sheaf OR . This implies, in particular, that (9.2)

H 1 (C, N ) ∼ = H 1 (C, N ) .

We then see that the third version (8.13) of Petri’s statement is really an assertion about N , and, in a sense, N is better behaved than N . We shall devote this section to the study of the skyscraper sheaf K and of the first-order deformations of φ corresponding to its sections. Write s νi pi , R= i=1

where the pi are distinct points. Choose a cover U = {Uα } of C by coordinate open sets in such a way that each pi lies in only one open set Uαi , that the local coordinate zαi on Uαi vanishes at pi , and that Uαi ∩ Uαj = ∅ if i = j. Then, clearly, every element of H 0 (C, K) comes from a cochain {aα } ∈ C 0 (U, T )

§9 Ramification

837

of the form $ (9.3)

∂ψ

i ci (zαi ) ∂zααi , aαi = zα−ν i

i = 1, . . . , s ,

aβ = 0 ,

β = αi , i = 1, . . . , s ,

i

where the ci are polynomials of degree at most νi − 1. Recalling the basic formula (8.7), we see that the Kodaira–Spencer class of the section of K we just constructed is given by

i bαi β = −zα−ν ci (zαi ) , i bαβ = 0 ,

i = 1, . . . , s , α = αi = β .

As we know from Chapter XI, Section 2, in case g > 0, ci (zαi ) = zανii−1 , and cj = 0 for j = i, this class is known as a Schiffer variation centered at the point pi . Such a class is a nonzero element of H 1 (C, TC ) which corresponds to a first-order deformation “changing the complex structure of C only at the point pi .” More geometrically, the Schiffer variations centered at pi describe the line in H 1 (C, TC ) which corresponds to the point pi under the bicanonical map φω2 : C → PH 1 (C, TC ). In a similar way the points in PH 1 (C, TC ) corresponding to the classes defined by bαi β = z −s , bαβ = 0 ,

1 ≤ s ≤ h + 1, α = αi = β ,

span the hth osculating space to the bicanonical curve at φωC2 (pi ). Let us look at the expressions (9.3) describing the elements of H 0 (C, K). The fact that aβ = 0 when β is not one of the αi can be interpreted as follows. Let φ˜ wM C p u Spec C[ε] be a first-order deformation of φ corresponding to a section of K. Set Γ = φ(C). Then φ˜ factors through the inclusion of Γ in M : C[

φ˜

wM ) ' ' ' /

[ ] [ Γ

838


To put it differently, the first-order deformations of φ corresponding to sections of K leave the image curve fixed (up to first-order, of course). The following simple example illustrates this phenomenon, which is local in character. Let x, y be affine coordinates in C2 . Let ψ(z) = (z 2 , z 3 ) ,

|z| < 1,

be the parametric equation of an ordinary cusp y 2 = x3 .

(9.4)

Then K is supported at z = 0, and its sections are all multiples of a(z) =

1 dψ = (2, 3z) . z dz

The corresponding first-order deformation is (9.5)

˜ ε) = (z 2 + 2ε, z 3 + 3εz) . ψ(z,

One checks that ψ˜ satisfies equation (9.4) modulo ε2 . We can therefore say that in general the presence of “cusps” in the image curve Γ = φ(C) implies that there exists, infinitesimally, more than one non-singular model of Γ. Looking again at the example of the ordinary cusp, if we interpret ε as a finite instead of an infinitesimal parameter in (9.5), we notice that for ε = 0, the curve ˜ ε) z → ψ(z, has a node and no cusps. This simple remark can be generalized in a way that will turn out to be very useful for our purposes. In order to explain this, we first have to analyze in more detail the local geometry of unibranch singularities (e.g., cuspidal points). Let Δ be the unit disc in C, and let ψ : Δ → Cn ,

n > 1,

be a nonconstant holomorphic map such that ψ(0) = 0 . We let O1 be the local ring of Δ at 0, O2 the local ring of Cn at 0, and set O = ψ ∗ (O2 ) , where ψ ∗ : O2 → O1 is the pullback homomorphism. We denote by m, m1 , m2 the maximal ideals of O, O1 , O2 , respectively. We choose a

§9 Ramification

839

minimal set of generators for m as follows. Let k1 ≥ 1 be the minimum among the integers h such that , m ∩ mh1 ⊂ mh+1 1 and let g1 be an element of m ∩ mk11 which does not belong to m1k1 +1 . If g1 does not generate m, let k2 > k1 be the minimum among the integers h such that m ∩ mh1 ⊂ mh+1 + Og1 , 1 and let g2 be an element of m∩mk12 which does not belong to m1k2 +1 +Og1 . If g1 , g2 do not generate m, we let k3 > k2 be the minimum among the integers h such that + Og1 + Og2 , m ∩ mh1 ⊂ mh+1 1 and so on. This process ends after at most n steps, and we are left with a minimal set of generators g1 , . . . , g of m such that (9.6)

ord0 (gi ) = ki , k1 < k2 < . . . < k ,

and, moreover, (9.7) for every i, ki is not a linear combination of k1 , . . . , ki−1 with nonnegative integral coefficients. Clearly, g1 , . . . , g are uniquely determined up to multiplication by units of O, hence, k1 , . . . , k are invariants of ψ. The number k1 − 1 is called the ramification index of ψ at 0. The point 0 ∈ Δ is called a ramification point of ψ if k1 > 1. We will say that this ramification point is nondegenerate in case > 1; we then call k2 the type of the ramification point. We may choose a minimal set of generators g1 , . . . , gn for m2 in such a way that ψ ∗ (gi ) = gi , ψ

∗

(gi )

= 0,

i = 1, . . . , , i > .

This means that we may choose local coordinates w1 , . . . , wn centered at 0 on Cn and a local coordinate z centered at 0 on Δ in such a way that, in these coordinates, ψ is given by w1 = z ki + higher-order terms, wi = 0 ,

i = 1, . . . , , i > ,

840


where k1 , . . . , k satisfy (9.6) and (9.7). Now set Δ2 = {(ξ, t) ∈ C2 : |ξ| < 1, |t| < 1} and let ψ : Δ2 → Cn be a holomorphic mapping. For each t, we set ψt (ξ) = ψ(ξ, t) . We assume that, for each t, ψt is not constant and has only one ramification point. Clearly, the index of this ramification point is independent of t; let it be equal to h − 1. Assume, moreover, that for general t, the ramification point of ψt is nondegenerate. Let V ⊂ Δ2 be the locus of the ramification points of ψt as t varies. Since V projects in 1–1 fashion onto {t ∈ C : |t| < 1}, this projection is a biholomorphic isomorphism. Then, in suitable coordinates, t, z = z(ξ, t) w1 , . . . , wn

for Δ2 , for Cn ,

the locus V is defined by the equation z = 0, and we may write ψ, in local coordinates, as (9.8)

wi = pi (z h , t) + γi (t)z k + [k + 1] ,

i = 1, . . . , n ,

where

(9.9)

⎧ pi (ζ, t) is a polynomial in ζ of degree m ≥ 1; ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ h does not divide k; k > hm; ⎪ ⎪ ⎪ the γi are not all identically zero; ⎪ ⎪ ⎩ [k + 1] stands for terms of order higher than k in z.

Clearly, when γi (t) = 0 for at least one i, k is the type of the ramification point of ψt . On the other hand, when γi (t) = 0 for every i, the ramification point of ψt is degenerate, or else its type is larger than k. If we agree to say that the type of a degenerate ramification point is equal to +∞, this shows that the type of ψt is an upper semicontinuous function of t. We are now in a position to understand the phenomenon of “disappearance of cusps” that we discussed in the previous example.

§9 Ramification

841 p

→ Δ = {t ∈ C : |t| < 1} be a smooth analytic Proposition (9.10). Let C − family of curves, and let φ:C→M be an analytic morphism of C into a complex n-dimensional manifold M , n ≥ 2. Set Ct = p−1 (t), φt = φ . Assume that Ct

a) φt is a birational map of Ct onto φ(Ct ) for all t; b) the number, the indices, and the types of the ramification points of φt are independent of t. Then the Horikawa class of (C, p, φ) at t = 0 does not belong to H 0 (C0 , Kφ0 ), unless it is zero. Proof. Let V be the locus traced out by the ramification points of the ψt . Assumption b) implies that V is a topological covering of Δ. We can then choose a coordinate cover {Uα } of C with coordinates (zα , t) such that V ∩ Uα = {zα = 0}. We may also assume that φ(Uα ) is contained in a coordinate patch Wα ⊂ M . Let wα = (wα1 , . . . , wαn ) be local coordinates on Wα , and let wα = ψα (zα , t) be the expression of φ in these coordinates. The Horikawa class of (C, p, φ) at t = 0 is the section s of Nφ0 represented by

∂ψα (zα , 0) . ∂t

Recalling (9.3), to say that this class belongs to H 0 (C0 , Kφ0 ) means that there are meromorphic functions fα such that ∂ψα ∂ψα (zα , 0) = fα (zα ) (zα , 0) . ∂t ∂zα To complete the proof, it is now sufficient to prove the following lemma. Lemma (9.11). Set Δ2 = {(z, t) ∈ C2 : |z| < 1, |t| < 1} and let ψ : Δ2 → Cn ,

n > 1,

be an analytic map. Assume that, for each t, ψt (z) = ψ(z, t) is injective, has only one ramification point, and that the index and type of this point are independent of t. Assume also that there exists a meromorphic function f (z) such that (9.12)

∂ψ ∂ψ (z, 0) = f (z) (z, 0) . ∂t ∂z

Then f (z) is holomorphic.

842


Proof. The statement of the lemma is clearly invariant under an arbitrary change of coordinates in Cn and a change of coordinates of the type z = z (z, t), t = t in Δ2 . Let V ⊂ Δ2 be the locus of the ramification points of the ψt . By assumption, V maps in a one-to-one fashion onto {t ∈ C : |t| < 1}. Therefore, V is a smooth subvariety of Δ2 , and we may assume that its equation is z = 0. Let h be the ramification index of ψt . With suitable choices of coordinates in Cn and Δ2 , we may assume that ψ = (ψ1 , . . . , ψn ) is given by (9.13)

ψi (z, t) = pi (z h , t) + γi (t)z k + [k + 1] ,

i = 1, . . . , n ,

where conditions (9.9) are satisfied. We can also assume that (9.14)

ψ1 (z, t) = α(t) + z h .

The fact that the type of the ramification point of ψt is constant implies that one of the γi , say γ2 , does not vanish at t = 0. Assumptions (9.12) and (9.14) give ∂α (0) = h f (z)z h−1 . ∂t Therefore f (z) = c z 1−h , where c is a constant. Using (9.12) and (9.13), we obtain ∂p2 h ∂γ2 (z , 0) + (0)z k + [k + 1] ∂t ∂t ∂p2 h =c h (z , 0) + kγ2 (0)z k−h + [k − h + 1] . ∂ζ Since γ2 (0) = 0, we must have c = 0. Therefore, in this coordinate system, f is identically zero; in particular, it is holomorphic. Q.E.D. If we had to give a capsule statement of Proposition (9.10), it could be the following. Suppose that we are given a deformation of φ:C→M, parameterized by a disc Δ = {t ∈ C : |t| < 1}, whose Horikawa class at t = 0 lies in H 0 (C, Kφ ); then, along such a deformation, the complexity of the ramification of φ decreases. As a first application of these techniques, we wish to prove the following consequences of Petri’s statement (1.3).

§9 Ramification

843

Proposition (9.15). Let C be a general curve of genus g. Let r and d be nonnegative integers such that r ≥ 2 and r =d−g

if d ≥ 2g − 1 ,

r ≥d−g ρ = g − (r + 1)(g − d + r) ≥ 0

if d ≤ 2g − 2 .

Let L be a line bundle on C corresponding to a general point of Wdr (C). Then i) |L| has no base points; ii) If φ : C → Pr is the morphism corresponding to |L|, then φ is not composed with an involution and is a local immersion. Proof. For g = 0, 1, the result is obviously true; therefore, from now on we assume that g ≥ 2. By Petri’s statement (1.3), every component of Wdr (C) has dimension ρ. On the other hand, again by the same result, the sublocus of Wdr (C) consisting of gdr ’s with base points has dimension at most equal to r (C) + 1 < ρ . dim Wd−1 Therefore |L| has no base points. In Proposition (8.33) we proved that φ is not composed with an involution, with the possible exception of the case in which g is even, d = g + 2, r = 2, and |L| is composed 1 . But this case cannot occur because such an L does not with a g(g+2)/2 2 correspond to a general point of Wg+2 (C). In fact, the third version of Petri’s statement says that there is only a finite number of such L’s, 2 (C) is equal to g. We now proceed to whereas the dimension of Wg+2 show that φ is a local immersion. Suppose it is not and let s be a nonzero section of Kφ . Since C is general, H 1 (C, Nφ ) vanishes by Petri’s statement in its fourth version (Theorem (8.13)). It then follows from Theorem (8.15) that any section of H 0 (C, Nφ ) can be integrated to a deformation of φ parameterized by a one-dimensional disk. Therefore, there exists a deformation φ of φ, parameterized by a one-dimensional disk, whose Horikawa class at t = 0 is s. By Proposition (9.10) the number, the index, and the type of the ramification points of φt cannot be constant in any neighborhood of 0. This contradicts the assumption that C and L are general. Q.E.D. By degenerating to a rational cuspidal curve, Eisenbud and Harris prove a much stronger theorem (Theorem (1.8) in [197]), showing that,

844


under the same assumption of Proposition (9.15), when r ≥ 3, the morphism φ is actually a smooth embedding. This is the theorem we stated in Section 1 of Chapter V. Let us finally remark that, in case r = 2, it can be proved that, under the assumptions of Proposition (9.15), the only singularities of φ(C) are nodes. In fact, in Theorem (10.7) we shall prove a much stronger statement which will not involve the generality of C in the sense of moduli but just the generality of the pair (C, gd2 ) as a point of an irreducible component of Wd2 . In Proposition (9.15) we saw that, when both the curve C and the line bundle L are general, the ramification divisor of the morphism attached to L is equal to zero. The following proposition gives a bound for the degree of the ramification divisor for any morphism from a general curve to projective space. Proposition (9.16). Let C be a general curve of genus g, and let φ : C → Pr ,

r ≥ 2,

be a nondegenerate morphism which is not composed with an involution. Let d be the degree of φ(C), and let R be the ramification divisor of φ. Then deg R ≤ ρ , where, as usual, ρ = g − (r + 1)(g − d + r) . In particular, when r = 2, (9.17)

deg R ≤ 3d − 2g − 6 .

Proof. We argue by contradiction, assuming that a general curve of genus g admits a map to projective space as in the statement whose ramification divisor has degree strictly bigger than ρ. By Theorem (8.15) there exist a family of smooth genus g curves q : Y → U such that the image of the moduli map U → Mg is open and a morphism ψ : Y → Pr with the following properties. For u ∈ U , denote by Yu the fiber q −1 (u) and by ψu the restriction of ψ to Yu ; then, for every u ∈ U , the morphism ψu is nondegenerate and not composed with an involution, and the degree of its ramification divisor is strictly larger than ρ. In particular, if u is a general point of U , the coboundary map δ : H 0 (C, Nψu ) → H 1 (Yu , TYu ) is onto. We may also assume that, as u varies in U , the number, index, and type of the ramification points of ψu do not change. We then fix a general point u in U , set C = Yu , φ = ψu , and we denote by R the ramification divisor of φ. It follows from Proposition (9.10) that δ(H 0 (C, Kφ )) = 0

§10 Plane curves

845

and hence that the surjective map H 0 (C, Nφ ) → H 1 (C, TC ) factors through H 0 (C, Nφ ), so that (9.18)

h0 (C, Nφ ) ≥ 3g − 3 + (r + 1)2 − 1 .

On the other hand, Nφ is of rank r − 1, and from (9.1) and (8.9) it follows that ∧r−1 (Nφ ) = Lr+1 ⊗ ωC (−R) , where L is the pullback via φ of the hyperplane bundle on Pr . Since C is general, we have h1 (C, Nφ ) = h1 (C, Nφ ) = 0. Therefore, by the Riemann–Roch theorem for vector bundles, h0 (C, Nφ ) = (r + 1)d + 2g − 2 − deg R + (r − 1)(1 − g) . Putting this equality together with (9.18), we get deg R ≤ ρ . Q.E.D. 10. Plane curves. In this section we try to answer the question: what is the dimension of Gd2 ? Our discussion will parallel the treatment of Gd1 presented at the 2 end of Section 6. The situation here is complicated by the fact that Mg,d need not be irreducible, contrary, as we shall see in the next section, to 1 what occurs for Mg,d . Theorem (10.1). Let p:C→U be a smooth family of curves of genus g parameterized by a smooth connected variety U . Suppose that ψ : C → P2 is a morphism such that the restriction of ψ to each fiber of p is not composed with an involution and that, for general u ∈ U , the homomorphism Tu (U ) → H 0 (C, Nφ ) is injective, where C = p−1 (u), φ = ψ

C

. Then

dim U ≤ 3d + g − 1, where d is the degree of φ(C). In particular, if g ≥ 2 and X is a component of Gd2 whose general point corresponds to a curve C of genus

846


g equipped with a basepoint-free gd2 which is not composed with an involution, then dim X = 3g − 3 + ρ = 3d + g − 9. Proof. If dim U ≤ g, there is nothing to prove; therefore, we assume that dim U ≥ g + 1. The assumption that u is a general point of U implies, by Proposition (9.10), that g + 1 ≤ dim U ≤ h0 (C, Nφ ) . Therefore Nφ is nonspecial. On the other hand, from (9.1) and (8.9) we get (10.2)

Nφ = ωC L3 (−R) ,

where R is the ramification divisor of φ, and L is the pullback via φ of the hyperplane bundle. Thus, dim U ≤ h0 (C, Nφ ) ≤ 3d + g − 1 . As for the second statement, by Proposition (3.21) it suffices to prove that the dimension of X does not exceed 3d + g − 9. Let U be the bundle of projective frames for the universal gd2 over X. Then there is a family of curves and morphisms w P2 X u U satisfying all the assumptions of the first part of our theorem. Thus, dim X = dim U − dim(Aut(P2 )) ≤ 3d + g − 9 . Q.E.D. Remark (10.3). In the above theorem the restriction to gd2 ’s which are not composed with an involution is a very natural one. In fact, let Y be a component of the variety of isomorphism classes of pairs (C, W ) with C a smooth curve of genus g and W a gd2 on C of the form φ∗ (W ), where φ : C → C is a degree n morphism to a curve of genus g , and W is a base-point-free gd2 on C which is not composed with an involution. Of course, d = nd . We denote by w the degree of the ramification divisor of φ. When g ≥ 2, it is clear from Theorem (10.1) that (10.4)

dim Y = 3d + g − 9 + w ,

and the reader may easily verify that the same is true also for g = 0, 1. Thus (10.4) is a lower bound for the dimension of the component of Gd2 containing Y and may well be strictly larger than the expected dimension

§10 Plane curves

847

3d + g − 9. In fact, using the Riemann–Hurwitz formula, it is immediate to check that this happens precisely when w > (n − 1)(6d + 2g − 2) or equivalently when (10.5)

g > (2n − 1)(g − 1) + 3d (n − 1) + 1 .

This provides plenty of examples of components of Gd2 of dimension larger than the expected one and hence of components whose general member is necessarily composed with an involution. It is interesting to notice that the two kinds of components of Gd2 , those whose general member is composed with an involution and those whose general member is not, may very well coexist for given g and d. In fact, the former exist if and only if g is not less that the right-hand side of (10.5) for some g ≥ 0 and some factorization d = nd with n > 1, while components of the second kind exist if and only if (10.6)

g≤

(d − 1)(d − 2) , 2

as we shall see later in this section. On the other hand, the right-hand side of (10.6) can be larger than the one of (10.5) since, for fixed n and g , it grows quadratically in d , as opposed to linearly. Theorem (10.1) is really a statement about continuous systems of plane curves. We recall that a plane curve of degree d can be thought of as a point in the projective space PN ,

N=

d(d + 3) . 2

A subvariety of this PN is then called a continuous system of plane curves of degree d. Let us denote by Σd,g ⊂ PN the continuous system of all irreducible plane curves of degree d whose normalization has genus g. Theorem (10.7). Let Σ be any irreducible component of Σd,g . Then dim Σ = 3d + g − 1. Moreover, a general point of Σ corresponds to a plane irreducible curve of degree d having (d − 1)(d − 2) δ= −g 2 nodes and no other singularity.

848


Proof. The dimensionality statement is essentially the second part of (10.1). To see it, we limit ourselves to the case g ≥ 2, leaving to the reader the easy task of modifying the proof to make it work when g = 0 or g = 1. If U is a sufficiently small neighborhood of a general point of Σ, then by simultaneously normalizing the curves corresponding to the points of U , we get a family of smooth curves of genus g p:C→U ⊂Σ and a morphism ψ : C → P2 such that for every x ∈ U , ψ(p−1 (x)) is the plane curve corresponding to x. By associating to each x ∈ U the curve p−1 (x) and the linear series giving the morphism ψx : p−1 (x) → P2 , we get an open morphism f : U → Gd2 whith fiber dimension equal to 8 = dim P GL(3). This, combined with the second part of (10.1) and the openness of f , gives dim (Σ) = 3d + g − 1. Having proved the dimensionality statement, we now prove the second part of (10.7). We first observe that, if φ : C → P2 is the normalization morphism for the plane curve corresponding to a general point x of Σ, noticing that dim Σ ≥ g + 1 and arguing as in the proof of (10.1), we may conclude that H 1 (C, Nφ ) vanishes. By Theorem (8.15), this implies that the elements of H 0 (C, Kφ ) are unobstructed. Using again the generality of x and Proposition (9.10), we conclude that Kφ = 0, proving that φ is an immersion and showing, at the same time, that Nφ is a line bundle of degree equal to 2g − 2 + 3d. Set Γ = φ(C). Suppose that there are two points p1 and p2 of C such that φ(p1 ) = φ(p2 ) and that the two branches of Γ in φ(p1 ) corresponding to p1 and p2 have a contact of order h > 1. By our previous remark about the degree of Nφ , we have that H 1 (C, Nφ (−p1 − p2 )) = 0 . Therefore, there is a section s of Nφ which vanishes at p1 but not at p2 . Since H 1 (C, Nφ ) = 0, the class s is unobstructed and hence corresponds to a one-parameter family of deformations of φ φ˜ w P2 C p u Δ

§10 Plane curves

849

where Δ = {t ∈ C : |t| < ε}, such that p−1 (0) = C, φ˜ = φ. Along this deformation the two branches C of Γ corresponding to p1 and p2 are deformed, for any t = 0, into analytic arcs which do not have contacts of order h (or higher), in contradiction with the generality of x. To see this, choose local coordinates (t, z1 ) and (t, z2 ) on C centered at p1 and p2 , respectively. Upon choosing suitable local coordinates centered at φ(p1 ) ∈ P2 , φ˜ can be represented, near p1 and p2 , by C2 -valued functions f1 (t, z1 ), f2 (t, z2 ) such that

(10.8)

(10.9) (10.10)

f1 (0, 0) = f2 (0, 0) = 0 , ∂f1 (t, z1 ) = 0 ∂z1 ∂f2 (t, z2 ) = 0 ∂z2 ∂f1 ∂f1 (0, 0) ∧ (0, 0) = 0 , ∂t ∂z2 ∂f2 ∂f2 (0, 0) = 0 . (0, 0) ∧ ∂t ∂z2

∀ t, z1 , ∀ t, z2 ,

From (10.10) it follows that, changing local coordinates in P2 , we may assume that f2 (t, z2 ) = (t, z2 ). Write f1 (t, z1 ) = (a(t, z1 ), b(t, z1 )). To say that the two branches of Γ under consideration have a contact of order h at φ(p1 ) means that (10.11)

∂ia (0, 0) = 0 , ∂z1i

i = 0, . . . , h − 1 ,

∂ha (0, 0) = 0 . ∂z1h We now argue by contradiction, assuming that for every t, there is a point (t, z(t)) (necessarily unique) such that z1 → f1 (t, z1 ) has a contact of order h with z2 → (t, z2 ) at f1 (t, z(t)). In other words, assume that there is z(t) such that a(t, z(t)) = t , i

∂a (t, z(t)) = 0 , ∂z1i

i = 1, . . . , h − 1 .

Since we must have ∂ ha (t, z(t)) = 0 ∂z1h

for small t ,

850


by the implicit function theorem z(t) is a holomorphic function of t. Differentiating the identity a(t, z(t)) = t and using (10.11), we get ∂a Since ∂z (0, 0) = 0, formula (10.9) implies that that ∂a ∂t (0, 0) = 1. 1 ∂f1 ∂z1 (0, 0) = 0, contradicting (10.8). Having established that all multiple points of Γ are ordinary, assume that Γ has an n-fold point q with n > 2. Let p1 , . . . , pn be the points of C which map to q and notice that d > n. Therefore, H 1 (C, Nφ (−p1 − · · · − pn )) = 0 , and there is a section of Nφ which vanishes at p2 , . . . , pn but not at p1 . Since H 1 (C, Nφ ) = 0, this class is not obstructed and corresponds to a family of deformations of φ along which the n-fold point q is transformed into finitely many points of strictly lower multiplicity. This, again, contradicts the generality of x. Q.E.D. The following theorem was established by Harris [349], proving a long-standing conjecture by F. Severi. Theorem (10.12). Σd,g is irreducible. We shall not deal with this result but will devote our attention to other, less difficult, aspects of the geometry of plane curves. In Theorem (10.1) we computed the exact dimension of any component of Gd2 whose general point corresponds to a morphism of a genus g curve in P2 which is not composed with an involution. The problem arises to determine whether such a component exists, i.e., to decide for which values of g and d there exist irreducible plane curves of degree d and geometric genus g (here and in what follows, by geometric genus we mean the genus of the normalization). The answer to this question was found by Severi [633], who, more specifically, showed that: Theorem (10.13). Given a positive integer d, for every value of g such that (d − 1)(d − 2) , 0≤g≤ 2 there exists an irreducible plane curve of degree d and geometric genus g having at most nodes as singularities. More generally, Tannenbaum [652] proves the following. Theorem (10.14) (Severi-Tannenbaum). Given any integer r ≥ 2 and any integer d such that d ≥ r, for every value of g such that 0 ≤ g ≤ π(d, r) , where π(d, r) is the Castelnuovo bound, there exists an irreducible nondegenerate curve in Pr of degree d and geometric genus g having at most nodes as singularities.

§10 Plane curves

851

For an extensive discussion of Castelnuovo’s bound and Castelnuovo extremal curves, we refer to Chapter I. Severi’s idea is to construct his curves by suitably smoothing the union of d general lines at a selected group of nodes. Tannenbaum follows a similar procedure on certain special kinds of smooth rational scrolls in Pr . The basic tool is the following lemma. Lemma (10.15). Let X be a smooth surface. Let C be a connected nodal curve in X. Assume that, for every component Γ of C, (ωX ·Γ) < 0. Let p1 , . . . , ph be the nodes of C. Then for every integer k, 0 ≤ k ≤ h, there exists a flat family Cy uπ

w X ×Δ

Δ where Δ = {t ∈ C : |t| < δ}, with the following properties: i) For any t = 0, Ct = π −1 (t) has exactly h − k nodes and no other singularities. ii) C0 = C. iii) The family C → Δ is analytically locally trivial at any point of C different from p1 , . . . , pk . The proof of this result, although conceptually quite simple, requires the introduction of a new tool, which can be thought of as an analogue, in the case of curves with nodes, of Horikawa’s theory. This in turn is a classical subject which has been revived, among others, by Wahl [677]. Following a brief intuitive remark, we shall give a sketchy presentation of those results of Wahl which are relevant to our purpose. The intuitive remark is simply that, if f (x, y, t) = 0 gives a family of plane curves Ct having nodes p1 (t), . . . , pn (t), then the curve given by ∂f ∂t (x, y, t) = 0 passes through the points pν (t). Formally, let X be a smooth surface, and let C be a reduced curve on X having, as its only singularities, nodes p1 , . . . , ph . Let k be an integer such that 0 ≤ k ≤ h. Set k = N

h %

mpi (OX (C) ⊗ OC ) ,

i=k+1

be the where mpi stands for the ideal sheaf of the point pi in OC . Let C normalization of C, and let φ : C → X be the normalization morphism may be disconnected). The connection between the sheaves (of course, C Nk and the normal sheaf Nφ is given by a natural isomorphism (10.16)

0 ∼ N = φ∗ (Nφ )

852


or, equivalently, by an isomorphism 0 ) = φ∗ O(C) − (qi + ri ) ∼ φ∗ (N = Nφ , where φ−1 (pi ) = {qi , ri }. One of Wahl’s result is the following. k ) Proposition (10.17). There is a natural bijection between H 0 (C, N and the set of isomorphism classes of flat families w X × Spec C[ε]

C u Spec C[ε]

such that C0 = C and C is locally trivial at any point of C different from p1 , . . . , pk . Moreover, setting Δ = {t ∈ C : |t| < δ}, he proves the following. k ) = 0, any element s of Proposition (10.18). In case H 1 (C, N 0 H (C, Nk ) arises, for any sufficiently small δ, from a flat family C

w X ×Δ

uπ Δ satisfying ii) and iii) (but not necessarily i)) in Lemma (10.15), via the inclusion Spec C[ε] → Δ given by t → ε. To prove (10.15), we consider the exact sequence k → T → 0 , 0 → N 0→N

(10.19)

where T is supported on {p1 , . . . , pk }. We will show that (10.20)

0 ) ∼ Nφ ) = 0 . H 1 (C, N = H 1 (C,

To see this, recall that −1 Nφ = φ∗ (ωX ) ⊗ ωC .

be any component of C and set Γ = φ(Γ). Since, by assumption, Let Γ (ωX · Γ) < 0, we obtain − 2 − (ωX · Γ) deg Nφ = 2g(Γ) Γ − 2. > 2g(Γ)

§10 Plane curves

853

k ) = 0 and that From (10.20) and (10.19) we deduce that H 1 (C, N 0 0 k ) surjects onto H (C, T ). Choose a section s of N k whose H (C, N image in H 0 (C, T ) does not vanish at any one of the points p1 , . . . , pk . Let w X ×Δ C u Δ be the family whose existence is guaranteed by (10.18). By the very choice of s, this family also satisfies property i) of Lemma (10.15). This concludes the proof of the lemma. Q.E.D. Following the classical terminology introduced by Severi, we shall call the nodes pk+1 , . . . , ph the assigned nodes of C, and the nodes p1 , . . . , pk the virtually nonexistent nodes of C. The reason for these names is that we allow the virtually nonexistent nodes to disappear under deformation, whereas we insist that the assigned nodes remain. We are now almost ready to prove Theorem (10.14). The only thing we need is a way of telling when the curves Ct constructed in Lemma (10.15) are irreducible. This is taken care of by the following simple observation of Severi. Remark (10.21). Let X, C, Ct , p1 , . . . , ph be as in Lemma (10.15). Let C be the partial desingularization of C at the assigned nodes pk+1 , . . . , ph . If C is connected, then Ct is irreducible for t = 0. This is essentially obvious. Blow up C along the curves traced out by the moving assigned nodes to obtain a flat family C → Δ whose central fiber is C . Thus Ct is connected for every t. On the other hand, for t = 0, Ct is the normalization of Ct . We now proceed to prove (10.14). We consider three separate cases. Let r = 2; in this case we choose C to be the union of d general lines, C = 1 ∪ · · · ∪ d . We number the nodes of C so that p1 , . . . , pd−1 are the intersections of 1 with the other lines in C. For any integer g such that 0 ≤ g ≤ π(d, 2) =

(d − 1)(d − 2) , 2

we set k = d−1+g. By Remark (10.21), when t = 0, the curve Ct , whose existence is guaranteed by Lemma (10.15), is an irreducible plane curve of degree d with exactly (d − 1)(d − 2) d(d − 1) −k = −g 2 2

854


nodes and no other singularity. New let us turn to the case of r ≥ 3. In this case set X = P 1 × P1 . Let L1 , L2 be curves belonging to the two rulings of X. We & view X as & L 2 &. embedded in Pr by means of the very ample linear system &L1 + r−1 2 Set ( ' ( ' r−1 d−1 d−1 , n=d−1− +1 . m= r−1 2 r−1 Let C be the union of the diagonal of X, m distinct members of |L1 |, and n distinct members of |L2 |. We number the nodes of C so that p1 , . . . , pm+n are the ones lying on the diagonal of X. For any integer g such that 0 ≤ g ≤ π(d, r) = m · n , we set k = m+n+g. Arguing as in the previous case, we find an irreducible curve Ct ∈ |C| of geometric genus g with exactly π(d, r) − g nodes as singularities. The degree of Ct ∈ |(m + 1)L1 + (n + 1)L2 | is easily seen to be d. Moreover, the assumption that d ≥ r implies that m + 1 ≥ 2, so that Ct cannot lie in a hyperplane of Pr . The last case, r even and r ≥ 4, can be handled in a similar manner and is left as an exercise. As a hint, we suggest the reader to try to construct the desired curve on the plane blown up r at one &point and & embedded in P by means of the very ample linear r system &E + 2 L&, where E is the exceptional divisor, and L is the proper transform of a line passing through the center of the blow-up. Q.E.D. A natural question to ask in connection with Theorem (10.14) is whether there exists a smooth connected nondegenerate curve in Pr of any given degree d and genus g with 0 ≤ g ≤ π(d, r), when r ≥ 3. In general the answer is no. Indeed, by the discussion at the end of Chapter I, any curve whose genus is close to the Castelnuovo bound necessarily lies on r−1 independent quadrics. When r = 3, such 2 a curve must lie on a quadric Q in P3 , and it is immediate to check that, for instance, there is no smooth curve of genus π(d, 3) − 1 on Q. 11. The Hurwitz scheme and its irreducibility. In this section we shall first recall a fundamental result due to L¨ uroth, Clebsch, and Hurwitz, concerning the irreducibility of M1g,d . We

§11 The Hurwitz scheme and its irreducibility

855

shall closely follow Fulton’s treatment [273]. Let C be a smooth curve of genus g. We recall that a ramified d-sheeted covering f : C → P1 is called simple if every ramification point of f has index equal to 2 and no two ramification points of f lie over the same point of P1 . We shall also say that the corresponding gd1 is simple. We denote by R(f ) ∈ Div(C) the ramification divisor of f and by Λ(f ) ∈ Div(P1 ) the branch locus of f , so that Λ(f ) = f∗ R(f ) if f is simple. Let w = 2d + 2g − 2 be the degree of the branch locus of f (w is the traditional symbol for the degree of the branch locus). Denote by Pw the open subset of the wth symmetric product of P1 consisting of unordered w-tuples of distinct points in P1 . For any A ∈ Pw , we let H(d, A) be the set of equivalence classes of d-sheeted ramified simple coverings (11.1)

f : C → P1

such that Λ(f ) = A . We recall that the covering (11.1) is said to be equivalent to a covering f : C → P1 if there exists an isomorphism φ : C → C such that f ◦ φ = f . We shall denote by [f ] the equivalence class of f . We now recall Riemann’s fundamental existence theorem. For this, we need to introduce a few definitions and some notation. If G and H are two groups, we set, as usual, Homext (G, H) = Hom(G, H)/ ∼ , where λ ∼ μ if there exists h ∈ H such that μ = hλh−1 . We denote by Sd the symmetric group on d letters. Now let A = {a1 , . . . , aw } be an element of Pw and choose a base point y ∈ P1 − A . We define a standard system of generators σ1 , . . . , σw of π1 (P1 A, y) as in the following picture:

856


Figure 1. We define the monodromy representation ) * φ(f ) : π1 (P1 A, y) → group of permutations of f −1 (y) by mapping a loop σ in P1 A with base point at y to the permutation of f −1 (y) obtained by associating to each point x of f −1 (y) the end point of the unique lifting of σ−1 having initial point at x. Numbering the points of f −1 (y) gives an identification between the group of permutations of f −1 (y) and Sd , canonical up to inner automorphisms of Sd . Therefore, the monodromy representation induces a well-defined map φ : H(d, A) → Homext (π1 (P1 A, y), Sd ),

(11.2)

obtained by associating to every element [f ] of H(d, A) the class of φ(f ). This being understood, we can state the following special instance of Riemann’s existence theorem. Theorem (11.3) (Riemann’s existence theorem). The map (11.2) is injective. Moreover, the image of (11.2) consists of those classes which are induced by irreducible representations ξ such that ti = ξ(σi ) , is a transposition and

i = 1, . . . , w ,

ti = 1.

That ti is a transposition is a reflection of the fact that we are dealing with simple coverings. In conclusion, via φ, we can identify H(d, A) with the set ⎡

Gd,w

⎤ set of conjugacy classes [t 1 , . . . , tw ] of w-tuples of = ⎣ transpositions such that ti = 1 and t1 , . . . , tw ⎦ . generate a transitive subgroup of Sd


857

We then consider the set /

H(d, w) =

H(d, A) .

A∈Pw

Given an element [f ] ∈ H(d, A), we say that [t1 , . . . , tw ] is the symbol of [f ] with respect to the basis {σ1 , . . . , σw }. There is a natural map Λ : H(d, w) → Pw given by [f ] → Λ(f ) . Since the fibers of Λ may all be identified with the finite set Gd,w , the set H(d, w) can be equipped with a unique complex structure which makes H(d, w) into a w-dimensional complex manifold and Λ into a topological covering. We shall call H(d, w) the Hurwitz space of type (d, w). It follows from the construction that there is, over the Hurwitz space, a family of d-sheeted simple coverings, in the following sense. There exist a complex manifold X and smooth maps X ψu

F

w P1

H(d, w) such that, for each s ∈ H(d, w), the fiber ψ −1 (s) = Xs is a smooth curve. Moreover, & Fs = F &Xs : Xs → P1 is a simple d-sheeted ramified covering such that [Fs ] = s . Clearly, by Riemann–Hurwitz’s formula, Xs is a curve of genus g, where w = 2g + 2d − 2 . Therefore, there is a natural morphism (11.4)

H(d, w) → Mg

given by s → [Xs ] . The fundamental result concerning H(d, w) is the following. ¨ roth–Clebsch–Hurwitz). H(d, w) is connected. Theorem (11.5) (Lu

858


Proof. Consider the topological covering Λ : H(d, w) → Pw and let A = {a1 , . . . , aw } be a point in Pw . The proof of Theorem (11.5) consists in showing that π1 (Pw , A) acts transitively on the fiber Λ−1 (A) = H(d, A) . To show this, let us consider loops Γi ,

i = 1, . . . , w ,

in Pw with endpoints at A of the form Γi (t) = {a1 , . . . , ai−1 , γi (t), γi (t), ai+2 , . . . , aw } , where

γi , γi : [0, 1] → P1 {a1 , . . . , ai−1 , ai+2 , . . . , aw }

are arcs such that

γi (0) = γi (1) = ai , γi (1) = γi (0) = ai+1 ,

as in Figure 2. It suffices to show that the subgroup Γ of π1 (P, A) generated by the Γi acts transitively on H(d, A). The following picture shows how to interpret the action of Γi on the set Gd,w to which H(d, A) has been identified.

Figure 2. How Γi acts. As t varies between 0 and 1, we have two varying loops σi (t) and σi+1 (t) with σi (0) = σi , σi+1 (0) = σi+1 ,

σi (1) = σi+1 , −1 σi+1 (1) ∼ σi+1 σi σi+1 .


859

Thus, if we start at t = 0 with a d-sheeted covering having symbol [t1 , . . . , tw ] with respect to the basis {σ1 , . . . , σi , σi+1 , . . . , σw }, we end up, at t = 1, with a d-sheeted covering having symbol [t1 , . . . , tw ] with respect to the basis {σ1 , . . . , σi−1 , σi+1 (1), σi+1 , σi+2 , . . . , σw } or, equivalently, with a d-sheeted covering having symbol [t1 , . . . , ti−1 , ti ti+1 ti , ti , ti+2 , . . . , tw ] with respect to the basis {σ1 , . . . , σi , σi+1 , . . . , σw }. Thus, the action of Γi on an element [t1 , . . . , tw ] ∈ Gd,w is as follows: Γi · [t1 , . . . , tw ] = [t1 , . . . , ti−1 , ti ti+1 ti , ti , ti+2 , . . . , tw ]. Thus, Γ−1 i · [t1 , . . . , tw ] = [t1 , . . . , ti−1 , ti+1 , ti+1 ti ti+1 , ti+2 , . . . , tw ]. It is now a combinatorial exercise to show that the subgroup Γ of π1 (P, A) acts transitively on Gd,w . Following the treatment of Enriques [215] (Vol. III, Libro V, Cap. 1, page 26), we will show that the orbit of any element of Gd,w under the action of Γ contains the element (11.6) [(1 2), (1 2), . . . , (1 2), (2 3), (2 3), (3 4), (3 4), . . . , (d − 1 d), (d − 1 d)] , 0 12 3 2g+2 times

where (α β) stands for the simple transposition that exchanges α with β. The proof will be broken up in several steps. Denote by G1 the set of transpositions (1 α) , 1 < α, and by G1 its complement in the set of all simple transpositions. Let τ be an element of Gd,w . Lemma (11.7). There is an h ≥ 1 such that the orbit of τ contains an element of the form [t1 , . . . , tw ] with t1 , . . . , th ∈ G1 ,

th+1 , . . . , tw ∈ G1 .

Such an h is necessarily even. Proof. Write τ = [s1 , . . . , sw ]. Since the subgroup generated by the si is transitive, sj = (1 α) for some j. If j > 1, Γ−1 j−1 acts by moving sj one place to the left without affecting the other si , except the one immediately preceding it. This kind of move can be applied repeatedly to push sj all the way to the left. The result is τ = [s1 , . . . , sw ], where s1 = sj = (1 α). Now we can apply the same procedure to s2 , . . . , sw , and so on, until the desired configuration is reached. That h is even

860


follows from the fact that

ti = 1, and hence t1 · · · th = 1,

since t1 , . . . , th are the only transpositions in τ involving 1. Q.E.D. Lemma (11.8). Let h be the minimum integer such that (11.7) holds. Then t1 = · · · = th . Proof. We know that h > 1. If we had th−1 = (1 α), th = (1 β), applying

Γ−1 h−1

α = β ,

would yield [t1 , . . . , th−2 , th , (α β), . . . ],

contradicting the minimality of h. Arguing by induction, we will be done if we can show that, whenever th−k+1 = · · · = th , with k ≥ 2, then −1 th−k = th as well. If k is even, applying Γ−1 h−1 · · · Γh−k to [t1 , . . . , tw ] yields [t1 , . . . , th−k−1 , th−k+1 , . . . th , th−k , th+1 , . . . , tw ] . −1 By the previous argument, th−k = th . If k is odd, applying Γ−1 h−2 · · · Γh−k to [t1 , . . . , tw ] yields

[t1 , . . . , th−k−1 , th−k+1 , . . . th−1 , th−k , th , . . . , tw ] , and again one concludes that th−k = th . Q.E.D. Using (11.8) and conjugation, if necessary, we may assume that (11.7) holds with t1 = · · · = th = (1 2) . Obviously, th+1 · · · tw = 1 , and th+1 , . . . , tw act transitively on {2, . . . , d}. It then follows by induction that the orbit of τ contains an element of the form [s1 , . . . , sw ] with s1 = · · · sh+1 = · · · · · · sh+h2 +···+hd−2 +1 = · · ·

= sh = (1 2) , = sh+h2 = (2 3) , · · · = sh+h2 +···+hd−1 = (d − 1 d) ,

where h, h2 , . . . , hd−1 are even and different from zero. We now want to reduce the number of (2 3)’s, (3 4)’s, etc., down to two. Suppose for example that [s1 , . . . , sw ] = [. . . , (1 2), (1 2), (2 3), (2 3), (2 3), (2 3), . . . ] .


861

to get: Apply Γ−1 h [. . . , (1 2), (2 3), (1 3), (2 3), (2 3), (2 3), . . . ] ; then Γh+1 : [. . . , (1 2), (2 3), (1 2), (1 3), (2 3), (2 3), . . . ] ; then Γh+2 : [. . . , (1 2), (2 3), (1 2), (1 2), (1 3), (2 3), . . . ] ; −1 then Γ−1 h+1 Γh :

[. . . , (1 2), (1 2), (1 2), (2 3), (1 3), (2 3), . . . ] ; then Γh+2 to get, at long last: [. . . , (1 2), (1 2), (1 2), (1 2), (2 3), (2 3), . . . ] . By repeatedly applying this procedure we finally deduce that the orbit of τ contains the element (11.6), where the number of (1 2)’s must be 2g + 2 by the Riemann–Hurwitz formula. This concludes the proof of (11.5). Q.E.D. A graphical representation of a d-sheeted ramified simple covering of P1 of type (11.6) can be given as follows.

Figure 3. A simple consequence of Theorem (11.5) is the following. 1 Corollary (11.9). Mg,d and Mg are irreducible.

Proof. By the Existence Theorem (2.3) in Chapter VII, 1 Mg = Mg,d

as soon as d ≥ (g/2) + 1. Therefore, the second assertion is a special case of the first one, and we may assume that d ≤ (g/2) + 1. Let X be the image of the Hurwitz space H(d, 2d + 2g − 2) in Mg via (11.4). Obviously, 1 . X ⊂ Mg,d

862


1 In view of Theorem (11.5), it suffices to show that Mg,d is in fact the closure of X in Mg . We shall prove that the general point of any component of Wd1 corresponds to a smooth curve C together with a complete, base-point-free, simple gd1 . That such a point corresponds to a (complete) gd1 follows from the fact that, on any smooth curve C, no component of Wd1 (C) is entirely contained in Wd2 (C) (cf. Lemma (3.5), Chapter IV). The fact that such a gd1 is base-point-free follows from a simple dimension count. In fact, we know (cf. Proposition (6.8)) that Wd1 is of pure dimension

dim Wd1 = 2d + 2g − 5 , whereas the dimension of the sublocus of Wd1 corresponding to gd1 ’s with r base points is not greater than 1 + r = 2d + 2g − 5 − r < dim Wd1 . dim Wd−r

To show that the general gd1 is simple, we proceed as follows. Let (11.10)

φ : C → P1

be the corresponding d-sheeted ramified covering. Suppose that φ has a ramification point p of index h > 2, so that in a neighborhood of p the map φ is given by w = zh . Let s ∈ H 0 (C, Nφ ) be a section of the normal sheaf to φ which vanishes to order exactly equal to 1 at p (this makes sense in the present context). Since H 1 (C, Nφ ) = 0 , s is the Horikawa class of an effective one-parameter deformation (cf. Theorem (8.15) or Remark (8.20)) {φt : Ct → P1 , t ∈ C, |t| < δ}. In local coordinates near p, φt is given by w = z h + t(a1 z + . . . + ah−2 z h−2 ) + [2] , where a1 = 0, and, as usual, [2] stands for terms of order at least 2 in t. Therefore, dw = hz h−1 + t(a1 + . . . + (h − 2)ah−2 z h−3 ) + [2] . dz

§12 Plane curves and gd1 ’s

863

If t is sufficiently small and t = 0, this function of z has zeros of order strictly less than h − 1 in a neighborhood of z = 0. This contradicts the assumption that (11.10) is general. Summing up, we have shown so far that all the ramification points of φ have index equal to two. Suppose now that two of them, p1 and p2 , lie over the same point of P1 . Arguing as before, we are led to a contradiction by taking an effective one-parameter deformation of φ corresponding to a section of Nφ which vanishes at p1 but not at p2 . Q.E.D. We finally remark that any smooth curve C can be realized as a d-sheeted ramified simple covering of P1 if d is sufficiently large. This can be seen as follows. Let φ : C → Pr ,

r ≥ 3,

be an embedding of C as a curve of degree d. For this, it suffices that d ≥ 2g + 1. Project generically C into a P2 . The image curve Γ will be a plane curve of degree d having only ordinary double points (nodes). Since Γ has only a finite number of flexes and bitangents, a general projection of Γ to a P1 yields a representation of C as a d-sheeted ramified simple covering of P1 . 12. Plane curves and gd1 ’s. In Section 6 we showed that, in genus g ≥ 2, dim Wd1 = 2d + 2g − 5 if 2 ≤ d < g + 2 (cf. Proposition (6.8). Thus, (12.1)

1 ≤ 2d + 2g − 5 . dim Mg,d

Our main goal in this section is to show that equality holds in (12.1) when the right-hand side is less than or equal to 3g −3, i.e., in case d ≤ g/2+1. This will be achieved as follows. Let p : C → B be a Kuranishi family of genus g smooth curves and set Wdr = Wdr (p), according to the general conventions (3.16). Then it will suffice to show that there is a point in Wd1 − Wd2 (which, by Proposition (6.8), is the smooth locus of Wd1 ) such that (12.2)

dη : T (Wd1 ) → Tη( ) (B)

is injective, where η : Wd1 → B is the natural forgetful morphism. In fact, the moduli map m : B → Mg is finite-to-one, and we proved in Section 11 1 is irreducible. We will then get the following result. that Mg,d

864


1 Theorem (12.3). If 2 ≤ d ≤ g/2+1, then Mg,d is irreducible of dimension 1 = 2d + 2g − 5 . dim Mg,d

As we just pointed out, it suffices to find a point ∈ Wd1 − Wd2 such that (12.2) is injective. The point will correspond to a line bundle L of degree d over a smooth curve C of genus g such that h0 (C, L) = 2. The kernel of dη is ker(dη) = TL Wd1 (C) . By Proposition (4.2), Chapter IV, TL Wd1 (C) = (Im μ0 )⊥ . Therefore, to show that this is zero, it suffices to show that, for L, dim(ker μ0 ) = −ρ = −(g − 2(g − d + 1)) = g − 2d + 2 . In view of the first part of Proposition (6.7), it then suffices to find a smooth curve C equipped with a degree d line bundle L = O(D) such that i) h0 C, O(D) = 2; ii) |D| has no base points; iii) h0 C, ωC (−2D) = g − 2d + 2. It has been proved by Beniamino Segre [618] that curves satisfying i), ii), and iii) can in fact be realized as plane curves of a very special kind. Here we will present a slightly modified version of Segre’s constructions. In order to carry these out, it is necessary to use a beautiful result of Castelnuovo [108] about linear systems of curves on regular surfaces. Castelnuovo originally stated his theorem only for rational surfaces, but in fact the only consequence of rationality that is needed in the proof is the regularity of the surface. We first need to introduce some notation and terminology. Let S be a smooth irreducible surface. Let Σ be a linear system on S without fixed components. We let φΣ : S S ⊂ Pr ,

r = dim Σ , S = φΣ (S) ,

be the rational map defined by Σ. We say that Σ is irreducible if its general member is an irreducible curve. Let Σ be an irreducible system, and C ∈ Σ a general member. Then the genus g(Σ) of Σ is defined to be the genus of the normalization of C. If Σ has s base points (both


865

ordinary and infinitely near*) of multiplicities ν1 , . . . , νs , respectively, it is well known that g(Σ) = dim H 1 (C, OC ) −

s νi (νi − 1) i=1

2

.

A positive divisor Γ on S is said to be fundamental for a linear system Σ if Γ is blown down by φΣ . Given a linear system Σ, a point p ∈ S and a nonzero tangent vector v ∈ Tp (S), we set (12.4)

Σp = {C ∈ Σ : p ∈ C}, Σp,v = {C ∈ Σp : v ∈ Tp (C)}.

Finally, given points p1 , . . . , pδ , we set (12.5) Σ2p1 ,... ,2pδ = {C ∈ Σ : C has multiplicity ≥ 2 at pi , i = 1, . . . , δ}. In proving the following result we shall make use of standard material from the elementary theory of linear systems of curves on surfaces, such as Bertini’s theorem and the Riemann–Roch formula. References for this are [213,610,690,55,52]. Theorem (12.6) (Castelnuovo). Let S be a smooth regular surface. Let Σ be an irreducible r-dimensional complete linear system on S, without fixed components and of genus g. Assume that there is a positive integer δ such that δ ≤ g ≤ r − 2δ − 1. Let p1 , . . . , pδ be general points on S. Then i) dim Σ2p1 ,... ,2pδ = r − 3δ; ii) Σ2p1 ,... ,2pδ is irreducible; iii) The genus of Σ2p1 ,... ,2pδ is g − δ. The intuitive motivation of the theorem is the following. Since it is three conditions for an irreducible curve to have a double point at a chosen point and since r ≥ 3δ + 1, one expects that there will be at least a pencil of curves of Σ having δ generically chosen double points. As a curve acquires a double point, either it becomes reducible, or its genus drops by one. Therefore, the condition g ≥ δ is the obvious necessary condition to get an irreducible curve with δ nodes. We first establish a number of lemmas. Lemma (12.7). Let S be a smooth regular surface, and let Σ be a complete irreducible r-dimensional linear system of genus g on S. If r ≥ g + 2, then φΣ is a birational map. *In classical terminology an infinitely near base point is an ordinary base point for the proper transform of Σ on a blowup of S.

866


Proof. Blowing up S, if necessary, we may assume that Σ has no base points. Let C be a general member of Σ. By Bertini’s theorem, C is smooth. To prove the lemma, it suffices to show that the restriction of φΣ to C is not composed with an involution. Since S is regular, the sequence 0 → H 0 (S, OS ) → H 0 S, OS (C) → H 0 C, OC (C) → 0 is exact, so that

h0 C, OC (C) ≥ r ≥ g + 2 .

By the Riemann–Roch theorem, this implies that (C)·2 ≥ 2g + 1 . An immediate consequence is that h1 C, OC (C)(−p − q) = 0 for any p, q in C, which shows that the restriction of φΣ to C is in fact an embedding. Q.E.D. (12.8)

Lemma (12.9). Let S, Σ, g, and r be as in the previous lemma. Let p be a general point on S, and v ∈ Tp (S) a general nonzero tangent vector. Let Σp and Σp,v be as in (12.4). Suppose that r ≥ g + 3. Then φΣp and φΣp,v are birational. Proof. Clearly, dim Σp = r − 1 and dim Σp,v = r − 2. Since Σp and Σp,v can be viewed as complete linear series on suitable blow-ups of S, the result for Σp follows from Lemma (12.7), and so does the one for Σp,v if r > g + 3. There remains the case r = g + 3. As in the proof of the previous lemma, we may assume that Σ has no base points and hence that a general member C of Σ is smooth. Let p be a general point of C, and let v be a nonzero tangent vector to C at p. Set L = O(C)⊗OC . By the Riemann–Roch theorem, deg L = 2g + 2. As we observed in the proof of Lemma (12.9), to show that φΣp,v is birational, it suffices to prove that its restriction to C is not composed with an involution. Since S is regular, Σp,v cuts on C the linear system |L(−2p)|, which is composed with an involution exactly when C is hyperelliptic and (12.10)

L(−2p) ∼ = O(gD) ,

where |D| is the g21 on C. On the other hand, (12.10) cannot hold for infinitely many p’s in C. Therefore, |L(−2p)| gives an embedding, by the generality of p. Q.E.D. Lemma (12.11). Let S be a smooth regular surface. Let Σ be an rdimensional linear system on S without fixed divisors and such that φΣ


867

is birational. Assume that Σ has positive genus. Let Σ ⊂ Σ be an (r − 1)-dimensional linear subsystem all of whose members are reducible. Let Γ be the fixed divisor of Σ . Set Σ = Σ + Γ. Then a) Γ is a fundamental curve for Σ; b) Σ is an irreducible system. Proof. After blowing up S, if necessary, we may assume that Σ has no base points. Let C be a general element of Σ. Since dim Σ = r − 1, Σ is generated by Σ and C. Since Σ has no base points, we must have (Γ · C) = 0, proving a). Now suppose that a general member of Σ is reducible. By Bertini’s theorem, there is then a pencil Λ such that every member of Σ is a sum of members of Λ. Since S is regular, all members of Λ are linearly equivalent. We now claim that (F · C) < 2 if F is a member of Λ. Arguing by contradiction, suppose that this is not the case. Then, given a general point p ∈ C, there is a point q ∈ C such that, if F is the unique member of Λ passing through p, then F cuts on C a divisor containing p + q. This means that every curve of Σ passing through p also passes through q. But then the same is true for the linear system Σ, which is generated by Σ and C. This contradiction shows that (F · C) = 1. But this is also impossible, since F moves in a linear system of positive dimension and C has positive genus. Q.E.D. Lemma (12.12). Let S and Σ be as in the preceding lemma. Let p be a general point of S, and let Σp be as in (12.4). Then every positive divisor which is fundamental for Σp is also fundamental for Σ. Proof. As usual, we can assume that Σ has no base points. Let π = πp : S˜ → S be the blow-up map of S at p. Set E = π−1 (p). Let C be a general member of Σ. We must show that, for any positive irreducible divisor Γ on S˜ with (Γ · (π ∗ C − E)) = 0, one has (Γ · π ∗ C) = 0. We first show that (Γ · E) < 2. Suppose not. Then the divisor cut by Γ on E contains a divisor of the form p1 + p2 . Since ((π ∗ C − E) · Γ) = 0, any member of |π ∗ C − E| passing through p1 contains Γ and hence also passes through p2 . This is absurd. In fact, since the point p is general on S, the linear system |π ∗ C − E| separates points on E. If the conclusion of the lemma is false, there is an algebraic family {Γp }p∈S such that (Γp · E) = (Γp · πp∗ C) = 1 and Γp is irreducible for general p. By construction, as p varies, the curves πp (Γp ) are all algebraically equivalent and hence linearly equivalent, since S is regular. Also, (πp (Γp )·C) = 1. This is absurd since C is not rational. Q.E.D. Proof of Castelnuovo’s theorem (12.6). Arguing by induction, we may as well assume that δ = 1. By blowing up S, if necessary, we may also

868


assume that Σ has no base points. Now let p = p1 be a general point of S, and v ∈ Tp (S) a general tangent vector. Let Σp , Σp,v , Σ2p be as in (12.4) and (12.5). We have Σp ⊃ Σp,v ⊃ Σ2p . Since p is a general point of S, we have (12.13) (12.14) (12.15)

dim Σp = r − 1, dim Σp,v = r − 2 , Σp,v = Σp,v if v = v .

Since Σp,v ⊃ Σ2p for every v, (12.14) and (12.15) give dim Σ2p ≤ r − 3. The reverse inequality is trivial. This proves i). We now prove ii). Suppose that Σ2p is reducible. By i) and (12.14), Σ2p has codimension 1 in Σp,v . Furthermore, since r ≥ g + 3 and p is general, φΣp,v is birational by Lemma (12.9). Therefore, by Lemma (12.11), Σ2p = (Σ2p ) + Γ, where Γ is fundamental for Σp,v , and (Σ2p ) is irreducible. This holds for every v ∈ Tp (S). Therefore Γ is fundamental for Σp . By Lemma (12.12), Γ is then fundamental for Σ. In particular, p ∈ Γ. This means that the curves of (Σ2p ) have a double point (at least) at p. Let Σ + Γ be the linear subsystem of Σ consisting of all curves of Σ that meet Γ. Since Σ contains (Σ2p ) , it is irreducible. Let g ≤ g be the genus of Σ . Then dim Σ = r − 1 ≥ g + 2 , Σ ⊃ Σ2p ⊃ (Σ2p ) , dim(Σ2p ) = r − 3 . By Lemma (12.7), φΣ is birational. Since the curves of (Σ2p ) have a double point at p, we get a contradiction by applying part i) to the system Σ . This proves ii). Finally we prove iii). Let ν1 , . . . , νs be the multiplicities of the base points of the irreducible system Σ2p . Let C be a general element of Σ2p . Let π : S˜ → S be a blow-up of S for which the proper transform of Σ2p has no base points. Let C˜ be the proper ˜ Let g˜ be the genus of C. ˜ We must show that transform of C in S. g˜ = g − 1. We have νi (νi − 1) ≤ g − 1, g˜ = g − 2 ˜ = dim Σ2p = r − 3 ≥ g ≥ g˜ + 1. dim |C|

§12 Plane curves and gd1 ’s We also have

869 C˜ ·2 = C ·2 −

νi2 .

By the Riemann–Roch theorem we then have νi (νi + 1) ·2 ·2 ˜ ˜ r − 3 = dim |C| = 1 + C − g˜ = 1 + C − g − 2 νi (νi + 1) νi (νi + 1) =r− . = dim |C| − 2 2 νi (νi +1) This means that = 3, proving that all the νi are equal to zero 2 except one, which is equal to two. Q.E.D We now come to Beniamino Segre’s existence theorem. Theorem (12.16) (B. Segre). Let g ≥ 2. For any integer d such that 2 ≤ d ≤ g/2 + 1, there exists a smooth curve C of genus g carrying a complete gd1 |D|, without base points and such that h0 (C, ωC (−2D)) = −(Brill–Noether number of the gd1 ) = g − 2d + 2 . Proof. When g = 2, any C will do; it suffices to take as |D| the canonical series. We therefore assume that g ≥ 3. In this case C, or rather a birational model of it, will be constructed as a curve lying on a smooth quadric in P3 , having nodes as singularities, and the gd1 will be cut out on C by one of the rulings of the quadric. Curves lying on quadrics have been encountered in Chapter III, Section 2, and provide a useful source of examples. Let Q be a smooth quadric in P3 . Let |L1 | and |L2 | be the two rulings of Q. For every pair of integers m and n, consider the complete linear system Σm,n = |mL1 + nL2 |. It is immediate to see that Σm,n = ∅ if min(m, n) < 0 , dim Σm,0 = m, dim Σ0,n = n if m ≥ 0, n ≥ 0 . Now suppose that m > 0, n > 0. It then follows from Bertini’s theorem that Σm,n is irreducible and that its general member is smooth. Recall that the canonical line bundle on Q is given by ωQ = O(−2L1 − 2L2 ).

870


Then, by adjunction, the genus of Σm,n is gm,n = (m − 1)(n − 1). Finally, by the Riemann–Roch theorem we have (12.17)

dim Σm,n = mn + m + n = gm,n + 2m + 2n − 1.

We notice that, given any fixed integer m ≥ 2, the intervals Im,n = [gm,n − m − n + 1, gm,n ],

n ≥ 1,

cover the half line [1, ∞]. This implies that, given integers g and d such that 2 ≤ d ≤ g/2 + 1 , there exists an integer h ≥ 4 such that g ∈ Id,h . Equivalently, there exist an integer h ≥ 4 and an integer δ such that 0≤δ ≤h+d−1 and g = gd,h − δ .

(12.18) By (12.17), this implies that

δ ≤ gd,h ≤ dim Σd,h − 2δ − 1. We may now apply Castelnuovo’s theorem to the system Σd,h and conclude that, given δ general points p1 , . . . , pδ on Q, there exists an irreducible curve Γ ∈ Σd,h = |dL1 + hL2 | having δ nodes at p1 , . . . , pδ and no other singularities. By construction, the genus of the normalization C of Γ is equal to g = gd,h − δ. The curve C comes naturally equipped with a gd1 , namely the one cut out on Γ by the ruling |L2 |; let D be a divisor in the gd1 . We will now compute h1 (C, O(νD)) for each ν ≥ 0. The residual series |ωC (−νD)| is cut out on Γ by the (possibly empty) linear system Σd−2,h−ν−2,δ consisting of those curves in Σd−2,h−ν−2 which pass through p1 , . . . , pδ . Since p1 , . . . , pδ are general points on Q, we have (12.19)

max(−1, dim Σd−2,h−ν−2,δ ) = max(−1, dim(Σd−2,h−ν−2 ) − δ).

From (12.17), (12.18), and (12.19) we conclude that max(0, g − νd + ν) if 0 ≤ ν ≤ h − 2 , h1 (C, O(νD)) = 0 if ν ≥ h − 1 .


871

Since 2 ≤ d ≤ g/2 + 1, we have, in particular, h1 (C, O(D)) = g − d + 1 , h1 (C, O(2D)) = g − 2d + 2 . The first equality tells us that the gd1 is complete. The second one tells us that h0 (C, ωC (−2D)) = −(Brill–Noether number of the gd1 ). Q.E.D. Returning to the discussion at the beginning of this section, this concludes the proof of Theorem (12.3). It may be amusing to remark that we proved that the morphism 1 m ◦ η : Wd1 → Mg,d 1 is generically finite by going to a point of Mg,d over which the fiber of m ◦ η is not finite (at least in some cases). In fact, referring to the proof of (12.16), notice that the two rulings of Q cut out on C a gd1 and a gh1 ; it may very well happen that h < d. In this case the curve C has an “isolated” gd1 and a continuous family of gd1 ’s of the form gh1 + p1 + · · · + pd−h . Segre’s theorem implies that, when d ≤ g/2 + 1, a smooth curve C 1 carries only a finite number which corresponds to a general point of Mg,d 1 of gd ’s. We shall now prove that in fact, when d < g/2 + 1, C carries a unique gd1 . More precisely, we shall prove the following result (cf. [24]).

Theorem (12.20). Let d, g be integers such that 2 ≤ d < g/2 + 1 . Let C be a smooth curve of genus g corresponding to a general point 1 . Then C carries a unique gd1 . Moreover, if C carries a baseof Mg,d point-free gh1 with h < g/2 + 1, then such a series is composed with the gd1 . Proof. We proceed by contradiction. Suppose that C carries a basepoint-free gh1 with h < g/2 + 1 which is not composed with the gd1 . Let f : C → P1 , f : C → P1 be the morphisms corresponding to the gd1 and gh1 , respectively. Since C 1 , we know that the gd1 has no base corresponds to a general point of Mg,d

872


points and, moreover, that f is a d-sheeted ramified simple covering (cf. the proof of Corollary (11.9)). Let φ : C → P1 × P1 = Q be the morphism defined by φ = (f, f ). We first show that φ is not composed with an involution. For this, we let p1 : Q → P1 be the projection onto the first factor so that f = p1 ◦ φ. Set Γ = φ(C). Since f is simple, either Γ projects isomorphically onto P1 via p1 , or else φ is not composed with an involution. The first case cannot occur. If it did, we would have that φ(p) = φ(q), and hence f (p) = f (q) whenever f (p) = f (q). Therefore, we would have f = λ ◦ f for some ramified covering λ : P1 → P1 , and this would mean that the gh1 is composed with the gd1 , contrary to our assumption. Now consider the normal sheaf N = Nφ to the morphism φ and the basic exact sequence 0 → K → N → N → 0 associated to φ (cf. (9.1)). Recall (cf. (9.2)) that h1 (C, N ) = h1 (C, N ). 1 Since C corresponds to a general point of Mg,d , by Lemma (9.10) we have 1 + dim(Aut(Q)) , h0 (C, Nφ ) ≥ dim Mg,d i.e., (12.21)

h0 (C, Nφ ) ≥ 2d + 2g + 1 .

Since Nφ is a line bundle, this implies that h1 (C, Nφ ) = h1 (C, Nφ ) = 0 . In turn, arguing as in the proof of (10.7), this implies that K = 0, so that 2 N = N ∼ = det(φ∗ (TQ )) ⊗ ωC ∼ = ωC L2 L , where L = f ∗ O(1), (12.22)

∗

L = f O(1). Thus, deg N = 2g − 2 + 2d + 2h .

Using (12.21), (12.22), and the Riemann–Roch theorem, we get 2h−2 ≥ g . This is a contradiction. Q.E.D 13. Unirationality results. In this section we will prove, among other things, the following classical result. Theorem (13.1). Mg is unirational if g ≤ 10.

§13 Unirationality results

873

We recall that an irreducible algebraic variety X is said to be unirational if there exists a dominant rational map g : PN X for some N . As we mention in the bibliographical notes, at the time of writing this book, the space Mg is known to be unirational for g ≤ 14. The situation in low genus contrasts with the one in high genus, where the following theorem, due first of all to Harris and Mumford [353], and then to Harris [348] and Eisenbud and Harris[204], holds. Theorem (13.2). Mg is of general type when g ≥ 24. We refer the reader to the bibliographical notes for a brief history of these results. In order to prove Theorem (13.1), we come to another interesting aspect of Segre’s theory of polygonal curves. Theorem (13.3) (B. Segre). Let d and g be integers such that 3 ≤ d ≤ g. Let C be a smooth curve of genus g equipped with a base-point-free gd1 such that the corresponding d-sheeted ramified covering f : C → P1 represents a general point of the Hurwitz space H(d, w), where w = 2g + 2d − 2. Then, for any integer n such that n≥

g+d + 1, 2

there exist a degree n plane curve Γ with an ordinary (n − d)-fold point − n + 1 − g nodes and no other singularities, and a p, δ = nd − d(d+1) 2 birational map φ : C → Γ ⊂ P2 such that the given gd1 on C is cut out on Γ by the pencil of lines through p. Following Segre, we will find it convenient to break up the proof of Theorem (13.3) into two propositions. The first result is really a mildly generalized version of Severi’s existence theorem (10.13). Proposition (13.4). Given integers g, d, n such that 0 ≤ g ≤ nd − 2 < d < n,

d(d + 1) − n + 1, 2

874


there exists an irreducible plane curve Γ of degree n and geometric genus g having an ordinary (n − d)-fold point p, (13.5)

δ = nd −

d(d + 1) −n+1−g 2

nodes, and no other singularities. Moreover, if (13.6)

n≥

g+d + 1, 2

it is possible to find a plane curve Γ as above which has no adjoint* curve of degree n − 4 having a point of multiplicity at least n − d at p. Before proving (13.4) we shall state and prove a second proposition, which, combined with the previous one, actually yields a slightly strengthened version of Segre’s Theorem. Proposition (13.7). Let g, d, n, δ be as in Proposition (13.4). Suppose that (13.6) is satisfied. Let V be an irreducible component of the variety of irreducible plane curves of degree n and genus g with an ordinary (n − d)-fold point at p and δ nodes. Assume that V contains a point corresponding to a curve Γ which has all the properties listed in Proposition (13.4). Then projection from p defines a dominant rational map V H(d, w) , where w = 2g + 2d − 2 . Proof. Let X be the blow-up of P2 at p, and let σ : X → P1 ˜ be the proper transform of Γ. Let be the projection from p. Let Γ φ:C →X,

˜ = φ(C), Γ

*For a plane curve C having ordinary singularities, the classical term adjoint curve means a curve having multiplicity at least ν − 1 at each ν-fold point of C. When C is irreducible, the adjoint curves of degree cut out on the normalization of C the complete linear system |ω(( − d + 3)D)|, where D is a divisor cut out by a line. When C has arbitrary singularities we define adjoint curves in the same way but where infinitely near multiple points are included; if C is irreducible, the above result concerning the system cut out by adjoint curves is still valid.


875

˜ Since φ is unramified, we have that be the normalization map for Γ. Nφ ∼ = ωC ⊗ φ∗ (ωX )−1 . ˜ < 0, we deduce that Since (ωX · Γ) H 1 (C, Nφ ) = 0. By Horikawa’s theory there exists a local universal family of deformations of φ φ˜ wX C π

u b0 ∈ B

˜C C = π −1 (b0 ); φ = φ|

where B is smooth, and Tb0 (B) = H 0 (C, Nφ ). Now consider the d-sheeted covering f = σ ◦ φ : C → P1 . Since the support of Nf is zero-dimensional, H 1 (C, Nf ) = 0 . Again, we have a local universal family of deformations of f C (13.8)

w P1

λ

u a0 ∈ A

C = λ−1 (a0 ); f = f˜|C

where A is smooth, and Ta0 (A) = H 0 (C, Nf ). Composing φ˜ with σ yields a family of deformations of f parameterized by (B, b0 ). By the universal property of (13.8), this family is induced (after shrinking B, if necessary) by a unique morphism h : (B, b0 ) → (A, a0 ). The differential of h at b0 is given by the homomorphism dh : H 0 (C, Nφ ) → H 0 (C, Nf )

876


induced by the corresponding map of sheaves in the commutative diagram

(13.9)

0

0

0

0

u L(Δ)

u L(Δ)

w TC

u w φ (TX )

u w Nφ

w0

w TC

dσ u w f ∗ (TP1 )

u w Nf

w0

u 0

u 0

∗

where L is the pullback to C of the hyperplane bundle of P2 , and Δ is the pullback of the exceptional divisor on X. To see that the kernel of dσ is actually L(Δ), it suffices to take determinants in the middle column of (13.9): −1 ∼ −2 ker(dσ) ∼ = L(Δ) . = f ∗ ωP1 ⊗ φ∗ ωX = L (2Δ) ⊗ L3 (−Δ) ∼

We now come to the central point of the argument, namely to the proof that the homomorphism dh is surjective. It suffices to show that h1 (C, L(Δ)) = h0 (C, ωC ⊗ L−1 (−Δ)) = 0 . This is clear since |ωC ⊗ L−1 (−Δ)| is cut out on Γ by adjoint curves of degree n − 4 having a point of multiplicity at least n − d at p and, by assumption, no such curve exists. We have thus proved that the morphism h is open. We have already remarked that a general point of A corresponds to a simple d-sheeted ramified covering of P1 (cf. the proof of (11.9)). This completes the proof of (13.7). Q.E.D. We now prove Proposition (13.4). Let 1 , . . . , d be d general lines in P2 , and let d+1 , . . . , n be n − d general lines through p. Let τ : X → P2 be the blow-up at p, and let Y1 , . . . , Yn be the proper transforms of 1 , . . . , n . Set Y = Y1 + . . . + Yn .


877

The divisor Y is reduced and has d(d + 1) δ¯ = nd − 2 nodes. We number the nodes of Y in such a way that p1 , . . . , pn−1 are the nodes lying on Y1 and that nodes belonging to Y1 or Y2 , or nodes (13.10) {p1 , . . . , p3n−d−3 } = . of the form Y3 ∩ Yi , i = d + 1, . . . , n Set k = n−1+g and notice that by assumption δ¯ ≥ k. Taking pk+1 , . . . , pδ¯ to be the assigned nodes, by Lemma (10.15) and Remark (10.21) the ˜ such that Γ = σ(Γ) ˜ is linear system |Y | contains an irreducible curve Γ an irreducible plane curve of degree n and geometric genus g having an ordinary (n − d)-fold point at p, δ = δ¯ − k nodes, and no other singularity. So far (13.10) has not been used. In case assumption (13.6) is satisfied, we get k ≤ 3n − d − 3 . Set qi = τ (pi ),

i = 1, . . . , δ¯ .

We wish to show that Γ can be constructed in such a way that it has no adjoint curves of degree n − 4 with a point of multiplicity at least n − d at p. By upper semicontinuity it suffices to show that there are no plane curves of degree n − 4 passing through pk+1 , . . . , pδ¯ and having a point of multiplicity at least n − d at p. In fact, such a curve would contain n points of each of the lines i , i = 4, . . . , d, and at least n − 1 points of each of the lines d+1 , . . . , n . It would then be a curve of degree n − 4 containing n − 3 distinct lines, which is absurd. Q.E.D. Remark (13.11). It is important to notice that when 2 < d < n, g+d + 1 ≤ n, 2 d(d + 1) 0 ≤ g ≤ nd − − n + 1, 2 Segre’s argument actually shows that if δ = nd −

d(d + 1) −n+1−g 2

and {p, q1 , . . . , qδ } is a general set of δ + 1 points in P2 , there is no curve of degree n − 4 passing through q1 , . . . , qδ and having a point of multiplicity at least n − d at p.

878


We end this chapter by proving a unirationality theorem for some Hurwitz spaces. Our proof will be based on Segre’s result (13.3) and on the following lemma, for a proof of which the reader is referred to [24]. Lemma (13.12). Let n, d, δ be positive integers such that n > d > 0, (n − 1)(n − 2) (n − d)(n − d − 1) − − δ ≥ 0, 2 2 n(n + 3) (n − d)(n − d + 1) − − 3δ ≥ 0. 2 2 Let {p, q1 , . . . , qδ } be a general set of δ + 1 points in P2 . Then there exists an irreducible plane curve of degree n having an ordinary (n − d)fold point at p, nodes at q1 , . . . , qδ , and no other singularity, with the exception of the case n = 6,

d = 4,

δ = 8.

In fact, the only sextic passing doubly through nine general points in P2 is a cubic counted twice. The result we wish to prove is the following. Theorem (13.13). Let d be an integer greater than or equal to 3. Then the Hurwitz space H(d, 2d + 2g − 2) of simple, genus g, d-sheeted ramified coverings of P1 is unirational in each of the following cases: i) d ≤ 5, g ≥ d − 1; ii) d = 6, 5 ≤ g ≤ 10 or g = 12; iii) d = 7, g = 7 . 1 is unirational. As a consequence, in each of these cases Mg,d

As the reader will certainly notice, some known cases of unirationality, as, for instance, the one of H(2, 2g +2), are not covered by our statement. This is due to our particular method of proof. To prove (13.13), let n be the minimum integer such that n≥ and set

g+d +1 2

d(d + 1) +1−g. 2 It is straightforward to verify that in each one of cases i)–iii) the assumptions of Lemma (13.12) are satisfied and that, moreover, the exception mentioned in the lemma cannot occur. Let {p, q1 , . . . , qδ } be a general set of δ + 1 points of P2 . Let Γ be the irreducible plane curve δ = nd − n −


879

whose existence is guaranteed by Lemma (13.12). By Remark (13.11), Γ has no adjoint curves of degree n − 4 with a point of multiplicity at least n − d at p. Let V be the irreducible component of the variety of irreducible plane curves of degree n with an ordinary (n − d)-fold point at p and δ nodes which contains Γ. By (13.7) there is a dominant rational map from V onto H(d, 2d + 2g − 2). On the other hand, by (13.12), there is a dominant rational map f : V Symδ (P2 ) given by Γ → {nodes of Γ}. The variety V is contained in the projective space PN of plane curves of degree n, and the fibers of f are linear subspaces of this PN ; therefore, V is unirational. This concludes the proof of (13.13). Since, by the fundamental existence theorem (3.2) and by the proof of (11.9), a general curve of genus g can be represented as a simple d-sheeted ramified covering of P1 when d≥

g+2 , 2

we get Theorem (13.1) from case ii) of Theorem (13.13). 14. Bibliographical notes and further reading. For the general theory of the Picard functor, we refer the reader to Kleiman’s beautiful Chapter 5 in [243] and its exhaustive bibliography. A natural problem, not treated in this book, is the one of compactifying the Picard varieties of stable curves. This has been the subject of intensive study starting with papers by D’Souza [157] and Altman and Kleiman [8,9], Oda and Seshadri [568], and Rego [605]. Among the many papers devoted to this problem, we may mention the ones by Caporaso [95], Pandharipande [583], Esteves [220], Esteves and Kleiman [222], Alexeev [6], and Melo [506]. A very useful overview on this subject and further bibliography can be found in Caporaso’s paper [98]. Our treatment of Brill–Noether varieties over moduli and their tangent spaces is based on [23], [25], and [26]. What is modernly called Petri’s conjecture is the statement by K. Petri quoted on p. 215 of our first volume. This statement was rediscovered by Arbarello and Sernesi [34]; this is where the μ1 map, although not thus denoted, first appeared (see p. 215 in [34]). The first result pointing to Petri’s conjecture is the weaker statement to the effect that, for every line bundle on a general curve, the Brill– Noether number ρ is positive. This was proved by the third author and

880


Joe Harris in [319]. The first proof of Petri’s conjecture is due to Gieseker [293]. Gieseker’s argument was simplified by Eisenbud and Harris in [196]. The three papers [319], [293], and [196] are all based on a degeneration argument by considering families of smooth curves degenerating to either a g-noded rational curve [293] or to suitable stable curves possessing g elliptic tails [196]. The first proof of Petri’s conjecture without degeneration to a singular curve is due to Lazarsfeld [464], who proved the theorem for curves lying on a K3 surface. For g > 11, these curves are far from being general curves in the sense of moduli. The idea that, in some sense, curves on a K3 surface behave like general curves has been very fruitful. In a different context it was used by Voisin in her breakthrough paper on Green’s conjecture about syzygies of canonical curves [674] (see also [675] and [16]). Building on Voisin’s remarks concerning the μ1 map [672], Pareschi [588] considerably simplified Lazarsfeld’s proof of Petri’s conjecture; this is the proof we present in Section 7. The link between the μ1 map and first-order obstructions to deformations of triples (curve, line bundle, space of sections) was established in [23]. The first author proposed to study the higher μ’s which are introduced in Exercises B-1–B-3. The link between the higher μ’s and higher obstructions to deformations of triples (curve, line bundle, space of sections) was extensively studied by the three authors of this book without success. The question has been revived and studied by Clemens in [126,127]. There is a completely straightforward generalization of the μ1 map. Given two line bundles L and L on a smooth curve C, let μ0,L,L : H 0 (C, L) ⊗ H 0 (C, L ) → H 0 (C, LL ) be the multiplication map and define μ1,L,L : ker μ0,L,L → H 0 (C, ωC LL ) by setting μ1,L,L ( i si ⊗ ti ) = i (si dti − ti dsi ). This map was studied by Wahl in [678] for the case L = L = ωC and in [679] for the general case; it is often referred to as the Gaussian map or the Wahl map. Fundamental progress on the properties of this map were made by Ciliberto, Harris, and Miranda [122] and by Voisin [672] (see also [585], [569], [400], [123], [85], and [587]). In our book we study only linear series on smooth curves. The idea of using degeneration to singular curves in order to prove properties of smooth curves goes back at least to Max Noether, Poincaré, and the Italian school of algebraic geometry. When dealing with linear series, there are many difficulties in making these arguments rigorous. In the case of one-dimensional linear series many of these difficulties can be circumvented by using admissible covers, which were first introduced by Beauville [54] (see also Knudsen and Mumford [425,426,427], Harris and Mumford [353], Chapter 3, Section G of Harris and Morrison [352], (14.1)


881

Abramovich, Corti, and Vistoli [2], and our Section 5 in Chapter XVI). However, for higher-dimensional linear series, degeneration arguments are vastly more complicated. The first modern treatment is due to Kleiman, who uses Hilbert-scheme techniques to tackle the problem. A different point of view is taken by Eisenbud and Harris in their theory of limit linear series [199]. Their method is very powerful and leads to a number of fundamental results, such as an alternative proof that Mg is of general type for g ≥ 24, the existence of Weierstrass points with given semigroup, the monodromy of Weierstrass points over moduli, the one of zero-dimensional Brill–Noether varieties, and many more [198,202,195]. The theory of limit linear series has also been studied by Esteves and his collaborators, e.g., in [224] and [223] (see also [580] and [89]). A third, natural approach to the study of linear series on singular curves is to define Brill–Noether subvarieties in the compactified Picard variety of, say, a stable curve, which is easier said than done. This construction, together with the first results on the Brill–Noether theory for stable curves, is given by Caporaso in [99]. Brill–Noether subvarieties of Mg , i.e., the subvarieties of Mg corresponding to curves carrying a linear series with negative Brill– Noether number ρ (or else having a special configuration of ramification points) have been extensively studied from the point of view of their existence, of their dimension, and of their cycle class. The case ρ = −1 is solved by Eisenbud and Harris in [205]. The case ρ = −2 is studied by Edidin in [188]. On a general curve one expects only finitely many linear series of given degree and dimension and with ρ = 0. Then the question of monodromy arises. This problem is addressed, and solved in the case of one-dimensional linear series, by Eisenbud and Harris in [200]. The computation of the cycle classes of Brill–Noether varieties in terms of tautological classes is of fundamental importance. The first step in this direction was taken by Mumford in [556], where he computed the cycle classes of the various strata of a flag of subvarieties of Mg defined in terms of Weierstrass points. Another instance of this type of computation can be found in the proofs by Harris and Mumford and by Eisenbud and Harris of their theorems on the Kodaira dimension of M g (cf. [353] and [204]). Yet other instances of this type of computation appear in Diaz’s work on the divisor of moduli space defined by exceptional Weierstrass points and in Cukierman’s study of families of Weierstrass points (cf. [172], [152] and also Gatto’s survey in [280]). Finally, Farkas in his study of the Gieseker–Petri divisor and Farkas and Popa in their study of the effective and ample cones of M g perform a large number of very instructive computations of cycle classes of Brill–Noether type (see [249], [253], [252],[257], and [257]). The Brill–Noether theory for vector bundles has been studied through the works of Narasimhan, Seshadri, Mukai, Le Potier, Hirschowitz, Newstead, Teixidor i Bigas, Ballico, and many others, leading to an

882


explosive development (see, for example, [44], [45], [46], [68], [70], [82], [83], [84], [87], [211], [289], [364], [417], [461], [468], [470], [500], [507], [508], [537], [539], [649], [596], [628], [651], [656], [657], [658], [659], [660], [661], and [687]). A nice overview of the Brill–Noether theory for vector bundles on curves is given by Grzegorczyk and Teixidor i Bigas in [81]. For the general theory of moduli of vector bundles on a fixed curve, we refer the reader to Le Potier [468,469], Ramanan [600], Seshadri [628,629], and to Chapter 10 in Mukai’s book [541]. Looijenga’s vanishing theorem appears in [489]. The fact that g −2 is a bound for the dimension of a complete subvariety of Mg (cf. Theorem (6.4), Chapter XVII) was originally established by Diaz in series of papers centered around the geometry of the moduli spaces of curves carrying a special Weierstrass point [169,172,170,171,173]. A nice discussion of Diaz’s theorem is in Chapter 6, section B of [352]. Complete subvarieties of the moduli space of principally polarized abelian varieties are studied in [234]. The reader can find a good survey of results on moduli of abelian varieties in the paper [282] by van der Geer and Oort. Weierstrass points have attracted the attention of many authors throughout a long period of time. From the point of view of existence of Weierstrass points with a given gap sequence, after the work of Pinkham [595] and Rim and Vitulli [606] the best results have been obtained by Eisenbud and Harris in [202]. The dimension of the subloci of Mg defined in terms of Weierstrass points was first determined by Rauch [603]. A flag of irreducible Weierstrass subvarieties of Mg was introduced in [19] (see also [556], Section 7). Weierstrass points on moving curves, their limits on stable curves and the cycles they define in the moduli space are studied by Lax, Laufer, Diaz, Cukierman, Laksov, Thorup, Esteves, Gatto, and many others (see, for example, [463], [458], [455], [171], [170], [172], [173], [153], [456], [457], [279], [280], [223], [131], [281], [225], and [156]). Horikawa’s theory is developed in [367], [368], [369], and [370]. The theorem of de Franchis on the finiteness of the number of nonconstant morphisms from a fixed curve to another, possibly variable, curve of genus ≥ 2 first appeared in [269]; it had been previously shown by Castelnuovo [110] and Humbert [375,376] that such morphisms are rigid. Our treatment is based, with modifications, on the ones by Samuel [611] and Howard and Sommese [371]. The decomposability of the normal bundle to a projective curve C and its link to the curve C being a complete intersection is studied by Griffiths-Harris and Harris-Hulek in [320] and [350]. Severi’s theorem on the existence of irreducible plane nodal curves of given degree and genus appears in [633], Anhang F , and is also discussed in Zariski’s book [690]. The existence of irreducible plane curves with


883

given number of nodes and cusps and their deformation theory is studied, among others, by Wahl [677], Tannenbaum [652,654], and Zariski [691]. Our discussion is based on [23], [24], and [25]. The theory culminated in Harris’ proof of the long-standing Severi conjecture [349]. The geometry of the Severi variety has been studied by Diaz and Harris [175,176]. Curves carrying a theta characteristic of dimension r ≥ 1 form a subvariety of Mg . These subvarieties and their dimensions have been studied, among others, by Teixidor i Bigas [655], Nagaraj [561], Colombo [128], Fontanari [266], and Farkas [249]. Brill–Noether varieties have been studied from the point of view of the Clifford index by Coppens and Martens [134,135,136,137,138], Coppens, Keem, and Martens [132,133], and Eisenbud, Lange, Martens, and Schreyer [206]. The Clifford index is at the center of yet another very important development in the theory of linear series on algebraic curves. This entire new field was initiated by Green and Lazarsfeld [312,313] and led to their well-known conjectures connecting the Clifford index of a smooth curve to the syzygetic resolution of its canonical ring via Koszul cohomology. In that direction we already mentioned Voisin’s breakthrough papers [674,675], but for the vast literature on this subject, we refer the reader to the beautiful book by Aprodu and Nagel [14] and its exhaustive bibliography. The Hurwitz scheme was introduced by Fulton [273] to give an algebraic proof of the irreducibility of the moduli space of curves of given genus. Our discussion is partly based on his paper. The combinatorial argument due to L¨ uroth and Clebsch is taken from the book [214] by Enriques and Chisini. In general, there at least two main difficulties in studying the r . First of all, geometry of the Brill–Noether varieties Gdr , Wdr , and Mg,d they are singular, but, most importantly, it is very hard to extend their definition, in a tractable way, to the case of stable curves. This difficulty is in fact the main motivation for Eisenbud and Harris’ theory of limit linear series, which, in essence, shortcuts the problem of compactification. However, when r = 1, the situation is more favorable. One advantage is that Gd1 is smooth, as we proved in Proposition (6.8). The other advantage is that, instead of working with Gd1 or Wd1 , one may work with the Hurwitz schemes, which can be compactified using admissible covers. All of this makes the theory of one-dimensional linear series more manageable. Earlier in these bibliographical notes, we pointed out the many instances in which admissible covers proved to be an extremely useful tool, such as the papers by Beauville [54], Knudsen and Mumford [425,426,427], Harris and Mumford [353], Harris [348], Diaz [172], Harris and Morrison [351], and Abramovich, Corti, and Vistoli [2]. The (co)homology of Hurwitz spaces has been studied, among others, by Diaz and Edidin [174] and Kazarian and Lando [406].

884


Another way of compactifying the Hurwitz scheme is given by Ekedahl, Kazarian, Lando, and Vainshtein in [209]. In this paper, the four authors describe Hurwitz numbers, i.e., the number of ramified covers of the sphere with given degree and ramification, in terms of top intersections of the moduli space of curves. As we already observed in the bibliographical notes to Chapter XX, their work has been extremely influential and has been used, especially in the study of the intersection theory of M g,n , by many authors including Okounkov [570], Okounkov and Pandharipande [572,577], Shadrin [639], Zvonkine [695,694], Lando and Zvonkine [460], and Kazarian and Lando [407]. Another point of view from which one may study Hurwitz numbers is taken by Fantechi and Pandharipande [245] who use Gromov–Witten theory. In the bibliographical notes to Chapter XVII we already mentioned the work of Goulden, Jackson, and Vakil [303,304,305], Bertram, Cavalieri, and Todorov [66], Cavalieri [113], and Liu and Xu [481]. All these papers involve, in one way or another, the Hurwitz scheme. The idea for the proof of the unirationality of Mg for g ≤ 10 is due to Severi [633] (see also Segre [619]). His argument was incomplete in more than one respect, and particularly because of the illicit (at that time) assumption of Harris’ theorem on the irreducibility of the variety of curves of given degree and genus. A modern treatement of Severi’s argument is given by Arbarello and Sernesi [35]. The proof given here is taken from [24], which was inspired by Segre [618]. As with many of his works, and certainly due to a change of style in the subject, Segre’s paper contained fundamental ideas but was apparently overlooked. The Bourbaki seminar [676] by Voisin and Farkas’ paper [254] offer beautiful overviews of unirationality results for moduli spaces of curves. Many of these results are based on the so-called Mukai’s models for Mg when g ≤ 9 [534,536,538,540]. The unirationality of Mg has been established for g ≤ 14, in genus 12 by Sernesi [622], in genera 11 and 13 by Chang and Ran [114], and in genus 14 by Verra [669] via an inventive argument which uses ideas close to one of Mukai [540]. A proof that M11 is uniruled is due to Mori and Mukai [517]. Bruno and Verra [90] prove that M15 is rationally connected. At the other extreme, in their breakthrough paper [353], Harris and Mumford showed that Mg is of general type for all odd g ≥ 25; this was later extended to all g ≥ 24 by Harris [348] and Eisenbud and Harris [204]. An alternative proof of these results is offered by Farkas [254], who uses the syzygetic resolution of the canonical ring of a curve and Koszul cohomology; in the same paper, Farkas proves that M22 is of general type as well. At the time of this book’s publication, there has been very interesting progress on the Kodaira dimension of moduli spaces. In [259], Farkas and

§15 Exercises

885

Verra prove that C g,n is unirational for g < 10 and n < g, that C 10,n is uniruled for n = 10, that C 11,n is uniruled for n = 11, that the Kodaira dimension of C 11,11 equals 19, and finally that the Kodaira dimension of C g,g equals 3g − 3 when g ≥ 12. In the same vein, Bini, Fontanari, and Viviani [71] prove that, if (d + g − 1 and 2g − 2 are relatively prime, then the Kodaira dimension of Picdg is equal to −∞ for 4 ≤ g ≤ 9, to 0 for g = 10, to 19 for g = 11, and to 3g − 3 for g ≥ 12. All these results give a new, deeper meaning to the g = 11 threshold. Genus g hyperplane sections of polarized K3 surfaces depend on g + 19 parameters. A naive count of moduli predicts that the general curve of genus g ≤ 11 should be realized as such a hyperplane section. As Mukai shows in [540], this is true, with the exception of the genus 10 case. In any event, for g ≥ 12, a hyperplane section of a K3 surface is far from being a general curve of genus g. On the other hand, in Lazarsfeld’s proof of Petri’s conjecture we saw that hyperplane sections of K3 surfaces, from the point of view of special divisors, behave as curves with general moduli. This idea has been efficiently used by Voisin in her work on Green’s conjecture. Curves on K3 surfaces are also used by Farkas and Popa [257] to give counter-examples to the slope conjecture. Good references and extensive bibliographies on this circle of ideas are again to be found in Voisin [676], Farkas [254], and Aprodu and Farkas [13].

15. Exercises. A. Low genus A-1. Do the Brill–Noether theory for genus g ≤ 5. B. Higher μ’s In this series of exercises we are introducing homomorphisms μk , k ≥ 0, generalizing the homomorphisms μ0 and μ1 . B-1. Let L be a line bundle on a smooth curve C. Consider sections si of L and ri of ωC L−1 . By covering C with coordinate disks Uα , with local coordinate zα in each Uα , write si = {si,α } and ri = {ri,α } with si,α and ri,α holomorphic in Uα and i = 1, . . . , l. Suppose that (15.1)

dh si,α dzαh

ri,α = 0 ,

h = 1, . . . , k − 1 ,

886


for every α. Show that the cochain $

dk si,α i

dzαk

ri,α

4 dzαk+1

is, in fact, a section μk (ψ) ∈ H 0 (C, ω k+1 ) and that one gets an array of homomorphisms H 0 (C, L) ⊗ H 0 (C, ω ⊗ L−1 )

(15.2)

μ0

w H 0 (C, ω)

∪ ker μ0

μ1

w H 0 (C, ω 2 )

∪ ker μ1

μ2

w H 0 (C, ω 3 )

∪ .. .

.. .

B-2. Show that if (15.1) holds, then dk si,α i

dzαk

ri,α = −

dk−1 si,α dri,α i

= (−1)k

dzαk i

si,α

dzα

= ···

dk ri,α . dzαk

B-3. Looking at a suitable Wronskian, show that ker μk = 0 if k ≥ r. B-4. Let Δ : C → C × C be the diagonal morphism and set D = Δ(C). Denote by π1 , π2 the projections of C × C onto the two factors and set M = π1∗ (L) ⊗ π2∗ (ω ⊗ L−1 ). Show that: i) Δ∗ M ω. ii) The homomorphism H 0 (C, L) ⊗ H 0 (C, ω ⊗ L−1 ) ∼ = H 0 (C × C, M ) → H 0 (C, ω) induced by Δ is just μ0 . iii) Show that the map μh can be identified, up to a multiplicative constant, with h+1 H 0 (C × C, M (−hD)) → H 0 (C, Δ∗ (M (−hD))) H 0 (C, ωC ).

§15 Exercises

887

iv) Show that, under the identifications H 0 (C, L) ⊗ H 0 (C, ω ⊗ L−1 ) H 0 (C × C, M ) k+1 H 0 (C, Δ∗ (M (−kD))) H 0 (C, ωC ),

k = 0, 1, 2, . . . ,

ker μh−1 = H 0 (C × C, M (−hD)), and μh is a nonzero multiple of the restriction map Δ∗ : H 0 (C × C, M (−hD)) → H 0 (C, Δ∗ M (−hD)) . B-5. It is classical that curves on C × C can be interpreted as correspondences on C, and the exercises above suggest that we may take this point of view in analyzing the spaces ker μh . To this end, we set C1,p = π1−1 (p) = {p} × C , C2,p = π2−1 (q) = C × {q} and agree to consider divisors on C1,p , C2,q as divisors on C × C. Consider a curve T ∈ |M (−(k + 1)D)|.

(15.3)

Show that, for general p and q, we have T · C1,p ∈ |L(−(k + 1)p)| , (15.4) T · C2,q ∈ |ωL−1 (−(k + 1)q)|. B-6. Show that the mapping & k+2 & & P(ker μk ) → &ωC induced by μk+1 sends each correspondence T satisfying (15.3) to the cycle of its united points, i.e., points p such that p ∈ T · C1,p . B-7. Consider a base-point-free pencil |L| corresponding to a branched Define the symmetric correspondence covering f : C → P1 . E = {(p, q) : q ∈ f −1 (f (p)) − p} . i) Show that T =E+

C1,pi

i

is a sum of E plus “vertical” fibers.

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ii) From the second condition in (15.4) we obtain pi ∈ |ωL−2 | . (15.5) i

Using exercise B-3 for r = 1, show that each member of the linear system |M (−D)| consists of the fixed curve E plus the variable curve C1,pi , where (15.5) is satisfied. Deduce that i

ker μ0 H 0 (C, ωL−2 ) . B-8. Now suppose |L| is a base-point-free linear system of dimension 2 and degree d, and let f : C → P2 be the corresponding morphism. i) Assume, for the moment, that f is an immersion. The exercise will consist in showing that ker μ1 = 0 . Argue by contradiction and let T ∈ |M (−2D)|. The first of the two equations (15.4) reads T · C1,p ∈ |L(−2p)| . Define the tangential correspondence E = {(p, q) : q ∈ f −1 (p · f (C)) − 2p} . Show, as before, that T =E+

C1,pi .

i

The inverse correspondence to E is given by E −1 (q) = {p ∈ C : q ∈ f −1 (p · f (C)) − 2p} = ramification divisor of the composition of f with the projection from q. Use the Riemann–Hurwitz formula, E −1 (q) ∈ |ωL2 (−2q)| , and the second condition in (15.4) to (15.6) pi ∈ |L−3 | , i

which is absurd.

§15 Exercises

889

ii) Now, suppose that f is birational onto its image but is not necessarily an immersion. Let Z be the divisor of zeros of the differential of f . Show that (15.6) has to be replaced by

pi ∈ |L−3 (Z)|.

This is still absurd when (15.7)

deg(Z) < 3d . Conclude that if (15.7) holds, Wd2 is smooth and of dimension g + 3d − 9 at L → C.

C. Curves on quadrics In this series of exercises we will give examples of linear systems for which ker μ1 is not zero. We shall use the notation α = codimension of Wdr in P icd at L → C, β = h0 (C, L) · h1 (C, L), γ = dim(Image μ), so that β = “expected” codimension of Wdr at L → C, γ = codimension of the Zariski tangent space to Wdr at L → C. We have the following obvious inequalities: (15.8)

β ≥α≥γ.

We shall consider smooth curves of type (m, n) on a smooth quadric Q ⊂ P3 , i.e., curves C belonging to the linear system |mL1 + nL2 |, where L1 and L2 are lines in the two rulings of Q. C-1. Show that the genus g and the degree d of C are d = m + n, g = (m − 1)(n − 1) , and that ωC ∼ = OQ ((m − 2)L1 + (n − 2)L2 ) ⊗ OC . C-2. From now on, we shall assume that m, n ≥ 3. Show that (15.9)

α ≤ 3(m − 1)(n − 1) − 2m − 2n + 4 ,

890


the inequality being due to the fact that there might be curves of degree d and genus g not lying on a quadric. C-3. Show that E = {(p, q) : q ∈ Pp · C − 2p} is a symmetric correspondence on C × C and that E ∈ |π1∗ L ⊗ π2∗ L(−2D)|. On the other hand, |M (−2D)| = |π1∗ L ⊗ π2∗ (ωL−1 )(−2D)|. Moreover, D is a fixed component of |M (−2D)|. It follows that, for m, n ≥ 3, dim(ker μ1 ) = h0 (C, ωL−2 )∨ = h0 (C, OQ ((m − 4)L1 +(n − 4)L2 ) ⊗ OC) = (m − 3)(n − 3). We may “explain” this equality as follows. Our curve C on Q may 1 and a gn1 . The be considered as an abstract curve having a gm infinitesimal conditions imposed on the moduli of C to keep both these pencils are given by the vector space of quadratic differentials vanishing on the sum of the ramification divisors of the branched 1 and to the gn1 . This is just coverings of P1 corresponding to the gm 0 −2 H (C, ωL ). C-4. Show that (15.10)

γ = β − (m − 3)(n − 3)

and β = 4(g − d + 3) = 4(m − 1)(n − 1) − 4m − 4n + 12 . Using (15.8), (15.9), and (15.10), conclude that γ =α≤β, α> d, and g >> md, study the subvariety m {(C, Lm ) | (C, L) ∈ Wd1 } ⊂ Wdm .

r ’s. E. Some Mg,d

In this sequence of exercises we will study some of the subvarieties r 2 . In exercises 1–12 we consider the locus M10,6 ⊂ M10 of genus 10 Mg,d 2 curves having a g6 .

892


E-1. Let C be a curve of genus 10 and D a g62 on C. Show that if φD : C → P2 is birational onto its image, then D is base-point-free, and C = φD (C) is a smooth plane sextic. Show also that if φD fails to be birational, then either i) D is base-point-free, and φD is 2-1 onto a cubic curve; ii) D is base-point-free, and φD is 3-1 onto a conic; iii) D has two base points, and φD is 2-1 onto a conic; 2 E-2. Conclude that M10,6 is the union of the loci Σ1 = {smooth plane sextics}, Σ2 = {trigonal curveds}, Σ3 = {hyperelliptic curves}, Σ4 = {elliptic-hyperelliptic curves}.

E-3. Show that the loci Σ2 , Σ3 , and Σ4 are all disjoint. Show also that Σ1 is disjoint from Σ2 and Σ3 , and that Σ1 is disjoint from Σ4 . E-4. Show that the dimension of the loci Σi are: dim Σ1 = 19, dim Σ2 = 21, dim Σ3 = 19, dim Σ4 = 18. E-5. Show that the loci Σ3 and Σ4 are closed, while Σ2 = Σ2 ∪ Σ3 . E-6. Show that the closure Σ1 of Σ1 contains some but not all trigonal curves. Can you say explicitely which trigonal curves lie in the closure of Σ1 ? That is, describe Σ1 ∩ Σ2 . (Answer (due to Lazarsfeld): Σ1 ∩ Σ2 consists of those curves C with a g31 E such that O(6E) = ωC .) E-7. Show similarly that Σ1 meets Σ3 but does not contain it. Can you describe the locus Σ1 ∩ Σ3 ? E-8. Show that Σ1 ⊃ Σ4 . 2 has exactly two irreducible E-9. Conclude from the above that M10,6 components Σ1 and Σ2 . Note that for the latter of these two components, the g62 on a general curve is not birational, while on the smaller component it is. 2 . We now turn our attention to the variety W26 = W10,6 2 E-10. Show that the fiber of the map W26 → M10,6 over a point C ∈ Σ1 ∪ Σ2 is just one point; while for C ∈ Σ3 , it is twodimensional, and for C ∈ Σ4 , it is one-dimensional.

E-11. Letting Ωi be the inverse image of σi in W26 , conclude that dim Ω1 = 19, dim Ω2 = 21,

§15 Exercises

893

dim Ω3 = 21, dim Ω4 = 19, and hence that the closures Ωi are all irreducible components of 2 . W10,6 E-12. Show that Ω1 ∩ Ω4 = {(C, L) : C ∈ Σ4 , L3 ∼ = ωC } and Ω2 ∩ Ω3 = {(C, L) : |L| = 2g21 + 2p} . To summarize some of the conclusions of the above exercises, 2 (15.12) M10,6

has 2 irreducible components, of dimesions 19 and 21 ,

(15.13) 2 has 4 irreducible components, of dimesions 19, 21, 21, and 19 . W10,6 3 3 We now turn our attention on M10,9 and W93 = W10,9 .

E-13. Suppose that C is a curve of genus 10 and D a g93 on C. Assuming that φD is birational, show that the image curve C0 = πD (C) is either: i) a smooth curve of type (3, 6) on a quadric Q ⊂ P3 , ii) a curve of type (3, 4) on a quadric Q ⊂ P3 , having two nodes (or a specialization thereof). iii) the smooth complete intersection of two cubic surfaces in P3 . (Hint: compute h0 (C, O(3)) to conclude that C0 lies on at least two cubic surfaces; ask whether C0 lies on a quadric) E-14. Suppose now that φD fails to be birational. Show that if C is nonhyperelliptic, then either i) D is base-point-free, and φD is 3-1 onto a twisted cubic; or ii) D has one base-point, and φD is 2-1 onto an aliptic quartic curve. Show that if C is hyperelliptic, then φD is 2-1 onto either a twisted cubic, a rational quartic (i.e., a curve of type (3, 1) on a quadric), or a singular curve of type (2, 2) on a quadric if D is incomplete. 3 E-15. Conclude from the above that the locus M10,9 is the union of the loci Σ2 = {trigonal curves}, Σ3 = {hyperelliptic curves}, Σ4 = {elliptic-hyperellipticcurves}, Σ5 = {curves of type (3, 6) on a quadric}, Σ6 = {curves of type (4, 5) on a quadric}, Σ7 = {complete intersections of two cubics in P3 }.

894


(The notation is chosen to be consistent with the definitions of 2 .) Σi ⊂ M10,6 E-16. Count dimensions to show that dim Σ2 = 21, dim Σ3 = 19, dim Σ4 = 18, dim Σ5 = 21, dim Σ6 = 21, dim Σ7 = 21. E-17. Show that if C is a general trigonal curve and |D| the trigonal series on C, then O(6D) = ωC . Show that r(ω(−4D)) ≥ 1 and that the map φ = φD × φω(−4D) : C → P1 × P1 = Q embeds C as a curve of type (3, 6) on a quadric. Conclude that Σ5 is a dense open set in Σ2 . E-18. Show that the remaining loci Σi are disjoint. E-19. Show that a curve C of genus 10 with a g93 , denoted by D, with D ⊂ |D|, of type E-13, iii) above, then ωC = OC (2D), that if (C, D) is of type E-13, ii), then ωC = OC (2D) if and only if the two nodes of C0 lie on the same line of the ruling of Q meeting C0 four times, and that if (C, D) is of type E-13, i), that is, C is trigonal with a g31 , denoted by E, then ωC = OC (2D) if and only if ωC = OC (6E). E-20. Show that hyperelliptic curves and elliptic–hyperelliptic curves of genus 10 both possess semicanonical gd3 ’s. Notation (15.14). We denote by T hrg the locus in Mg of curves possessing a theta characteristic of dimension r. Thus, there is a divisor D on C with . T hrg = [C] ∈ Mg : OC (2D) ∼ = ωC and dim |D| ≥ r + 1 E-21. Conclude from the previous two exercises that T h310 = Σ7 and in particular that Σ3 ∪ Σ4 ⊂ Σ7 . Also use this to describe the intersections Σ2 ∩ Σ7 and Σ6 ∩ Σ7 . 3 E-22. Conclude from he above that M10,9 has exactly three irreducible components: Σ3 = Σ5 , Σ6 , and Σ7 .

E-23. Suppose that C ∈ Σ2 and that |D| is the g31 on C. Show that W93 (C) = {u(3D)} ∪ {u(ω(−3D))}. E-24. Show that if C ∈ Σ6 , then W93 (C) consists of exactly two points.

§15 Exercises

895

3 of Σi , show that E-25. If Ωi is the inverse image in W93 = W10,9 dim Ω2 = 21, dim Ω3 = 22, dim Ω4 = 20, dim Ω6 = 21, dim Ω7 = 21.

Show moreover that Ω2 has exactly two irreducible components, Ω 2 and Ω 2 , while the other Ωi are irreducible. E-26. Conclude that Ω4 ⊂ Ω7 (and hence an alternative proof that Σ4 ⊂ Σ7 ). E-27. Combining the above exercises, show that, in genus 10, the variety W93 has five irreducible components Ω 2 , Ω 2 , Ω3 , Ω 6 , and Ω7 , whose general points are given, respectively, as pairs i) (C, L) : C has a g31 , |D|, and L = OC (3D); ii) (C, L) : C has a g31 , |D|, and L = ωC (−3D); iii) (C, L) : C has a g21 , |D|, and L = OC (3D + p1 + p2 + p3 ); iv) (C, L) : φL is of type E-13, ii); v) (C, L) : φL is of type E-13, iii). Conclude, in summary, that (15.15) In genus 10, W93 has 5 components of which: 3 - two map birationally to the same component of M10,9 , 3 - one maps 2-1 onto another component of M10,9 , 3 , - one maps birationally to a thin component of M10,9 - two map birationally, with fiber dimensions 2 and 3, to subvarieties 3 of the third component of M10,9 .

Finally, following the notation in Section 2, Chapter III, we let r of extremal curves. g = π(d, r) and consider the locus Σrd ⊂ Mg,d E-28. In case r = 3, show that dim Σrd = g + 2d − 7 and that Σrd = is irreducible. (Note: this requires a little argument in case d is odd.) Observe that for d ≥ 8, this dimension count violates the naive estimate. r E-29. In the case d = 8, g = 9, show that Mg,d is irreducible but that r Wd has three components of dimensions 17, 18, and 19.

E-30. Show that if r ≥ 4, then Σrd is irreducible of dimension dim Σrd =

(m + 1)(m + 2) (r − 1) + (m + 2)(r − 3 − ε) − 7 . 2

896


E-31. Continuing the preceding exercise, in the case ε = 0, show that Σrd has exactly two irreducible components Σ1 and Σ2 of dimensions dim Σ1 =

(m + 1)(m + 2) (r − 1) + (m + 2)(r − 3) − 7 , 2

dim Σ2 =

m(m + 1) (r − 1) + 2(m + 2) − 7 . 2

E-32. In the case r = 4, d = 10 what is the intersection of Σ1 and Σ2 ? What is it in general?

F. Exercises on theta-characteristics and μ0 . In this exercises we will use the terminology from Appendix B of Chapter VI. ∼ ωL−1 ) with h0 (C, L) ≥ 3. F-1. Let L be a theta-characteristic (i.e., L = 2

Considering Λ = ∧H 0 (C, L) as a subspace of H 0 (C, L)⊗H 0 (C, ωL−1 ), show that Λ ⊂ ker μ0 and Λ ∩ ker μ1 = 0. Show that T h2g has codimension 3 in Mg . F-2. Show that for g = 7, 8, the locus T h2g is exactly the locus of hyperelliptic curves of genus g, and so T h27 has codimension 5, and T h28 has codimension 6. F-3. Let C be a curve of genus 9 with a g83 denoted by D, and let D ⊂ |D|. Show that if φD is birational, the φD embeds C in P3 as the intersection of a quadric and a quartic surface, and hence that ωC ∼ = OC (2D). F-4. Show that if φD fails to be birational, then φD either maps C in a 2-1 fashion onto a quartic or onto a twisted cubic. Conclude that C is either hyperelliptic or elliptic–hyperelliptic. F-5. Show that the loci Γ1 = {hyperelliptic curves}, Γ2 = {elliptic-hyperelliptic curves}, Γ3 = {intersections of a quadric and a quartic in P3 } in Mg have dimensions 17, 16, and 18 respectively. Conclude that T h39 = Γ3 and in particular that T h39 is irreducible of codimension 6 in Mg . F-6. Observe that Γ1 , Γ2 ⊂ Γ3 . Can you describe explicitly a family of curves Cλ in Γ3 specializing to a hyperelliptic curve C0 in Γ1 ? F-7. Show that in genus 10, T h310 is exactly the closure of the locus of intersections of pairs of cubic surfaces in P3 . Conclude that T h310 is irreducible of codimension 6 in M10 .

§15 Exercises

897

F-8. Let Γ ⊂ M11 be S ⊂ P3 and such are skew lines in codimension 6 in

the locus of curves C lying on a quartic surface that OS (C) = OS (2)(L1 + L2 ), where L1 and L2 P3 lying on S. Show that T h311 is irreducible of M11 .

F-9. It may be conjectured, on the basis of the examples above, that for g ≥ 8, T h3g has codimension 6 in Mg ; and more generally that in for any r and g >> 0, the subvariety T hrg has codimenion r(r+1) 2 Mg . Is this true?

G. Miscellaneous exercises on the normal sheaf. G-1. Let us go back to Exercises C-1–C-4 and compute directly the dimension of the kernel of μ1 for a smooth curve C of type (m, n), m, n ≥ 3, on a smooth quadric Q in P3 . i) Show that there is an exact sequence 0 → A → N → B → 0, where A = OC (mL1 + nL2 ) ,

B = OC (2L1 + 2L2 ) .

ii) Show that h1 (C, A) = 0. iii) Show that dim ker μ1 = h1 (C, N ) = h1 (C, B) = h0 (Q, O((m − 4)L1 + (n − 4)L2 )) = (m − 3)(n − 3) . This is in accordance with Petri’s statement, together with the well-known fact that the only smooth curves of genus g ≥ 3 with general moduli lying on a smooth quadric in P3 are curves of type (3, 3). G-2. Reconsider Exercise B-8 from the point of view of the normal bundle. First of all, let φ : C → P2 be a morphism corresponding to a gd2 on C which is not composed with an involution. Denote by L the line bundle C corresponding to the gd2 . Show that (15.16)

3d > deg Z

=⇒ H 1 (C, Nφ ) = 0 .

898


In particular, if one wishes to prove directly Petri’s statement (8.13) for gd2 ’s, it suffices to show that (15.16) always holds when C is general; that is, it suffices to show that any plane model of a general curve of genus g does not have too many cusps (cf. (9.16)). G-3. This exercise proves Petri’s statement (8.13) when r = 2. Let C be a general curve of genus g. Consider a gd2 on C corresponding to a nondegenerate morphism φ : C → P2 . Assume that H 1 (C, Nφ ) = 0 . Construct a deformation φ

C

w P2

p u S inducing a commutative diagram ρs

Ts (S)[

[ ] h [

w H 1 (C, TC )

δ

0

H (C, Nφ ) with ρ surjective. Use Proposition (9.10) to reach a contradiction. G-4. It is natural to ask if the arguments used in Exercise G-3 can be generalized to the case r ≥ 3 The answer, as matters stand, is no. By squeezing hard the arguments used in Exercise G-3, show that, given a nondegenerate morphism φ : C → P3 with C general, one has h1 (C, Nφ ) ≤ 1. G-5. Looking at smooth curves of type (m, n) on a smooth quadric Q, show that the analogue of Theorem (10.1) for Wdr is definitely false if r > 2. G-6. Complete the proof of (10.14): r even and r ≥ 4. G-7. Construct singular plane curves with nonzero μ1 . H. Mumford’s example revisited We go back to Mumford’s example in Section 6 of Chapter IX. Keeping the notation of that section, we study curves lying on a smooth

§15 Exercises

899

cubic surface F ⊂ P3 . We denote by L one of the 27 lines on F and by X the hyperplane section, we fix integers n and m such that n − 1 > m ≥ 2, and we let C be a general (smooth) member of |X n Lm | of degree d = 3n + m and genus g = 3(n2 − n)/2 − (m2 − m)/2 + nm + 1. Finally, we let N denote the normal bundle of C in P3 . H-1. i) Show that there is an exact sequence 0 → A → N → B → 0, 3 where A = H n Lm |C and B = H|C . 1 ii) Show that h (C, A) = 0 and H 1 (C, N ) ∼ = H 1 (C, B). 1 2 3−n −m L ) = h0 (F, X n−4 Lm ). iii) Show that h (C, N ) = h (F, X 1 Deduce that h (C, N ) is never zero and is equal to one only when n = 4, m = 2.

H-2. Show that h1 (C, N ) = g − 3d + 18 if n ≥ m + 3 and h1 (C, N ) = g − 3d + 19 if n = m + 2. H-3. Show that h0 (C, X|C ) = 3, so that X|C determines a point w ∈ Wd3 Wd4 . Set α = codimension of Wd3 in Picd at w , β = expected dimension of Wd3 = 4g − 4d + 12 , γ = codimension of Tw (Wd3 ) in Tw (Picd ) , α = codimension in Picd of the subvariety corresponding to the family of all smooth curves belonging to |H n Lm | for some cubic surface F, and recall that γ = β − h1 (C, N ). H-4. Show that

γ=

3g − d − 6 if n ≥ m + 3 ,

3g − d − 7 if n = m + 2 , β = 4g − 4d + 12 , α = 3g − d − 6 . In particular, β > α with the only exception β = α in the case n = 4, m = 2. H-5. Now restrict to Mumford’s case: n = 4, m = 2. Show that when 3 whose codimension in Picd is g = 24, there is a component Z of W14 3 4 , the codimension of Tw (Z) 52, whereas, for any point w ∈ W14 W14 d in Tw (Pic ) is equal 51. This means that Z is nowhere reduced.

900


In theses two series of exercises, we will give another approach to the proof of the unirationality of the Hurwitz spaces H(3, b) and H(4, w). I. Unirationality of H(3, w) I-1. Let C be a smooth curve of genus g, and f : C → P1 a map of degree 3. Show that we have an inclusion 0 → OP1 → f∗ OC and that the quotient sheaf E is locally free of rank 2. moreover that c1 (E) = −g − 2.

Show

I-2. Show that we have a natural map α : C → PE and that this map is in fact an embedding—in other words, every smooth trigonal curve may be embedded in a Hirzebruch surface Fn for some n ≡ g (mod 2). I-3. If the surface PE in the preceding problem is the Hirzebruch surface Fn , show that the image α(C) ⊂ PE is a divisor of class C ∼ 3e0 +

g − 3n + 2 g + 3n + 2 f = 3e∞ + f, 2 2

where f is the class of a fiber of PE → P1 , and e0 and e∞ are the classes of sections of PE → P1 of self-intersection n and −n, respectively. I-4. Now suppose the map f : C → P1 above is a general 3-sheeted cover of genus g (equivalently, if g ≥ 4, the curve C is a general trigonal curve of genus g). Show that PE ∼ = F1 = F0 if g is even and PE ∼ if g is odd. Show moreover that the locus of those simple covers for which PE ∼ = Fn with n > 1 is codimension n − 1 in the Hurwitz space H(3, 2g + 4) (if nonempty). I-5. Now let PN be the projective space parameterizing the linear system |3e0 + mf | on F0 (respectively, F1 ). Show that a general point in PN corresponds to a smooth curve of genus g and deduce from the above that we have a dominant rational map PN → H(3, 4m) (respectively, H(3, 4m + 6)) and hence that H(3, b) is unirational for any (even) b. J. Unirationality of H(4, w) Here we give another approach to the unirationality of the Hurwitz spaces H(4, b). The analysis is similar to that of the preceding series of

§15 Exercises

901

exercises; this time we realize a general tetragonal curve as a complete intersection in a rational threefold. J-1. Let C be a smooth curve of genus g, and f : C → P1 a map of degree 4. As before, we have an inclusion 0 → OP1 → f∗ OC ; show that the quotient sheaf E is locally free of rank 3 and c1 (E) = −g − 3. Show as well that the natural map α : C → PE is in fact an embedding. J-2. For the time being, we will restrict ourselves to the case where g is divisible by 3 and write g = 3k. In this case, show that if the map f : C → P1 above is a general 4-sheeted cover of genus g, then E ∼ = OP1 (−k − 1)⊕3 , and in particular

PE ∼ = P1 × P2 .

J-3. Continuing in this case (in particular, continuing under the assumption that f : C → P1 is general), let ω and η ∈ A1 (P1 × P2 ) denote the pullbacks to P1 × P2 of the hyperplane classes on P2 and P1 , respectively; let E and D denote the corresponding divisors restricted to C. Show that E = ωC − (k − 1)D. Deduce in particular that the degree of E is 2k + 2 and hence that the class of the curve C in P1 × P2 is [C] = 4ω 2 + (2k + 2)ωη. J-4. Now, suppose further that g ≡ 3 mod 6, so that k is odd; write k + 1 as 2. Compare the dimensions of the linear series |OP1 ×P2 (, 2)| and its restriction |OC (D + 2E)| to show that the curve C ⊂ P1 × P2 lies on a pencil of divisors of type (, 2) on P1 × P2 ; deduce moreover that C is the base locus of this pencil. J-5. To finish this case, let PN be the projective space parameterizing the linear system |OP1 ×P2 (, 2)|, and G(1, n) the Grassmannian

902


parameterizing lines in PN . Show that for a general pencil {Xλ } ⊂ |OP1 ×P2 (, 2)|, the base locus ∩Xλ is a smooth curve of genus g and deduce from the above that we have a dominant rational map G(1, n) → H(4, 2g + 6) and hence that H(4, b) is unirational for any b ≡ 0 (mod 12). J-6. Suppose now that g ≡ 0 (mod 6), so that k is even; in this case write k = 2. Show, analogously to Exercise 4, that the curve C ⊂ P1 × P2 is a complete intersection of divisors of type (, 2) and ( + 1, 2) on P1 ×P2 ; and deduce similarly that H(4, b) is unirational for any b ≡ 6 (mod 12). J-7. Finally, in case g ≡ 1 or 2 (mod 3) and f : C → P1 is general, describe the vector bundle E and corresponding P2 -bundle PE; show that the curve C ⊂ PE is a complete intersection of two divisors (as in the case g ≡ 0 (mod 3), the classes of these divisors may depend on the class of g (mod 6)), and deduce that H(4, b) is unirational for all remaining congruence classes of b (mod 12).

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Index

G-covers, 504, 525 admissible, family of, 527 automorphism of, 526 limits of, 526 G-linearization, 340 V -cover, 274 Γ-marking, 315 weak, 314 μ1 map, 808 ω-coordinate, 463 ω-geodesic, 473 ball, 476 horizontal, 473 ray, 474 vertical, 473 ω-length, 473 ω-metric, 473 2-category, 280 Abel–Jacobi map, 446 relative, 790 Abikoff, William, 498 Abramovich, Dan, 562, 881, 883 Adapted charts, 56 functions, 56 metric, 57 metric on a C m family of vector bundles, 215 partition of unity, 57 relative form, 57 section, 57 Additivity property of the κ1 class, 377, 427 of the Hodge class, 365, 427 Adem, Alejandro, 323 Adjunction isomorphism for Deligne pairing, 375 Admissible G-cover, 504, 525, 556

G cover of a stable pointed curve, 556 Beltrami differential, 466 covers, family of, 526 quasi-diffeomorphism, 468 Ahlfors, Lars, 498 Alexeev, Valery, 879 Algebraic Index Theorem, 422 Algebraic space, 251, 270, 307 groupoid presentation of, 306, 307 normalization of, 308 separated, 270 Altman, Allen, 879 Ample cone of Mg , 439 locally free sheaf, 229 Ampleness Nakai’s criterion of, 424 of bλ − δ + ψ,425 of κ1 + aλ + bi ψi , 435 of κ1 + aλ, 425 of Mumford’s class κ1 , 398, 425 of the relative dualizing sheaf ωf , 424 Seshadri’s criterion of, 230, 426 Andreotti, Aldo, 248 Aprodu, Marian, 883, 885 Arakelov, Suren Ju., 424, 435, 438 Arbarello, Enrico, 397, 879, 884 Arc complex, 609, 613 Arrows of a Lie groupoid, 275 Arsie, Alessandro, 397 Artin, Michael, 323 Asymptotic expansion, 736 in more than one variable, 739 of the partition function, 744 Atiyah, Michael, 769

E. Arbarello et al., Geometry of Algebraic Curves, Grundlehren der mathematischen Wissenschaften 268, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-540-69392-5,

946 Atlas and descent data, 329, 337 for an algebraic space, 251, 270 for an orbifold, 277 for a Deligne–Mumford stack, 300 relative C m , 56 Automorphisms of admissible G-covers, 536 Average or expectation value, 734 stretching of a quasidiffeomorphism, 480 Axis of a hyperbolic transformation, 631 Baer, R., 497 Baily, Walter L., Jr., 438 Ballico, Edoardo, 881 ´ Barja, Miguel Angel, 438 Barth, Wolf, 161 Base change and ampleness, 231 and stable reduction, 105 compatibility of Deligne pairing with, 331, 369, 371 compatibility of Hilbert scheme with, 46 compatibility of Hodge line bundle and points bundles with, 344 compatibility of Mumford’s class κ1 with, 377 compatibility of relative dualizing sheaf with, 98 compatibility of the boundary divisor with, 363 compatibility of the determinant of the cohomology with, 358 compatibility of Riemann–Roch isomorphism with, 379 faithfully flat, 292 in cohomology, 1, 8, 12, 13, 121, 388, 788

Index

property stable under, 300 Beauville, Arnaud, 880, 883 Behrend, Kai, 769 Beilinson, Alexander A., 397 Beltrami differential, 466 equation, 445, 466 Benedetti, Riccardo, 665 Bernoulli number, 585, 751, 765 Bers, Lipman, 497, 498, 665 Bertram, Aaron, 605, 884 Bessis, Daniel, 771 Bini, Gilberto, 773, 885 Bipartition of a pair (integer, finite set), 95, 100 stable, 100, 261, 312, 339, 571 Birman, Joan, 497 Biswas, Indranil, 397 Blow-up and stable reduction, 106 of CN , real oriented, 488 real oriented, 149 Boggi, Marco, 562 Borel, Armand, 683 Bost, Jean-Benoˆıt, 438 Bott, Raoul, 769 Boundary of moduli, as a determinant, 361 of moduli space, 81, 261, 279 of moduli space; irreducible components of, 262 pullback under clutching, 347 Boundary class, 339, 571, 676, 717 for the moduli stack of stable hyperelliptic curves, 391 Boundary divisor, 261, 262, 312, 313, 339 as a determinant 331, 361 contribution from, in Witten’s conjecture, 724 in Mumford’s formula for the canonical class of moduli space, 386

Index

pullback of, under clutching, 583, 584 of M 0,P , 599, 601–604, 608 Boundary strata of moduli space of stable curves, 312, 321 of moduli stack of stable curves, 312 pullback of, under clutching, 582 Bowditch, Brian H., 665 ´ Brézin, Edouard, 772 Brill, Alexander von, 779, 883 Brill–Noether matrix, 790 number, 779, 795, 808, 813, 827, 869 subloci of Mg , 793 theory, 779 theory, dimension theorem, 835 varieties, 780, 788–793 varieties, tangent spaces to, 807 Bruno, Andrea, 884 Bryan, Jim, 605 Brylinski, Jean-Luc, 562 Canonaco, Alberto, 323 Canonical class of M g,n , 386 of Mg,n , 386 of Mg,P , 332, 344 Caporaso, Lucia, 879, 881 Cartan, Henri, 209, 257 Cartesian morphism, 280 Castelnuovo, Guido, 851, 864, 882 Category fibered in groupoids, 279, 294, 332, 335 Catenacci, Roberto, 397 Cattani, Eduardo, 594 Cavalieri, Renzo, 605, 884 Cellular decomposition of Teichm¨ uller space, 609, 614, 623, 643, 690 and combinatorial expression

947 for ψ-classes, 694 and vanishing theorems for homology of moduli spaces, 671 extension to bordification, 614, 652 Chang, Mei-Chu, 884 Characteristic exterior homomorphism, 510 homomorphism, 821 subgroup, 510 Characteristic linear system, 3, 32, 65, 243 Characteristic map, 32, 243 Chasles, Michel, 766 Chern character, 382, 586 Chern classes of the boundary divisors, 339, 571, 676, 717 of the Hodge bundle, 334, 572 of the point bundles, 335, 572, 694, 717 of the sheaf of relative K¨ ahler differentials, 383 Chern, Shiing-Shen, 497 Chisini, Oscar, 883 Chow ring Gorenstein conjectures, 597 of a moduli stack, 570 of a quotient of a smooth variety by a finite group, 570 of M g , 565, 570, 605 of M 0,P , 599 Chow variety, 70 open, 70 Ciliberto, Ciro, 880 Classes boundary, 339, 391, 396, 571, 602, 676, 678, 710, 713, 717, 721 Mumford’s, 332, 572, 721 Mumford–Morita–Miller, 572, 721 point-bundle, 335, 572, 717

948 Classes (cont.) tautological, 382, 384, 565, 570, 572, 573, 581, 596, 604, 669, 676, 680, 710, 713, 717, 721 Cleavage, 281 Clebsch, Alfred, 854, 883 Clemens, C. Herbert, 161, 880 Clutching, 81, 126, 187, 254, 311– 323, 330, 345, 396, 565, 570, 581–585, 589, 752 Codimension of a regular embedding, 36 Coherent topology, 615 Cohomology base change in. See Base change determinant of the. See Determinant equivariant, 754–759 of moduli spaces, 445, 485, 565, 599, 668, 670–689, 708, 710 of orbifolds, 278 rational, of Γg , 82 PL, 696 Collar Lemma, 635 Colombo, Elisabetta, 883 Commutative diagram in a category fibered in groupoids, 280 Composition in a Lie groupoid, 275 Connection, 224 compatible with hermitian product, 225 Gauss–Manin, 220, 593 hermitian, 225 Conormal sheaf, 31 Continuous system of plane curves, 847 Contraction of a graph, 314 Contraction functor, 125 Coolidge, Julian L., 65 Coppens, Marc, 883 Cornalba, Maurizio, 397, 438 Corti, Alessio, 562, 881, 883

Index

Cukierman, Fernando, 881, 882 Curvature form of a connection, 224 Curve hyperelliptic stable, 101, 192 nodal, 83 nodal n-pointed, 94 nodal P -pointed, 94 nodal, with marked points, 92 of compact type, 90 semistable, 100 stable, 99 Cusp, 630 Cycle rings of moduli stacks of curves, 570 D’Souza, Cyril, 879 Date, Etsur¯ o, 773 De Concini, Corrado, 397 de Franchis’ theorem, 830 de Franchis, Michele, 780, 882 de Jong, Aise Johan, 161, 562 de Jonquières, Ernest, 766, 768 de Rham complex, 591 Deformation continuous, of a compact complex manifold, 213 differentiable, of a compact complex manifold, 213 first-order, 172 first order embedded, 27, 42 first order, of a morphism, 819, 836 first order, of a pair (curve, line bundle), 803 first order, of an admissible G-cover, 557 infinitesimal, 167–171, 197, 201, 242, 769, 835 of a morphism, 819 of an analytic space, 172 of a nodal curve, 178 of a scheme, 172 Dehn twist, 82, 145–158, 445, 460, 483, 491, 493, 535

949

Index

Dehn, Max, 497 Dehn–Nielsen realization, 443 theorem, 454, 459 Deligne pairing, 367, 369 as a product of determinants, 371 Deligne, Pierre, 323, 396, 397, 562, 604, 669, 674, 675, 686, 709 Deligne–Gysin spectral sequence, 669, 685 Descent construction of the stack [X/G], 297 data, 89, 253, 289, 294 data defining line bundles on moduli stacks, 336–343 data, effective, 295 faithfully flat, for quasicoherent sheaves, 288–294 theory, 253, 323 Determinant boundary of moduli as, 361 Hodge line bundle as, 355, 357, 359 of a finite complex, 330, 350, 703 of a vector bundle as a Z/2graded line bundle, 348 of the cohomology, 330, 354, 357, 396 of the hypercohomology, 331, 357 Determinantal curve, 75 variety, generic, 792 Di Francesco, Philippe, 720, 745, 771 Diaz, Steven, 566, 598, 882, 883 Dickey, Leonid A., 773 Dijkgraaf, Robbert, 726, 772 Dilatation, 469 minimal, 469 Dilaton equation, 574, 723 Dimension of Brill–Noether varieties, expected, 795

of Gd2 , 846 1 for 2 ≤ d ≤ g/2 + 1, of Mg,d 864 1 , expected, 813 of Mg,d of the Severi variety, 847 of the Hilbert scheme, lower bound on, 33, 54 of Wd1 , 811 Divisor admissible, 356–363 boundary. See Boundary divisor Cartier, 123, 329, 335, 339, 356, 366, 422, 783 class 365, 391, 599, 606 effective, 177, 361, 367, 373, 387, 435, 788, 818 nef, 426, 433–438 of sections of a family of curves, 95 relative, 243, 367, 371, 375– 377, 785, 800 theory of characteristic system for, 243 universal, 243, 784, 789 universal, effective, 784 with normal crossings, 106, 149, 152, 161, 279, 487, 669, 685, 709 zero, 27, 36, 98, 131 Dolgachev, Igor, 437 Douglas, Michael R., 772 Dualizing sheaf, 90, 97, 101 logaritmic, relative, 377, 572 relative, 97, 572 relative, direct image of, 234, 334 relative, nefness of, 435 relative, positivity properties of, 417–421, 424 Edge disconnecting, 95 nondisconnecting, 95 of a graph, 93

950 Edidin, Dan, 323, 709, 881, 883 Eguchi, Tohru, 770 Eisenbud, David, 439, 843, 873, 880–884 Ekedahl, Torsten, 685, 771, 884 Eliashberg, Yakov, 685 Enriques, Federigo, 65, 859, 883 Epstein, David B. A., 497, 665 Equivalence λ-, of stable curves, 436 of categories, 280, 282, 284, 289, 337 of deformations, 172 of deformations of n-pointed curves, 176 Equivalence relation in the context of groupoids, 276 quotient of, 270 quotient of, Grothendieck’s theorem, 784 relation defining Deligne pairing, 367 schematic, 268 Esteves, Eduardo, 879, 881, 882 Euler sequence, 35, 197, 813, 822 Euler–Poincaré characteristic, 63, 361, 382, 527 virtual, 693, 721, 754, 758–766, 773, 777 Exceptional chain, 111 divisor, 110, 371, 600, 714, 854, 876 Excess intersection, 321, 330, 346, 396, 582 Expectation value, 734 Expected dimension of Brill–Noether varieties, 795 1 , 813 of Mg,d Exterior differentiation, along the fibers, 219 homomorphism, 454, 501, 508, 509, 514 isomorphism, 455, 459

Index

Faber, Carel, 566, 580, 597, 605, 750, 773 Faithfully flat descent, 288–294 algebra, 87 module, 291 morphism of schemes, 289 Family C m , of compact complex manifolds, 62, 213–216 uller C m , of curves with Teichm¨ structure, 450 C m , of differentiable manifolds, 56 C m , of differentiable vector bundles, 57, 215 C m , of projective varieties, 63 flat, of subschems of PN , 1, 5, 12, 22, 26 isotrivial, of curves, 418 Mumford’s, of curves in P3 , 40–43 of curves on quadrics, 74 of curves with general moduli, 794 of curves with level G structure, 508 of curves with level m structure, 503, 538 of curves with level ψ structure, 511 of curves with Teichm¨ uller structure, 444, 449, 471 of elliptic curves, semistable reduction of, 161 of formally self-adjoint, strongly elliptic differential operators, 215 of Γ-marked stable curves, 315 of hyperelliptic curves, 418, 606 of hypersurfaces in PN , 8 of k-planes, 10 of gdr ’s, 792 of ν-log-canonically embedded curves, 288

Index

of nodal curves, 83 of semistable curves, stable model of, 124 of P -pointed nodal curves, 95, 101 of quadrics in 3-space, 55 of rational normal curves, 73 of semistable curves, 101 of smooth cubics in P3 , 41, 76 of smooth Beltrami differentials, 468, 471 of stable n-pointed curves, 81, 101 of stable curves, isomorphisms of, 113–117 of subschemes in the fibers of a morphism, 43, 53 of subschemes of a scheme, 4 of subschemes of an affine scheme, 66 of zero-dimensional subschemes, 10 universal, on Hilbert scheme, 25 universal, on moduli space, lack of, 266, 267, 283, 286 universal, on moduli space, surrogate for, 267, 307 universal, on moduli stack, 310 universal, on Teichm¨ uller space, 449 Fantechi, Barbara, 248, 323, 769, 884 Farkas, Gavril, 439, 881, 883–885 Farkas, Hershel, 812 Fenchel–Nielsen coordinates, 445, 485, 487, 494, 497 Feynman diagram, 734–744, 776 move, 621, 692, 701 Fiber product of stacks, 299, 303 symmetric, of a family of curves, 242, 784, 797 symmetric, of the universal curve, 675

951 Fiorenza, Domenico, 771 Fitting ideal, 196, 788–790 Flag Hilbert scheme, 48 Flat R-module, 4 coherent sheaf, 5 family of subschemes, 5 morphism, 5 Fogarty, J., 437 Fontanari, Claudio, 883, 885 Ford, Lester R., 665 Formally self-adjoint, strongly elliptic differential operator. See Family of formally selfadjoint, strongly elliptic differential operators Fricke, Robert, 443, 461 Fuchsian group, 627 Fujiki, Akira, 579 Fulton, William, 439, 566, 855, 883 Functor contraction Contr, 125 deformation, 248 essentially surjective, 282 fully faithful, 282 p(t) hilbX/S , 43 Isom, 253, 296 Hilbert, 2, 6, 25 moduli, 285, 504 Picard, 782, 879 projection Pr, 125 projection, for a category fibered in groupoids, 279 representable, 2, 25, 285 represented by Gdr (p), 793 represented by Wdr (p), 789 stable model StMd, 124 Teichm¨ uller, 450 Fundamental region for a Fuchsian group, 629 improper side of, 630 improper vertex of, 629 Funnel, 634

952 G¨ ottsche, Lothar, 323 GAGA, 87, 172 Galatius, Søren, 684, 685 Gardiner, Friederick P., 498 Gatto, Letterio, 881, 882 Gauss–Bonnet, 476, 478, 628, 644 Gaussian measure, 719, 734–742 map, 880 Gelfand–Dikii form of KdV hierarchy, 726 Geodesics for the hyperbolic metric. See Geodesics for the Poincaré metric for the metric induced by a quadratic differential, 473– 479 for the Poincaré metric, 611, 623, 628, 633, 637, 658 Geometric realization of a graph, 93 Gervais, Sylvain, 460, 497 Getzler, Ezra, 566, 605, 769, 773 Ghost components, 649 Gibney, Angela, 439 Gieseker, David, 438, 880 Gillet, Henri, 323 Givental, Alexander B., 769, 773 Gorchinskiy, Sergey, 397 Gorenstein conjecture, 597 Gorenstein graded algebra, 597 Goulden, Ian P., 605, 884 Graber, Tom, 323, 396, 604, 605, 769, 772 Graph P -marked, 93 connected, 93 dual, 88, 90, 93, 126, 160, 311– 323, 545, 548, 555, 582, 648–653, 694 numbered, 93 ribbon. See Ribbon graph semistable, 100 stable, 99

Index

Grauert, Hans, 248 Green operator, 215 Green, Mark, 248, 880, 883, 885 Griffiths, Phillip, 709, 882 Gromov, Mikhail, 766, 884 Gross, David J., 772 Grothendieck Riemann–Roch formula, 382, 585 formula, for the determinant of the cohomology, 379 theorem, 415, 416, 565, 585, 588 Grothendieck, Alexander, 64, 248, 323, 396, 498, 580, 668, 784 Groupoid, 251 complex orbifold, 277 contravariant functor as a, 283 isomorphisms of, 280 Lie, 275 moduli, 286 moduli space as a, 281 morphisms of, 280 orbifold, 276 presentation of a Deligne– Mumford stack, 304 presentation of an algebraic space, 307 proper étale Lie, 276 quotient, 286 represented by a scheme, 283 scheme as a, 253 sections of a, 281 Grzegorczyk, Ivona, 882 Gysin homomorphism, 686–689 Hain, Richard, 605, 685 Half-edge of a graph, 88, 93, 118, 126, 322, 345, 363, 517, 581 of a ribbon graph, 616, 700, 738 Halpern, Noemi, 665 Harer, John, 671, 683, 685, 708, 721, 773 Harish-Chandra, Mehrotra, 746

Index

Harmonic projector, 215 Harris, Joseph, 397, 438, 781, 843, 850, 873, 880–884 Hartshorne, Robin, 27, 64, 90, 248 Hassett, Brendan, 439 Hermitian matrix model, 740 Hilbert functor. See Functor point, 22, 63, 207, 399, 406– 409, 414–416, 430, 438 polynomial, 1, 4–26, 41, 43, 48, 67, 72, 112, 195 Hilbert scheme, 2, 6, 25, 43, 46 and base change, 46 of morphisms, 47 of isomorphisms, 3, 48 flag, 48 non-reduced, 40 of complete intersections, 73 of curves on quadrics, 74 and determinantal curves, 75 of k-planes in Pr , 10 projectivity of, 26 quasi-complete intersections, 75 sections of, 73 tangent space to, 33, 49–56 lower bound on dimension, 33, 54 universal property, 25 universal property with respect to analytic families, 26 universal property with respect to C m families, 63 universal family on, 25 variants of, 43 of ν-log-canonically embedded stable n-pointed genus g curves, 196 of automorphisms of fibers of a standard Kuranishi family, 209 of closed subschemes of projective space with given Hilbert polynomial, 7

953 of hypersurfaces in projective space, 7 of space conics, 67 of twisted cubics, 68 of zero-dimensional subschemes, 10, 33, 72 restricted, 69 the Grassmannian as a, 10 Hilbert, David, 438 Hilbert–Mumford numerical criterion, 404 Hirschowitz, André, 881 Hodge bundle, 226, 572, 585, 591 on the moduli stack of stable curves, 334 semipositivity of, 233, 237 Hodge class, 334, 585, 750 additivity of, 365 generalized, 334 higher, 572 higher generalized, 573 nefness of, 433 Hodge line bundle, 334, 344, 359 ampleness on the Satake compactification of Mg , 435–437 Homology equivariant, 755 of a group with integral coefficients, 754 of Mg,n , vanishing of, 671 Hori, Kentaro, 770 Horikawa class of a first-order deformation of a morphism, 821 Horikawa, Eiji, 438, 780, 819, 824 Horizontal trajectory, 480 vector field, 479 Horocycle, 611, 632 region, 632 region, standard, 632 Howard, Alan, 882 Hubbard, John Hamal, 161, 498 Hulek, Klaus, 161, 882

954

Index

Humbert, Georges, 882 Humphries, Stephen, 497 Hurwitz numbers, 771, 772 scheme, 854 space, 857 space, irreducibility of, 857 space, unirationality of, 878, 900 Hurwitz, Adolf, 854, 883 Huybrechts, Daniel, 64 Hyeon, Donghoon, 439 Hyperbolic spine, 611, 623, 640, 659, 660, 697, 730

of a groupoid presentation of a Deligne–Mumford stack, 304 Itzykson, Claude, 720, 745, 771 Ivanov, Nikolai V., 683 Izadi, Elham, 605

Iarrobino, Anthony, 65 Igusa, Kiyoshi, 771 Illusie, Luc, 323 Imayoshi, Yˆ oichi, 498, 665 Immersion closed, of Deligne–Mumford stacks, 304, 340 open, of Deligne–Mumford stacks, 304 of algebraic spaces, 307 Index of a node of a stable genus zero curve, 392 of a ramification point, 839 of an admissible cover at a node, 505, 527 Infinitesimal automorphism, 116 Inverse function theorem, 57 in a Lie groupoid, 275, 306, 323 Ionel, Eleny-Nicoleta, 605 Isomorphism of categories fibered in groupoids, 280 of deformations, 172 Isotrivial family of curves, 418, 419, 422, 431 Isotropy group in an orbifold groupoid, 276

Kähler differentials, 95 relative, 95, 365 Kac, Victor, 397, 773 Kaku, Michio, 772 Kaplan, Aroldo, 594 Kashiwara, Masaki, 773 Kawamoto, Noboru, 397 Kazakov, Vladimir A., 772 Kazarian, Maxim, 772, 883, 884 KdV (Korteweg de Vries) hierarchy, 726, 774 Gelfand–Dikii form, 726 Keel, Seán, 323, 439, 566, 599 Keem, Chango, 883 Keen, Linda, 665 Kempf, George, 242, 248 Khosla, Deepak, 439 Kirwan, Frances, 685 Kleiman, Steven, 879, 881 Kleiman, Steven L., 323, 788 Kleppe, Jan O., 65 Knudsen, Finn Faye, 161, 323, 396, 438, 880, 883 Knutson, Donald, 323 Kodaira, Kunihiko, 32, 65, 167, 215, 248 Kodaira–Spencer class, of a first-order deformation of a manifold, 173

Jackson, David M., 605, 884 Jacobian variety of a nodal curve, 89 relative, 786 Jacobian locus, 461 Jacobson ring, 16 Jenkins, James A., 771 Jimbo, Michio, 773 Jost, J¨ urgen, 665

Index

class, of a first-order deformation of a nodal curve, 178 class, of a first-order deformation of an n-pointed nodal curve, 183 class, of a first-order deformation of a line bundle, 201 class, of a first-order deformation of a pair (curve, line bundle), 804 homomorphism, 175, 178 homomorphism, in a Kuranishi family, 188 homomorphism, in a versal family, 192 homomorphism, and the differential of the period map, 217 Koll´ ar, J´ anos, 64, 248, 323, 438 Konno, Kazuhiro, 438 Kontsevich’s matrix model, 743, 745–750 Kontsevich, Maxim, 397, 612, 702, 709, 717, 743, 761, 768, 771 Korn, Arthur, 497 Kouvidakis, Alexis, 709 Kuranishi family action of automorphism group on, 189 for a morphism, 824 for a curve with Teichm¨ uller structure, 448 for admissible G-covers, 530– 535, 557 for an n-pointed stable curve, 188 standard, 208 standard algebraic, 207 standard, of hyperelliptic stable curve, 210, 211 universal property with respect to continuous deformations, 212–216 Kuranishi, Masatake, 248

955 L¨ uroth, Jacob, 854, 883 Laksov, Dan, 882 Lando, Sergei K., 771, 772, 883, 884 Lange, Herbert, 883 Laplace–Beltrami operator, 214 Laufer, Henry B., 882 Laumon, Gérard, 323 Lax, Robert F., 882 Lazarsfeld, Robert, 780, 814, 880, 883, 885 Lazarsfeld–Mukai bundle, 814 Le Potier, Joseph, 881, 882 Lefschetz, Solomon, 161 Leg of a graph, 93, 126, 313, 347, 363, 581, 648 Lehn, Manfred, 64 Leida, Johann, 323 Level Jacobi structure of level m, 512 Teichm¨ uller structure of level G, 508, 511 structure associated to a surjective exterior homomorphism, 511 structure dominating another one, 514 Li, Jun, 769 Lichtenstein, Leon, 497 Lickorish, William B. R., 460, 497 Lie groupoid, 275 proper étale, 276 Line bundle even, 348 graded, 348 odd, 348 on a nodal curve, 89 on a Deligne–Mumford stack, 333 Hodge, 334 nef, 229–231 point, 334 G-equivariant, 343 Poincaré, 781, 782, 785, 786

956 Linear differential operators C m family of, 215 smooth dependence of solutions on parameters, 216 Linearly reductive linear algebraic group, 401 Linearly stable curve in projective space, 408 Liu, Kefeng, 605, 884 Liu, Xiaobo, 769 Local complete intersection (l.c.i.) morphism, 86, 97, 578 Local criterion of flatness, 28 Local Torelli theorem, 223, 420, 461 for hyperelliptic curves, 224, 420 Log-canonical sheaf, 92, 99, 195 relative, 377, 572 Looijenga, Eduard, 498, 562, 566, 598, 604, 605, 668, 684, 685, 708, 771, 796, 882 M¨ obius transformation, 627 dilatation of, 627 elliptic, 627 hyperbolic, 627 parabolic, 627 translation, 627 Madsen, Ib, 684, 685 Manetti, Marco, 248 Manin, Yuri˘ı I., 397, 773 Mapping class group, 144, 450, 451, 454, 458, 459 generators of, 460 action on bordification of Teichm¨ uller space, 491 action on the arc complex, 614 Marking weak Γ-, 314 Γ-, 314 of a ribbon graph, 619 of a P -pointed stable curve, 490

Index

Martellini, Maurizio, 397 Martens, Gerriet, 883 Martens, Henrik, 812 Martin-Deschamps, Mireille, 65 Matelski, Peter J., 665 Matsmura, Hideyuki, 96 Matsuzaki, Katsuhiko, 665 Max Noether’s theorem, 223, 241 Mayer, Alan, 161 Melo, Margarida, 879 Mestrano, Nicole, 709 Metric conformal, 628 intrinsic, 633 Poincaré, 627, 628 Metric topology, 615 Migdal, Alexander A., 772 Miller, Edward, 604, 684 Miranda, Rick, 880 Mirzakhani, Maryam, 772 Mishachev, Nikolai M., 685 Miwa, Tetsuji, 773 Module with descent data, 292 Moduli map finite, onto moduli, 268, 307 of a family of curves, 261 Moduli space of d-gonal curves, irreducibility and dimension, 864 coarse, for a stack, 302 for admissible G-covers, 505, 535, 556 of stable genus g curves, 104 of elliptic curves, 254–257, 266 of stable n-pointed genus g curves, 257, 259, 260 of stable n-pointed genus zero curves, 264, 265, 599 of curves with level structure, 508 of curves with ψ-structure, 510 of curves with level structure, compactification of, 522 of stable ribbon graphs, 664 of stable maps, 767

Index

Moduli space of curves as an analytic space, 259, 260 boundary of, 261 completeness, 268 as an algebraic space, 271 as an orbifold, 277 as a Deligne–Mumford stack, 300 Picard group, 379 projectivity, 425 irreducibility, 462, 861 unirationality in low genus, 872 Moduli stack of admissible G-covers, 505, 535 of stable n-pointed genus g curves, 138, 300 Mondello, Gabriele, 665 Monodromy group, local, 523 representation, 856 representation, local, 522 Moret-Bailly, Laurent, 323 Morgan, John, 709 Mori, Shigefumi, 323, 884 Morita, Shigeyuki, 604, 605, 684 Moriwaki, Atsushi, 438 Morphism of (categories fibered in) groupoids, 280 of deformations, 172 of families of nodal curves, 95 of orbifold groupoids, 276 of stacks, 296 representable, of stacks, 299 Morrey, Charles B., 497 Morrison, Ian, 439, 880, 883 Mukai, Shigeru, 437, 881, 882, 884, 885 Mulase, Motohico, 771, 773 Multidegree, 89 Mumford class, 572, 721 κ1 , 332, 377 κ1 , ampleness of, 425

957 Mumford’s example, 40–43 Mumford’s formula, 384 Mumford’s relations for Hodge classes, 586–592 Mumford, David, 12, 65, 161, 323, 396, 397, 435, 437, 438, 562, 565, 566, 591, 604, 605, 665, 683, 708, 812, 873, 881, 883, 884 Mumford–Morita–Miller classes, 572, 721 Murri, Riccardo, 771 Nœther’s theorem, 223, 241, 461 Nag, Subhashis, 397, 498 Nagaraj, Donihakkalu S., 883 Nagel, Jan, 883 Nakano, Shigeo, 579 Namikawa, Yukihiko, 397 Narasimhan, Mudumbai S., 881 Newstead, Peter, 881 Nielsen extension, 634, 658 Nielsen kernel, 634 Nielsen, Jakob, 497 Nirenberg, Louis, 248 Nitsure, Nitin, 64, 323, 784 Node, 83 assigned, 853, 877 nonseparating, 94, 100 nonseparating, on a stable hyperelliptic curve, 102, 390 separating, 95, 100 separating, on a stable hyperelliptic curve, 102, 390 virtually nonexistent, 853 Noether, Max, 779, 880, 883 Norm map, 366, 375 Normal sheaf, 31 of a regular embedding, 38 to a clutching morphism, 346 and Petri’s statement, 824 to a morphism, 345, 819 Normalization of a Deligne–Mumford stack, 305

958 Normalization (cont.) of an algebraic space in an extension of its function field, 308 Obstructed tangent vector, 53 Oda, Tadao, 879 Okounkov, Andrei, 769, 771, 884 Oort, Frans, 161, 248, 882 Orbicellular decomposition of moduli, 614, 623, 661, 690 extension to compactification, 614, 662 Orbifold local chart, 275 groupoid, 276 groupoid, complex, 277 groupoid, quotient of, 278 quotient, of a manifold by a finite group, 277, 323 structure, 277 structure, on the moduli space of n-pointed genus g curves, 277 Orbit in an orbifold groupoid, 276 in a ribbon graph, 617, 618 Orientation form on the combinatorial moduli space of curves, 699 Outer automorphisms, 454, 540 Pandharipande, Rahul, 323, 396, 566, 604, 605, 769, 771, 879, 884 Pants decomposition, 485, 497 Papadopol, Peter, 161, 498 Pareschi, Giuseppe, 814, 880 Partition function, 721 asymptotic expansion of, 744 Penner, Robert, 665, 771 Period map, 217 holomorphicity of, 217, 220 differential of, 217

Index

Period matrix, 217 Perrin, Daniel, 65 Persson, Ulf, 438 Peters, Chris A. M., 161 Petri’s condition, 794, 808, 815 Petri’s statement, 780, 824 for gd1 ’s, 810 for gd2 ’s, 845 Petri, K., 779, 811, 879, 885 Petronio, Carlo, 665 Pfaffian, 702 Picard functor, relative, 782, 879 Picard group of Mg,n , 379, 381, 484, 713 of a Deligne–Mumford stack, 333 of a stack quotient, 343 of the moduli stack of stable hyperelliptic curves, 391, 396 Picard variety. See Relative Picard variety Picard–Lefschetz, representation, 145, 483, 523 transformation, 144 transformation and Dehn twists along vanishing cycles, 148, 158–160 transformation in the context of G covers, 539 Pikaart, Martin, 562, 684 Pinkham, Henry, 882 Poincaré duality, on an orbifold, 279 line bundle, 781, 782, 785, 786 metric, 627, 628 Poincaré, Henri, 880 Point bundles, 334, 344 nefness of, 434 Point bundle classes, 335, 572, 717 combinatorial expression for, 697 intersection numbers, 721 Polishchuk, Alexander, 773

Index

Polyakov, Alexander M., 397 Popa, Mihnea, 439, 881, 885 Positive component, 649, 653 Presentation of the mapping class group, 460 Primitive sublattice, 546, 547 Procesi, Claudio, 397 Projection functor, 125 morphism, 311, 560 Projectivity of Hilbert scheme, 26 of the moduli space of stable curves, 425 Propagator, 735 Pullback in a category fibered in groupoids, 281 of boundary classes under projection, 581 of boundary classes under clutching, 581–584 of Mumford–Morita–Miller classes under projection, 581 of Mumford–Morita–Miller classes under clutching, 581, 582, 584 Pushforward of the fundamental class of a stack, 569 Qing Liu, 96 Quadratic differential, 462 Teichm¨ uller deformation associated to a, 463–465 Teichm¨ uller map associated to a, 463–465 canonical, on a Teichm¨ uller deformation, 465 metric attached to a. See ω-metric Quadrics through the canonical curve, 248 Quasi-complete intersection, 75 Quasi-diffeomorphism, 468 admissible, 468

959 Quotient effective, of a schematic equivalence relation, 270 of a schematic equivalence relation, 270 groupoid, 286 orbifold, 277, 323 stack, quasi-coherent sheaves on, 343 Raina, Ashok K., 773 Ramanan, Sundaraman, 882 Ramification, 835 Ramification divisor, 836 Ramification index, 839 Ran, Ziv, 884 Ratcliffe, John G., 665 Rational functions on an irreducible algebraic space, 271, 308 Rational tail, 574 moduli space of curves with, 598 Rauch, H. Ernest, 882 Reduced degree of a curve in projective space, 408 Rego, C. J., 879 Regular embedding, 36, 38, 54, 87, 97 Regular sequence, 35–39 Reina, Cesare, 397 Relative Picard variety, 781, 785, 787 Relative C m atlas, 56 Relative dualizing sheaf, 97, 572 direct image of, 234, 334 nefness of, 435 positivity properties of, 417– 421, 424 Relatively minimal fibration, 438 Reynolds operator, 258, 402 Ribbon graph, 616 associated to a proper simplex, 621 dual of, 619

960 Ribbon graph (cont.) embedded, 620 genus and boundary components of, 617 half-edge of, 616, 700, 738 half perimeters of, 619 in connection with Gaussian integrals, 741 isomorphism, 619 marking of, 619 oriented edges of, 617 stable P -marked, 648 stable P -marked, moduli space of, 664 topological surface attached to, 617 unital metric on, 619 unital metric on stable P marked, 648 Riccati equation, 726 Riemann surface, 151, 167, 216, 466, 473, 479 compact n-pointed, 462, 469, 559, 672 hyperbolic, 624 of finite type, 627, 629, 633 parabolic, 624 simply connected, 624 with boundary, 489 Riemann’s existence theorem, 799, 856 Riemann’s extension theorem, 257, 260 Riemann’s moduli count, 828, 834 Rim, Dock Sang, 882 Rosenlicht, Maxwell, 397 Ruan, Yongbin, 323, 769 Samuel, Pierre, 882 Sard’s lemma for flatness, 18 Sard’s theorem, 809 Satake compactification of the moduli space of genus g curves, 437 Satake, Ichir¯ o, 323, 439

Index

Schechtman, Vadim V., 397 Schiffer variation, 175–177, 533, 837 Schlessinger, Michael, 248 Schmid, Wilfried, 594, 709 Schreyer, Frank-Olaf, 883 Schubert, Hermann, 768 Section of a category fibered in groupoids, 281 Segal, Graeme, 773 Segre, Beniamino, 65, 813, 864, 869, 873, 884 Seifert, Herbert, 455, 497 Semipositive locally free sheaf, 229, 230 Semistable graph, 100 curve, 100 point, in the sense of Geometric Invariant Theory, 401 Sernesi, Edoardo, 64, 248, 879, 884 Serre, Jean-Pierre, 87, 172 Seshadri’s criterion of ampleness, 230, 426 Seshadri, Conjeeveram S., 879, 881, 882 Severi, Francesco, 65, 811, 812, 850, 882–884 Shadrin, Sergey, 772, 773, 884 Shapiro, Michael, 771 Sheaf G-equivariant quasi-coherent, 340 quasi-coherent, on a Deligne– Mumford stack, 333, 337 Shenker, Stephen H., 772 Siegel upper half-space, 217 Simplex in the arc complex, 613 proper, 613 Singularity An , 109 of a quadratic differential, 463

961

Index

Slope inequality, 417, 438 of a divisor in moduli space, 439 Smith, Roy, 248 Socle, 597 Sommese, Andrew J., 882 Soulé, Christophe, 396 Source of an arrow of a Lie groupoid, 275 Specialization of a graph, 314, 319 of an automorphism of a graph, 314, 319 Spencer, Donald C., 32, 65, 167, 215, 248 Springer, George, 665 Stability of the cohomology of moduli, 683 Stabilization functor, 128 as inverse of the contraction functor, 138 Stable graph, 99 curve, 99 P -marked ribbon graph, 648 point, in the sense of Geometric Invariant Theory, 401 Stable model of a family of semistable curves, 124 of a semistable curve, 118 Stable reduction, 104–113 for admissible G-covers, 528 theorem, 113 uniqueness of, 114, 116 Stack, 295 Artin, 300 cycle ring of, 570 Deligne–Mumford, 300 Standard coordinate patch for the Teichm¨ uller space, 448 Standard system of parameters, 152

Steenbrink, Joseph, 248 Stoppino, Lidia, 438 Strebel, Kurt, 771 Stretching function, 479 String equation, 574, 723, 747 Strongly characteristic quotient, 510, 541 subgroup, 510, 541 Subgraph, 313 Substack, 304, 339, 340 Sullivan, Dennis, 397 Symbol map, 804 Tangent space to Gdr , 805, 807 to Wdr , 807 to Hilbert scheme, 33, 49–56 to the Hilbert scheme of ν-logcanonical stable curves, 198, 202 to the moduli space of admissible G-covers, 531 to relative Picard variety, 805 Taniguchi, Masahiko, 498, 665 Tannenbaum, Allen, 850, 883 Target of an arrow of a Lie groupoid, 275 Tautological class. See Classes, tautological relation, 382, 386, 565, 573 ring, 565, 584, 587, 591, 605, 796 Teichm¨ uller deformation, 463 map, 462, 463, 465, 469, 470 marking of a stable curve, 490 modular group, 144, 441, 450, 453, 454, 459, 483, 683, 757 space, 441, 446, 453, 454, 459, 471, 483, 509, 614, 757 space, bordification of, 490– 497, 614 space, cellular decomposition of. See Cellular decomposition of Teichm¨ uller space

962 Teichm¨ uller (cont.) structure of level G, 508 structure on a pointed curve, 445 strucure on a family of pointed curves, 447 theory, 167 uniqueness theorem, 469, 479– 483 Teichm¨ uller, Oswald, 441 Teixidor i Bigas, Montserrat, 881–883 Thorup, Anders, 882 Thurston, William, 665 Tian, Gang, 769 Tillmann, Ulrike, 684, 685 Toda lattice, 772 Todd class, 382, 588 Todorov, Gueorgui T., 605, 884 Topology coherent, 615 metric, 615 of the bordification of Teichm¨ uller space, 494– 496, 655 Topological covering of the stack Mg,n , 483 Topological surface nodal P -pointed, 649 stable P -pointed, 649 Torelli group, 460 Torelli theorem, 216 local. See Local Torelli theorem Torsion of a complex, 704 Total degree of a line bundle, 89 Totally unimodular lattice, 552 matrix, 552 Transverse family of stable curves, 152, 155, 157, 492 Triangle decomposition, 455 Tromba, Anthony J., 498 Tsuchiya, Akihiro, 397 Type of a ramification point, 839

Index

of a separating node, 95, 261 of node of hyperelliptic stable curve, 102, 211, 389, 390 Uniformization, 624–626 Unimodular lattice, 551 Unirational variety, 326 Unit of a Lie groupoid, 275 Universal deformation of a map, 825 Universal effective divisor, relative of degree d, 784 Vainshtein, Alek, 771, 884 Vakil, Ravi, 65, 605, 772, 884 Van de Ven, Antonius, 161 van der Geer, Gerard, 882 Vanishing cycle, 146, 158, 493, 497, 524, 546, 549, 555 of the homology of Mg,n in high degree, 671 theorem, for the tautological ring, 796 Varley, Robert, 248 Vector field horizontal, 479 vertical, 479 Verlinde, Erik Peter, 726, 772 Verlinde, Herman, 726, 772 Verra, Alessandro, 884 Versal family, 192 Vertical coordinates, 56 derivatives, 56 vector field, 479 Vertices of a graph, 93, 126, 313, 549, 581, 616, 637, 648 Veselov, Vladimir, 161, 498 Viehweg, Eckart, 439 Virasoro algebra, 725 equations, 718, 726, 773 operators, 718, 722 Vistoli, Angelo, 323, 397, 562, 604, 881, 883

963

Index

Vitulli, Marie, 882 Viviani, Filippo, 397, 885 Voisin, Claire, 780, 813, 880, 883–885 Wahl, Jonathan, 851, 880, 883 Weierstrass point, 262, 388, 798, 881 Weil reciprocity, 366, 396 Weil, André, 248 Weiss, Michael S., 684, 685 Weyl, Hermann, 438 Wick’s lemma, 735 Wilson, George, 773 Witten, Edward, 612, 709, 717, 766, 771, 884 Xiao, Gang, 438 Xiong, Chuan-Sheng, 770

Xu, Hao, 605, 884 Yamada, Yasuhiko, 397 Yamaki, Kazuhiko, 438 Yoneda lemma, 2-categorical, 284, 335 Zagier, Don, 721, 773 Zariski’s connectedness theorem, 308 Zariski’s main theorem, 254, 435 Zariski, Oscar, 65, 882, 883 Zeuthen, Hieronymus, 766, 768 Zuber, Jean Bernard, 720, 745, 771 Zucconi, Francesco, 438 Zvonkin, Alexander, 771 Zvonkine, Dimitri, 772, 773, 884

Geometry of Algebraic Curves: Volume II

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