Combinatorial Aspects of Commutative Algebra and Algebraic Geometry: The Abel Symposium 2009

ABEL SYMPOSIA Edited by the Norwegian Mathematical Society For other titles published in this series, go to www.spring...

Author: Gunnar Fløystad | Trygve Johnsen | Andreas Leopold Knutsen

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ABEL SYMPOSIA Edited by the Norwegian Mathematical Society

For other titles published in this series, go to www.springer.com/series/7462

Participants of the Abel Symposium 2009, Voss, Norway. Photo Credits: David Eisenbud

Gunnar Fløystad r Trygve Johnsen Andreas Leopold Knutsen

r

Editors

Combinatorial Aspects of Commutative Algebra and Algebraic Geometry The Abel Symposium 2009

Editors Gunnar Fløystad Andreas Leopold Knutsen University of Bergen Dept. Mathematics Johannes Brunsgate 12 5008 Bergen Norway [email protected] [email protected]

Trygve Johnsen University of Tromsø Dept. Mathematics & Statistics 9037 Tromsø Norway [email protected]

ISBN 978-3-642-19491-7 e-ISBN 978-3-642-19492-4 DOI 10.1007/978-3-642-19492-4 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011928558 Mathematics Subject Classification (2010): 13-06, 14-06 © 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: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface to the Series

The Niels Henrik Abel Memorial Fund was established by the Norwegian government on January 1. 2002. The main objective is to honor the great Norwegian mathematician Niels Henrik Abel by awarding an international prize for outstanding scientific work in the field of mathematics. The prize shall contribute towards raising the status of mathematics in society and stimulate the interest for science among school children and students. In keeping with this objective the board of the Abel fund has decided to finance one or two Abel Symposia each year. The topic may be selected broadly in the area of pure and applied mathematics. The Symposia should be at the highest international level, and serve to build bridges between the national and international research communities. The Norwegian Mathematical Society is responsible for the events. It has also been decided that the contributions from these Symposia should be presented in a series of proceedings, and Springer Verlag has enthusiastically agreed to publish the series. The board of the Niels Henrik Abel Memorial Fund is confident that the series will be a valuable contribution to the mathematical literature. Ragnar Winther Chairman of the board of the Niels Henrik Abel Memorial Fund

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Preface

The title of the Abel Symposium 2009 was “Combinatorial aspects of Commutative Algebra and Algebraic Geometry”. The last couple of decades has seen a shift in commutative algebra and algebraic geometry towards more concrete problems, often related to combinatorics. The above was therefore a natural choice for the title of the symposium. In fact many of the topics of the symposium have risen to prominence only over the last decade. Among those are cluster algebras, tropical geometry and the recent breakthrough in the studies of syzygies of graded modules over polynomial rings, called Boij–Söderberg theory. But also more classical topics were the subjects of talks, like Hilbert schemes, with the more recent approach where the parametrized schemes are multigraded, real algebraic geometry, toric and homogeneous varieties, and cohomology rings and Gromow–Witten invariants. A more recent topic is binomial ideals, and their primary decomposition. The present volume is fortunate to contain a comprehensive introductory survey on this topic by Ezra Miller, exploring its applications to such varied topics as hypergeometric series, combinatorial game theory, and chemical dynamics. The local organizers of the symposium were: Gunnar Fløystad, University of Bergen, Trygve Johnsen, University of Tromsø, Andreas L. Knutsen, University of Bergen. The scientific advisory committee consisted of: Aaron Bertram, University of Utah, Salk Lake City, David Eisenbud, University of California, Berkeley, Frank-Olaf Schreyer, University of Saarbrücken, Mike Stillman, Cornell University, Ravi Vakil, Stanford University. The symposium is intended to be a smaller event, limited upwards to forty speakers and participants. There were nineteen invited speakers, leading experts in their fields. The speakers and titles of the talks were as follows. vii

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Preface

(a) Aaron Bertram, Deconstructing Coherent Sheaves (b) Mats Boij, Hilbert Functions and Betti Numbers up to Multiples and Parameter Spaces (c) Anders Buch, Quantum K-Theory of Grassmannians (d) Aldo Conca, Syzygies of Veronese and Koszul Algebras (e) David Eisenbud, Boij–Söderberg Theory: Introduction and possible Future Perspectives (f) Sergey Fomin, Enumeration of Plane Curves and Labeled Floor Diagrams (g) William Fulton, Character Formulas (h) Jürgen Herzog, Powers of Componentwise Linear Ideals (i) Joel Kamnitzer, Equivalences of Derived Categories from Geometric sl(2) actions (j) Dan Laksov, A Formalism for Equivariant Cohomology (k) Diane Maclagen, Connected Multigraded Hilbert Schemes (l) Ezra Miller, Applications of Binomial Commutative Algebra (m) Sam Payne, Boundary Complexes and Weight Filtrations (n) Frank-Olaf Schreyer, Boij–Söderberg Theory for Coherent Sheaves on Pn (o) Jessica Sidman, Syzygies of the Secant Varieties of Curves (p) Mike Stillman, Green’s Conjecture and high rank Syzygies on low rank Quadrics (q) Rekha Thomas, The Convex Hull of a Real Algebraic Variety (r) Jerzy Weyman, Quivers with Potentials and their Mutations (s) Andrei Zelevinsky, Cluster Algebras via Quivers with Potentials The Abel Symposium 2009 was held Monday June 1. to Thursday June 4. in the town of Voss in Western Norway. Most of Wednesday was devoted to a trip by bus, boat, and train to the spectacular Nærøyfjord, which is on UNESCO’s world heritage list, and the Aurlandsfjord. In chilly but clear weather we went ashore from the boat to a small farm by the Nærøyfjord and had an ample lunch. At the end of the Aurlandsfjord we took the Fl˚am railway, a 20 kilometer trip ascending 866 meter to Myrdal, and from there back to Voss. We would like to express our gratitude to the Abel foundation and the Norwegian Mathematical Society for giving us the opportunity to organize the Abel Symposium 2009. We also thank the administration at the Department of Mathematics, University of Bergen for their kind help and assistance with practical matters, we thank Bjørn Theisen for doing a solid job with typesetting the articles, and we thank Springer Verlag and especially Ruth Allewelt for an excellent job in following up these proceedings. Bergen and Tromsø, 22. December 2010

Gunnar Fløystad Trygve Johnsen Andreas L. Knutsen

Contents

The Cone of Betti Diagrams of Bigraded Artinian Modules of Codimension Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Mats Boij and Gunnar Fløystad 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Betti Diagrams and the Multigraded Herzog–Kühl Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The Equivariant Resolution . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 The Linear Space of Betti Diagrams of Multigraded Artinian Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 The Linear Space in the Case of Two Variables . . . . . . . . . 6 3 The Positive Cone of Bigraded Betti Diagrams . . . . . . . . . . . . . . . . . 7 3.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Decomposing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Existence of Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5 Resolutions of Trigraded Artinian Modules of Codimension Three 14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Koszul Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Winfried Bruns, Aldo Conca and Tim Römer 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation and Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Bounds for Koszul Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Green–Lazarsfeld Index for Segre–Veronese Rings . . . . . . . . . . . . . 5 Generating Koszul Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 18 21 26 29 32

Boij–Söderberg Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 David Eisenbud and Frank-Olaf Schreyer 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2 Betti Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ix

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3 Facets of the Cone and Cohomology Tables . . . . . . . . . . . . . . . . . . . 4 Sheaves on a Subvariety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Several Line Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 42 43 47

Powers of Componentwise Linear Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jürgen Herzog, Takayuki Hibi and Hidefumi Ohsugi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Chordal Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Modules with 1-Dimensional Socle and Components of Lusztig Quiver Varieties in Type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joel Kamnitzer and Chandrika Sadanand 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Preprojective Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Socle of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Lusztig Quiver Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Modules with One-Dimensional Socle . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Maya Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Computation of Hom Spaces . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Connection with MV Polytopes . . . . . . . . . . . . . . . . . . . . . . 4 Description of Irreducible Components . . . . . . . . . . . . . . . . . . . . . . . 4.1 Savage’s Description of the Components . . . . . . . . . . . . . . 4.2 Description of the Components by Hom Spaces . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Realization Spaces for Tropical Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eric Katz and Sam Payne 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Realization and Compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Representability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Tropical Realization of Matroid Fans . . . . . . . . . . . . . . . . . . . . . . . . . 5 Murphy’s Law for Tropical Realization Spaces . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 51 55 60 61 61 62 62 62 63 63 64 64 64 65 68 69 70 70 71 72 73 73 76 78 79 83 87

A Relation Between Symmetric Polynomials and the Algebra of Classes, Motivated by Equivariant Schubert Calculus . . . . . . . . . . . . . . . . . . . . . . . . 89 Dan Laksov 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Contents

3 Factorial Schur Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Factorial Schur Polynomials over Polynomial Rings . . . . . . . . . . . . 5 Factorial Schur Polynomials and Schubert Classes . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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90 93 94 97

Theory and Applications of Lattice Point Methods for Binomial Ideals . . . 99 Ezra Miller 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 2 Affine Semigroups and Prime Binomial Ideals . . . . . . . . . . . . . . . . . 100 2.1 Affine Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 2.2 Affine Semigroup Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.3 Prime Binomial Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3 Monomial Ideals and Primary Binomial Ideals . . . . . . . . . . . . . . . . . 108 3.1 Monomial Primary Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.2 Congruences on Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.3 Binomial Primary Ideals with Monomial Associated Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4 Binomial Primary Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.1 Monomial Primes Minimal over Binomial Ideals . . . . . . . 116 4.2 Primary Components for Arbitrary Given Associated Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.3 Finding Associated Primes Combinatorially . . . . . . . . . . . 121 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5 Hypergeometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.1 Binomial Horn Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.2 True Degrees and Quasidegrees of Graded Modules . . . . . 128 5.3 Counting Series Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6 Combinatorial Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.1 Introduction to Combinatorial Game Theory . . . . . . . . . . . 134 6.2 Lattice Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.3 Rational Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.4 Misère Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7 Mass-Action Kinetics in Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.1 Binomials from Chemical Reactions . . . . . . . . . . . . . . . . . . 147 7.2 Global Attractor Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 149 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Equations Defining Secant Varieties: Geometry and Computation . . . . . . . 155 Jessica Sidman and Peter Vermeire 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 2 Computing Secant Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 2.1 Secant Varieties via Elimination . . . . . . . . . . . . . . . . . . . . . 159 2.2 Secant Varieties via Prolongation . . . . . . . . . . . . . . . . . . . . 160 2.3 Compute the Ideal of a Smooth Curve . . . . . . . . . . . . . . . . 161

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Secant Varieties as Vector Bundles: Terracini Recursion . . . . . . . . . 164 3.1 Secant Varieties of Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 3.2 Blowing up a Rational Normal Curve of Degree 3 . . . . . . 166 3.3 Blowing up a Rational Normal Curve of Degree 4 . . . . . . 167 3.4 Cohomology Along the Fibers . . . . . . . . . . . . . . . . . . . . . . . 169 Appendix A Code for Computing Prolongations . . . . . . . . . . . . . . . . . . . . . 171 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Contributors

Mats Boij Institutionen för matematik, KTH, S-100 44 Stockholm, Sweden, [email protected] Winfried Bruns Universität Osnabrück, Institut für Mathematik, 49069 Osnabrück, Germany, [email protected] Aldo Conca Dipartimento di Matematica, Universitá di Genova, Via Dodecaneso 35 16146 Genova, Italy, [email protected] David Eisenbud Dept. of Mathematics, University of California, Berkeley, Berkeley CA 94720, USA, [email protected] Gunnar Fløystad Matematisk Institutt, Johs. Brunsgt. 12, 5008 Bergen, Norway, [email protected] ¨ Jurgen Herzog Fachbereich Mathematik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany, [email protected] Takayuki Hibi Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 5600043, Japan, [email protected] Joel Kamnitzer Department of Mathematics, University of Toronto, Toronto, Canada, [email protected] Eric Katz Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA, [email protected] Dan Laksov Department of Mathematics, KTH, S-100 44, Stockholm, Sweden, [email protected] Ezra Miller Mathematics Department, Duke University, Durham, NC 27708, USA, [email protected] Hidefumi Ohsugi Department of Mathematics, College of Science, Rikkyo University, Tokyo 171-8501, Japan, [email protected] xiii

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Contributors

Sam Payne Stanford University, Mathematics, Bldg. 380, 450 Serra Mall, Stanford, CA 94305, USA, [email protected] Tim Römer Universität Osnabrück, Institut für Mathematik, 49069 Osnabrück, Germany, [email protected] Chandrika Sadanand Department of Mathematics, Stony Brook University, Stony Brook, USA, [email protected] Frank-Olaf Schreyer Facultät für Mathematik und Informatik, Campus E 2 4, Universität des Saarlandes, D-66123 Saarbrücken, Germany, [email protected] Jessica Sidman Department of Mathematics and Statistics, Mount Holyoke College, South Hadley, MA 01075, USA, [email protected] Peter Vermeire Department of Mathematics, 214 Pearce, Central Michigan University, Mount Pleasant, MI 48859, USA, [email protected]

The Cone of Betti Diagrams of Bigraded Artinian Modules of Codimension Two Mats Boij and Gunnar Fløystad

Abstract We describe the positive cone generated by bigraded Betti diagrams of artinian modules of codimension two, whose resolutions become pure of a given type when taking total degrees. If the differences, p and q, of these total degrees are relatively prime, the extremal rays are parametrized by order ideals in N2 contained in the region px + qy < (p − 1)(q − 1). We also consider some examples concerning artinian modules of codimension three.

1 Introduction In [2], D. Eisenbud, J. Weyman, and the second author gave for every sequence of integers d : d0 < d1 < · · · < dn a construction of pure resolutions of graded artinian modules over a polynomial ring S = k[x1 , . . . , xn ] S(−d0 )β0 ← S(−d1 )β1 ← · · · ← S(−dn )βn assuming that char k = 0. Moreover these resolutions are GL(n)-equivariant, and so in particular invariant under the diagonal matrices and hence Zn -graded. The existence of pure free resolutions were an essential ingredient of the conjectures in [1] by the first author and J.Söderberg. The conjectures were subsequently proven by the construction of pure resolutions mentioned above, and by the work of Eisenbud and F.-O. Schreyer in [3]. In the case when S = k[x1 , x2 ], the first author and J. Söderberg in [1, Remark 3.2] gave a different construction of pure resolutions of artinian bigraded modules. These have different bigraded Betti diagrams than the equivariant resolutions above. Mats Boij Institutionen för matematik, KTH, S-100 44 Stockholm, Sweden e-mail: [email protected] Gunnar Fløystad Matematisk Institutt, Johs. Brunsgt. 12, 5008 Bergen, Norway e-mail: [email protected] G. Fløystad et al. (eds.), Combinatorial Aspects of Commutative Algebra and Algebraic Geometry, Abel Symposia 6, DOI 10.1007/978-3-642-19492-4 1, © Springer-Verlag Berlin Heidelberg 2011

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M. Boij and G. Fløystad

Example 1.1. Suppose d1 − d0 = 2 and d2 − d1 = 3. The equivariant resolution has the following form where we have written the bidegrees of the generators below the terms. S 3 ← S5 ← S2 . (1) (2, 0) (1, 1) (0, 2)

(4, 0) (3, 1) (2, 2) (1, 3) (0, 4)

(4, 3) (3, 4)

Let β1 be its bigraded Betti table. The resolution in [1] is of a quotient of a pair of monomial ideals. For the type above the resolution has the following bidegrees. S 3 ← S5 ← S2 .

(4, 0) (2, 2) (0, 4)

(6, 0) (4, 2) (3, 3) (2, 4) (0, 6)

(2)

(6, 3) (3, 6)

Denote by β2 be its Betti diagram. This indicated that there may be many types of multigraded Betti diagrams of Zn -graded artinian modules of codimension n whose resolutions become pure of a given type when taking total degrees. In [4] the second author showed that the multigraded Betti diagram of the equivariant resolution has a fundamental position. This diagram and its twists with a ∈ Zn form a basis for the linear space generated by multigraded Betti diagrams of artinian Zn -graded modules whose resolutions become pure of the given type when taking total degrees. Even more natural than describing the linear space, is describing the positive cone generated by the multigraded Betti diagrams. In this paper we to this in the case when S = k[x1 , x2 ]. Let e1 = d1 − d0 and e2 = d2 − d1 . We describe all the extremal rays of the positive cone P(e1 , e2 ) generated by bigraded Betti diagrams of artinian bigraded modules of codimension two whose resolutions become pure when taking total degrees, and where the differences of these total degrees are e1 and e2 . In the example above the two resolutions, or rather their Betti diagrams, are essentially the full story in the sense that the extremal rays in P(2, 3) are exactly the rays generated by β1 (a) and β2 (a) for a ∈ Z2 . To explain the general situation, assume here for simplicity that e1 and e2 are relatively prime. Let R(e1 , e2 ) be the integer coordinate points in the region of the first quadrant of the coordinate plane bounded by the line e1 x + e2 y < (e1 − 1)(e2 − 1). There is a partial order on N2 given by (a1 , a2 ) ≤ (b1 , b2 ) if a1 ≤ a2 and b1 ≤ b2 , and the region R(e1 , e2 ) inherits this. An order ideal in R(e1 , e2 ) corresponds to a partition λ . We give a construction which to each partition λ in R(e1 , e2 ) associates a bigraded resolution Se2 ← Se2 +e1 ← Se1 . Let βλ be the bigraded Betti diagram of this complex. The following is our main result in the case that e1 and e2 are relatively prime.

Cone of Betti Diagrams of Bigraded Artinian Modules

3

Theorem. The extremal rays in the cone P(e1 , e2 ) are the rays generated by the βλ (a) where a varies over Z2 and λ ranges over partitions contained in the region R(e1 , e2 ). The general case is formulated in Theorems 3.8 and 4.2. In the region R(e1 , e2 ) there are two distinguished partitions, the maximal one and the empty one. It turns out that the maximal one corresponds to the equivariant complex and the empty one corresponds to the bigraded resolution of a quotient of monomial ideals constructed in [1]. The organization of the paper is as follows. Section 2 contains preliminaries. First we give the multigraded Herzog–Kühl equations which give strong restrictions on Betti diagrams of multigraded artinian modules. We recall the equivariant resolution, and the result of [4] that its twists generate the linear space of multigraded Betti diagrams of artinian Zn -graded modules of codimension n whose resolution becomes pure when taking total degrees. This give us a very simple alternative description of the positive cone P(e1 , e2 ). This is used in Sect. 3 where we show that the extremal rays of the positive cone P(e1 , e2 ) are generated by the Betti diagrams βλ (a) for a ∈ Z2 , provided these diagrams really come from resolutions. And that such resolutions really exist is established in Sect. 4. In Sect. 5 we briefly discuss the positive cone in the case of three variables, providing an example.

2 Preliminaries Let S = k[x1 , . . . , xn ] be the polynomial ring over a field k. We shall study Zn -graded free resolutions of artinian Zn -graded S-modules F0 ← F1 ← · · · ← Fn . For a multidegree a = (a1 , a2 , . . . , an ) in Zn let |a| = ∑ ai be its total degree. We shall be interested in the case that these resolutions become pure resolutions if we make them singly graded by taking total degrees. In other words, there is a sequence d0 < d1 < · · · < dn such that Fi =

S(−a)βi,a .

a:|a|=di

2.1 Betti Diagrams and the Multigraded Herzog–Kuhl ¨ Equations The multigraded Betti diagram of such a resolution is the element {βi,a }ni=0, a∈Zn ∈

a∈Zn

Nn+1 .

4


A way of representing a multigraded Betti table which will be very convenient for us is to represent β = {βi,a } where i = 0, . . . , n and a ∈ Zn by Laurent polynomials Bi (t) =

∑n βi,a · t a

a∈Z

in variables t1 ,t2 , . . . ,tn . In this way we get an (n + 1)-tuple of Laurent polynomials B = (B0 , B1 , . . . , Bn ). Also the module ⊕a S(−a)βi,a may be conveniently denoted as S.Bi . For i = 1, 2, . . . , n, let ei = di − di−1 and let e = (e1 , e2 , . . . , en ) be the sequence of differences. Let L(e) be the linear subspace of ⊕a∈Zn Qn+1 generated by the multigraded Betti diagrams of Zn -graded artinian S-modules whose resolutions become pure when taking total degrees, and where the difference sequence given by these total degrees is e. Similarly let P(e) be the positive cone in ⊕a∈Zn Qn+1 generated by such Betti diagrams. There are some natural restrictions on L(e) coming from the multigraded Herzog–Kühl equations. If the resolution resolves the artinian module M, the multigraded Hilbert series of M is the polynomial hM (t) =

∑i,a (−1)i βi,a · t a n (1 − t ) , Πk=1 k

which gives n (1 − ti ). ∑(−1)i βi,at a = hM (t) · Πk=1

(3)

i,a

For each multidegree a ∈ Zn and each integer k = 1, . . . , n, let the projection πk (a) be (a1 , . . . , aˆk , . . . , an ), the n − 1-tuple where we omit ak . Now we have the multigraded analogs of the Herzog–Kühl (HK) equations. We obtain these by setting tk = 1 in (3) for each k. This gives for every aˆ in Zn−1 and k = 1, . . . , n an equation

∑

(−1)i βi,a = 0.

(4)

i,πk (a)=â

Let L (e) be the linear space of elements in ⊕a∈Zn Qn+1 which satisfy the multigraded HK-equations above, and which become pure diagrams when taking total degrees with the difference sequence of these total degrees equal to e. Also let P (e) be the cone in L (e) consisting of the elements with nonnegative coordinates. There are natural injections L(e) → L (e) and P(e) → P (e). In [4] the second author showed that the first injection is an isomorphism when char. k = 0, and moreover gave an explicit basis for L (e) which we now describe.


5

2.2 The Equivariant Resolution In [2] the second author together with D. Eisenbud and J. Weyman constructed GL(n)-equivariant pure resolutions of artinian modules when char. k = 0. For a partition λ = (λ1 , . . . , λn ) let Sλ be the associated Schur module, it is an irreducible representation of GL(n) (see for instance [5]). The action of the torus of diagonal matrices in GL(n) gives a decomposition of Sλ as a Zn -graded vector space. The basis elements are given by semi-standard Young tableau of shape λ with entries from 1, 2, . . . , n. All the nonzero graded pieces in this decomposition have total degree |λ | = ∑ni=1 λi . The free module S ⊗k Sλ then becomes a free multigraded module where the generators all have total degree |λ |. Now given the difference vector e, let

λi =

n

∑

(e j − 1)

j=i+1

and define a sequence of partitions for i = 0, . . . , n by

α (e, i) = (λ1 + e1 , λ2 + e2 , . . . , λi + ei , λi+1 , . . . , λn ). The construction in [2] then gives a GL(n)-equivariant resolution E(e) : S ⊗k Sα (e,0) ← S ⊗k Sα (e,1) ← · · · ← S ⊗k Sα (e,n)

(5)

of an artinian S-module. In the case of two variables, S = k[x1 , x2 ], the resolution takes the form E(e1 , e2 ) : S ⊗k Se2 −1,0 ← S ⊗k Se1 +e2 −1,0 ← S ⊗k Se1 +e2 −1,e2 .

(6)

2.3 The Linear Space of Betti Diagrams of Multigraded Artinian Modules For a multigraded Betti diagram β = {βi,a } and a multidegree t in Zn , we get the twisted Betti diagram β (−t) which in homological degree i and multidegree a is given by βi,a−t . If F· is a resolution with Betti diagram β , then F· (−t) is a resolution with Betti diagram β (−t). Also let Fr : S → S be the map sending xi → xir . Denote by S(r) the ring S with the S-module structure given by Fr . Given any complex F· we may tensor it with (r) − ⊗S S(r) and get a complex we denote by F· . Note that if F· is pure with degrees (r) d, then F· is pure with degrees r · d. In [4] we showed the following:

6


Theorem 2.1. Let m = gcd(e1 , . . . , en ) and let e = m·e . The space L (e) of diagrams satisfying the HK-equations (4) has a basis consisting of the βE(e )(m) (a) where a varies over Zn . Moreover when char. k = 0, the space L(e) is equal to L (e). This may also be formulated in terms of the associated (n + 1)-tuple of Betti polynomials. Corollary 2.2. Let s = (s0 , . . . , sn ) be the (n + 1)-tuple of Betti polynomials of E(e )(m) . Let B = (B0 , . . . , Bn ) be an (n + 1)-tuple of homogeneous Betti polynomials with difference vector of the total degrees equal to e, and which fulfill the HK-equations (4). Then B = p · s for some homogeneous Laurent polynomial p.

2.4 The Linear Space in the Case of Two Variables Now assume that S = k[x1 , x2 ] and let ξd (t, u) = t d−1 + t d−2 u + · · · + ud−1 be the cyclotomic polynomial. The first and last Betti polynomials of the equivariant resolution (6) are then given by

ξe2 (t, u),

(tu)e2 ξe1 (t, u),

respectively, and the middle Betti polynomial is given by

ξe1 +e2 = t e2 ξe1 (t, u) + ue1 ξe2 (t, u) = ue2 ξe1 (t, u) + t e1 ξe2 (t, u).

(7)

By Corollary 2.2 the space L (e1 , e2 ) may now be described by the following lemma. Lemma 2.3. Let e1 = mq and e2 = mp where m is the greatest common divisor of e1 and e2 . A triple of homogeneous Laurent polynomials B0 , B1 , B2 whose degrees have e1 and e2 as differences, is in L (e1 , e2 ) if and only if the following two equations hold: B2 (t, u) · ξ p (t m , um ) = (tu)mp B0 (t, u) · ξq (t m , um ), B1 (t, u) = u−pm B2 (t, u) + uqm B0 (t, u)

(8) (9)

= t −pm B2 (t, u) + t qm B0 (t, u). Proof. By Corollary 2.2 we have (B0 , B1 , B2 ) = f (t, u) · (ξ p (t m , um ), ξ p+q (t m , um ), (tu)mp ξq (t m , um )). This gives (8). Also (9) follows by (7). Conversely, if (8) and (9) hold, we may deduce that the equation above holds, so (B0 , B1 , B2 ) is in L (e1 , e2 ).

For a homogeneous Laurent polynomial f (t, u) denote by f dh (t) its dehomogenization with respect to u. If we now dehomogenize (8) we get an equation pm m Bdh · ξ p (t m ) = Bdh 2 /t 0 · ξq (t ).


7

Each of the first factors are uniquely determined by the other, and if the triple comes from an actual complex, the coefficients are non-negative. With some abuse of notation we also identify the cone P = P (e1 , e2 ) with the positive cone of pairs of Laurent polynomials (A(t), B(t)) in one variable t and with non-negative coefficients, such that B(t)ξ p (t m ) = A(t)ξq (t m ). We shall in the next section describe the cone P completely. Recall that we have an injective map P(e1 , e2 ) → P (e1 , e2 ). In Sect. 4 we show that this map is an isomorphism.

3 The Positive Cone of Bigraded Betti Diagrams In this section we describe completely the positive cone P (e1 , e2 ) of diagrams fulfilling the HK-equations (4). We shall show that there is a finite number of diagrams βλ parametrized by certain partitions λ such that the extremal rays in the positive cone are the one-dimensional rays generated by βλ (a) for a ∈ Z. Note. In the following we let e1 = mq and e2 = mp where m is the greatest common divisor of e1 and e2 .

3.1 Partitions Let N2 have the partial ordering where (a1 , a2 ) ≤ (b1 , b2 ) if a1 ≤ b1 and a2 ≤ b2 . An order ideal T in N2 (a subset closed under taking smaller elements) gives rise to two partitions. The first is given by

λ j+1 = 1 + max{i | (i, j) ∈ T }, j ≥ 0. The second is the dual partition

μi+1 = 1 + max{ j | (i, j) ∈ T }, i ≥ 0. (If for a given j no (i, j) is in T , we set λ j+1 = 0 and correspondingly for μi+1 .) Note that λ and μ are dual partitions. So μi+1 is the cardinality of { j | λ j > i}. We shall be interested in order ideals T which are contained in the region R(p, q) in the first quadrant bounded by the following strict inequality px + qy < (p − 1)(q − 1).

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Lemma 3.1. Let the order ideal T correspond to the partition λ . Then T is contained in the region above if and only if every aq − pλ p−a is nonnegative for 0 ≤ a < p. Correspondingly for the dual partition μ . Proof. First note that aq − pλ p−a ≥ 0 if and only if (p − 1 − a)q + (λ p−a − 1)p ≤ pq − p − q. Assume 0 ≤ a < p. If T is contained in R(p, q) then if λ p−a ≥ 1 it fulfills the second equation above and therefore the first. If λ p−a = 0 the first equation is also fulfilled. Suppose now that T fulfills the first equation. Then when λ p−a ≥ 1 the point (p −

1 − a, λ p−a − 1) is in R(p, q), so T is contained in R(p, q). The following easy lemma will be useful. Lemma 3.2. Let P(t) = ∑ cit i be a polynomial. Write P(t)ξd (t) = ∑ j∈Z α j t j . Then α j − α j−1 = c j − c j−d , for all j. Proof. This is clear from α j = ∑i= j−d+1 ci . j

The following result will essentially describe the extremal rays. Proposition 3.3. Suppose p and q are relatively prime and let T be an order ideal in R(p, q). Write p−1

AT (t) =

∑ t aq−pλ p−a ,

a=0

Then

q−1

BT (t) =

∑ t ap−qμq−a .

a=0

AT (t)ξq (t) = BT (t)ξ p (t).

Proof. Note that since p and q are relatively prime, the coefficient of each power t j in AT or BT is 0 or 1. Writing ∑ α j t j for the product AT (t)ξq (t) we see that when α j > α j−1 we have α j = α j−1 + 1. We shall show that the indices j for which this happens are exactly when j = 0 or j = pq− p−q−qu− pv where (u, v) is a maximal element in the poset T , i.e. (u, v) is in T , but neither (u + 1, v) nor (u, v + 1) is in T . Since the analog holds for the product BT (t)ξ p (t), these products must increase exactly at the same indices. An analogous argument also shows that they decrease at exactly the same indices, namely α j < α j−1 iff j = pq − qu − pv where (u, v) is not in T but (u − 1, v) and (u, v − 1) are either in T or have −1 as a coordinate. Hence the products are equal. Now α j > α j−1 when j = aq − pλ p−a for some a but (a − 1)q − pλ p−a is not a power in A(t). Thus either a = 0 or λ p+1−a < λ p−a . But this means that j = 0 or (u, v) = (λ p−a − 1, p − 1 − a) is a maximal element in T . We easily compute that j = aq − pλ p−a = pq − p − q − qu − pv.


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Remark 3.4. The empty poset T = 0/ corresponds to the polynomials A0/ (t) = ξ p (t q ) B0/ (t) = ξq (t p ). Via the correspondence at the end of Sect. 2.4 these corresponds to the Betti diagram of a resolution of an artin module. This is the module described in [1, Remark 3.2] which is the quotient I/J of two monomial ideals in k[x, y]: the ideal I = (x(p−1)q , x(p−2)q yq , . . . , y(p−1)q ) and the ideal J = (x pq , x p(q−1) y p , . . . , y pq ). Remark 3.5. There is also a maximal order ideal Tˆ in the region R(p, q) and this corresponds to the polynomials ATˆ (t) = ξ p (t) BTˆ (t) = ξq (t) which again via the correspondence at the end of Sect. 2.4 corresponds to the Betti diagrams of the GL(2)-equivariant resolutions E(p, q) constructed in [2].

3.2 Decomposing Now any polynomial A(t) may be written p−1

A(t) =

∑ ∑ αa,bt aq−bp .

a=0 b∈Z

For each a let λ p−a be the maximum of the set {b | αa,b = 0}. We may then write A(t) = Amin (t) + A+ (t) where

p−1

Amin (t) =

∑ αa,λ p−a t aq−λ p−a p .

a=0

Correspondingly we may write q−1

B(t) =

∑ ∑ βa,bt ap−bq .

a=0 b∈Z

For each a let μq−a be the maximum of the set {b | βa,b = 0}. We may then write B(t) = Bmin (t) + B+ (t) where

q−1

Bmin (t) =

∑ βa,μq−a t ap−μq−a q .

a=0

10


Proposition 3.6. Let p and q be relatively prime. Assume A(t) and B(t) are polynomials with nonnegative coefficients and nonzero constant terms. Suppose that A(t)ξq (t) = B(t)ξ p (t). Let λ and μ be the sequences corresponding to Amin (t) and Bmin (t). Then these sequences are partitions which are dual. Proof. Write the product above as ∑ α j t j . Let 0 ≤ b < p − 1 and assume bq − pλ occurs as a power in Amin (t), so λ = λ p−b . We want to show that λ p−1−b ≥ λ p−b . If (b + 1)q − pλ occurs as a power in A(t) then clearly λ p−1−b ≥ λ = λ p−b . So assume (b + 1)q − pλ does not occur in A(t). By Lemma 3.2 applied to A(t)ξq (t):

α(b+1)q−pλ < α(b+1)q−pλ −1 so (b + 1)q − p(λ + 1) is a power in B(t). We may now write (b + 1)q − p(λ + 1) = (q − λ − 1)p − q(p − b − 1). There will then be an a ≤ q − λ − 1 such that a p − q(p − b − 1) is in B(t) but (a − 1)p − q(p − b − 1) is not. By Lemma 3.2 applied to B(t)ξ p (t):

αa p−q(p−b−1) > αa p−q(p−b−1)−1 . Now we may write a p − q(p − b − 1) = (b + 1)q − p(q − a ) and recall that q − a ≥ λ + 1. Again by Lemma 3.2 we get that the number in this equation will occur as a power in A(t). But this means that

λ p−1−b ≥ q − a > λ = λ p−b . Since A(t) and equivalently B(t) has nonzero constant term, we have λ p = 0 so we get a partition λ . An analogous argument gives that the sequence of μi ’s also form a partition. Now let T be the order ideal corresponding to λ and T the order ideal corresponding to μ . We show that they are equal and so λ and μ will be dual partitions. Suppose λ p−b < λ p−1−b . Then bq − pλ p−1−b is not in A(t). By Lemma 3.2

α(b+1)q−pλ p−1−b > α(b+1)q−pλ p−b −1 . And this implies again by Lemma 3.2 that (b + 1)q − pλ p−1−b occurs as a power in B(t). Rewriting, this is (q − (λ p−1−b ))p − q(p − 1 − b), which means that (λ p−1−b − 1, r) is in T for some r ≥ p − b − 2. We conclude that T contains T . By symmetry, we also get the opposite inclusion so the order ideals are in fact equal.


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Corollary 3.7. The polynomials A(t) and B(t) can be written as A(t) = ∑T,i γT,i × t cT,i AT (t) and B(t) = ∑T,i γT,i t cT,i BT (t) where the sum is over order ideals T in R(p, q) and a running index i for each T . Proof. Let α be the minimal positive coefficient of Amin (t) and Bmin (t) and suppose these correspond to the order ideal T . Then we can subtract α AT (t) from A(t) to get A (t) and similarly subtract α BT (t) from B(t) to get B (t). Thus we get that A (t)ξq (t) = B (t)ξ p (t) and we may proceed inductively, since then new polynomials have no more terms than the original ones, and one of them has strictly less.

From this we obtain our goal of describing the extremal rays of the cone P described at the end of Sect. 2.4. Theorem 3.8. Let e1 = mq and e2 = mp where p and q are relatively prime. The rays generated by (t c AT (t m ),t c BT (t m )) where T is an order ideal in R(p, q) and c ∈ Z, are the extremal rays in the cone P (e1 , e2 ). In particular any element in this cone may be written as a positive linear combination of these elements. Proof. In the case m = 1 this follows immediately from Proposition 3.7. For m > 1, assume that we have A(t)ξq (t m ) = B(t)ξ p (t m ). m−1 i We may then write A(t) = ∑i=0 t Ai (t m ) and correspondingly for B(t). Hence we must have the equations

Ai (t m )ξq (t m ) = Bi (t m )ξ p (t m ) for each i. By Corollary 3.7 we may then conclude the proof.

Remark 3.9. Such a positive linear combination is in general not unique. Remark 3.10. We see that the extremal rays fall into classes, one for each order ideal T in R(p, q). These form a poset with a minimal element T = 0/ and a maximal element Tˆ . In Remarks 3.4 and 3.5 we showed that these correspond to Betti diagrams of well known resolutions.

4 Existence of Resolutions We will now show that for any extremal ray in P (e1 , e2 ) there is a resolution whose Betti diagram is in this extremal ray. This will show that P (e1 , e2 ) = P(e1 , e2 ). Given an order ideal T in R(p, q) where p and q are relatively prime. If e1 = mq and e2 = mp, we have the two polynomials AT (t m ) and BT (t m ). Homogenizing these we may construct an associated triple B0 , B1 , B2 fulfilling the equations of Lemma 2.3, with positive integer coefficients. These lie on an extremal ray in

12


P (e1 , e1 ). Note that in B0 and B2 each monomial occurs with coefficient 0 or 1 and similarly for B2 . We may therefore apply the following proposition whose proof will occupy this section. Proposition 4.1. Let k be any field, and (B0 , B1 , B2 ) be a triple of homogeneous Laurent polynomials over Z of increasing degrees, fulfilling the HK-equations (4). If the coefficients of each monomial of B0 and B2 is 0 or 1, there is a minimal free resolution β α (10) S.B0 (t, u) ←− S.B1 (t, u) ←− S.B2 (t, u) of an artinian S = k[x1 , x2 ]-module. As a consequence we get the following. Theorem 4.2. Let e1 = mq and e2 = mp where p and q are relatively prime. Let (B0 , B1 , B2 ) be the triple of homogeneous Laurent polynomials associated to an order ideal T in R(p, q), with t m as argument. Then this is a triple of Betti polynomials associated to a bigraded artinian module. Hence the cone P(e1 , e2 ) = P (e1 , e2 ), and so also L(e1 , e2 ) = L (e1 , e2 ) for any field k. Remark 4.3. Proposition 4.1 holds for any B0 and B2 with nonnegative integer coefficients. But for ease of demonstration we make the above assumptions. Remark 4.4. In the case of three variables it is not true that P(e1 , e2 , e2 ) is equal to P (e1 , e2 , e3 ). We provide an example where this is not so in the last section. We shall prove Proposition 4.1 towards the end of this section. But the following outlines what we need to show. Since ker α is a free module, ker α /im β will be either 0 or nonzero of codimension one or zero. But the latter is equivalent to coker β ∨ being nonzero of codimension one or zero. Hence we need to show the following. • coker α is of codimension two. • coker β ∨ is of codimension two. • The composition α ◦ β = 0. In the rest of this section we assume that (B0 , B1 , B2 ) fulfills the hypothesis of Proposition 4.1. First we have the following. Lemma 4.5. Given a bidegree (i, j) with i+ j ≥ deg B2 (t, u)−1. Then the dimension of the bigraded part S.B1 (t, u)i, j is the sum of the dimensions of S.B0 (t, u)i, j and S.B2 (t, u)i, j . Proof. Since the Bi fulfill the HK-equations, the following quotient is a polynomial: h(t, u) =

∑i,a (−1)i βi,a · t a1 ua2 . (1 − t)(1 − u)

1 , the coWrite h(t, u) as ∑ αi, j t i u j . Note that since S has Hilbert series (1−u)(1−t) efficient αi, j will be the alternating sum of the dimensions of the S.B p (t, u)i, j . We will show that αi, j = 0 for i + j ≥ deg B2 (t, u) − 1. Let (i, j) be a maximal pair (for the partial order on Nn ) such that αi, j is nonzero. The pair (i + 1, j + 1) must then


13

occur as a power in the numerator in the fraction above. But this implies in turn that

i + j + 2 is less or equal to the degree of B2 (t, u). To facilitate the discussion we now introduce some notation. Let s, e : [1, . . . , n] → [1, . . . , m] be two weakly increasing functions such that s(i) ≤ e(i). The subset D = {(i, j) | s(i) ≤ j ≤ e(i)} of [1, . . . , n] × [1, . . . , m] is a thick diagonal. We then write s = sD and e = eD . If s(1) = 1 and e(n) = m and s and e are strictly increasing we call D a strict thick diagonal. If s is only strictly increasing as soon as s(i) > 1 and e is only strictly increasing as long as e(i) < m, we call D semi-strict. 1 2 p Let B0 (1, 1) = p and B2 (1, 1) = q and write B0 (t, u) = ∑i=1 t ai uai where {a1i } is strictly increasing and {a2i } is strictly decreasing. Similarly for B2 (t, u) with pairs (c1k , c2k ) and for B1 (t, u) with pairs (b1j , b2j ) but now with the {b1j } only weakly increasing and the {b2j } only weakly decreasing. We may now note that the positions where α may have nonzero entries, i.e. those pairs (i, j) such that (a1i , a2i ) ≤ (b1j , b2j ), form a thick diagonal Dα of [1, . . . , p] × [1, . . . , p + q]. It has no zero rows because of the HK-equations (4): for each (a1i , a2i ) there is a (b1j , b2j ) with a1i = b1j . Similarly we have a thick diagonal Dβ ∨ in [1, . . . , q]× [1, . . . , p + q]. Lemma 4.6. a. sDα (i) = j if and only if j is the smallest index such that a1i = b1j . b. eDα (i) = j if and only if j is the largest index for which a2i = b2j . The analogous result holds for Dβ ∨ . Proof. Let sDα (i) = j and let j˜ be the smallest index such that a1i = b1j˜. Such an index exists by the HK-equations. Clearly j ≤ j˜. But if j < j˜ then b1j < b1˜ and so j

we could not have (a1i , a2i ) ≤ (b1j , b2j ) The other arguments are analogous.

Corollary 4.7. The thick diagonal Dα is strict. Similarly Dβ ∨ is strict. Proof. Since the a1i are strictly increasing, we get that sDα is strictly increasing. We thus need to show that sDα (1) = 1. Let sDα (1) = j. Suppose j is not 1. Then b11 < a11 . By the HK-equations there will be (c1k , c2k ) with c1k = b11 . But then again there will be a (b1j , b2j ) with b2j = c2k and this would have b1j < c1k = b11 which is impossible. Thus sDα (1) = j = 1.

Lemma 4.8. Let D be a semi-strict diagonal of [1, . . . , n] × [1, . . . , n + 1] with eD (1) > 1 and sD (n) < n + 1. Let A be a general matrix of type D (meaning that only the positions of D may have nonzero entries). Then there is a vector in the null space of A with nonzero first and last coordinates. Proof. If we omit the first column we get an n × n-matrix of semi-strict diagonal type. But a general such matrix is easily seen to be non-singular. Hence a null vector must have nonzero first coordinate. Similarly for the last coordinate.

Lemma 4.9. If α is nonzero in positions (i, sDα (i)) and (i, eDα (i)) for i = 1, . . . , p, then coker α has codimension two. Similarly for the map β ∨ .

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Proof. By Lemma 4.6, in the first position of each row there is a power of x2 . Hence for the matrix to degenerate we must have x2 = 0. Similarly there is a power of x1 in the last position, and so x1 = 0 when the matrix degenerates.

Now when α and β are composed, columns in β are multiplied with the rows of α . Motivated by this we have the following. Lemma 4.10. Let k be a column in Dβ which starts in position ( j0 , k) and ends in ( j1 , k). Then Dα restricted to [1, . . . , p] × [ j0 , j1 ] has j1 − j0 nonzero rows, say the interval [i0 , i1 ] where j1 − j0 = i1 − i0 + 1, and Dα restricted to [i0 , i1 ] × [ j0 , j1 ] is semi-strict with eDα (i0 ) > j0 and sDα (i1 ) < j1 . Proof. 1. That j1 − j0 = i1 − i0 + 1 follows from Lemma 4.5 by restricting to the bidegree (c1k , c2k ). 2. Now we show eDα (i0 ) > j0 . By the HK-equations there is a (b1j , b2j ) with b2j = c2k . Since the b2j are decreasing, this must happen for j = j0 . (This is the analog of Lemma 4.6 for β ∨ .) Clearly eDα (i0 ) ≥ j0 . If we have equality, by Lemma 4.6 we get a2i0 = b2j0 . But then a2i0 = c2k and by the HK-equations there must then be two (b1j , b2j ) with b2j = a2i0 = c2k . But this gives eDα (i0 ) > j0 . Similarly we can argue that sDα (i1 ) < j1 . 3. That the restriction is semi-strict follows from i) sDα and eDα are strictly increasing, ii) sDα (i0 ) ≤ j0 , and iii) eDα (i1 ) ≥ j1 . To show ii) note that if sDα (i0 ) > j0 then clearly sDα (i1 ) > j0 +i1 −i0 = j1 −1. But this contradicts that sDα (i1 ) ≤ j1 −1. Similarly we can show iii).

Proof of Proposition 4.1. We choose α to be a general matrix, homogeneous with respect to the multidegrees. It will be of type Dα and it degenerates in codimension two by Lemma 4.9. By Lemmata 4.10 and 4.9 we get for each column k in Dβ a vector in the kernel of α which is nonzero in positions sDβ ∨ (k) and eDβ ∨ (k). Hence these kernel vectors make up the columns of the matrix of a map β such that β ∨ degenerates in codimension two by Lemma 4.9. Also the composition α ◦ β = 0, and this is what we needed to show.

5 Resolutions of Trigraded Artinian Modules of Codimension Three In the case of trigraded artinian modules over the polynomial ring k[x, y, z] where the resolution has pure total degrees, we do not know much. The following are natural questions. • For Betti diagrams with given total degrees, are there, up to translation, only a finite number of extremal rays in the positive cone of such Betti diagrams?


15

• Suppose the above property holds. From Sect. 3 we know that the translation classes of extremal rays form a poset with a unique minimal member and a unique maximal member. Is there a maximal member in the translation classes in the three variable case also? We do not know the answer to these questions. A general fact we do know is that L(e) = L (e) when char. k = 0. However in three variables it is not the case that the injection P(e) → P (e) is an isomorphism. Let us consider as example the case of resolutions of type 0, 1, 2, 1. The equivariant resolution of this type has the form (we have listed the tridegrees of the generators below each free module) S3 ← S6 ← S6 ← S3 . 100 200 211 221 010 020 121 212 001 002 112 122 110 220 101 202 011 022

(11)

To facilitate notation write ∑i ki β (ai , bi , ci ) as ∑i [ki (ai , bi , ci )]β . Let β be the Betti diagram of the complex (11). One may check that [(2, 1, 0) + (0, 2, 1) + (1, 0, 2) − (1, 1, 1)]β gives a diagram with no negative entries (and it fulfills the HK-equations). But no multiple of this is the Betti diagram of a module: If F• is a complex with this diagram, then S(−3, −1, 0) is a term in F0 . But there is no term S(−3, −1, ∗) in F1 (but there is one in F3 ), and so the cokernel of F1 → F0 cannot have codimension three. In particular this diagram is in P (1, 2, 1) but not in P(1, 2, 1). However let α be the diagram [(2, 1, 0) + (2, 0, 1) + (1, 2, 0) + (0, 2, 1) + (1, 0, 2) + (0, 1, 2) − (1, 1, 1)]β . Claim 1. The diagrams β and α are Betti diagrams of resolutions of indecomposable artinian trigraded modules of codimension three, and they generate rays which are extremal rays in the cone P(1, 2, 1). Proof. That β is a Betti diagram is clear and that it resolves an indecomposable module is also immediate to see from the resolution. That α is a Betti diagram of a resolution of an indecomposable module, may be checked on Macaulay 2 by filling in general monomial matrices with the tridegrees of α . Now the only way α can decompose into nonnegative diagrams which are not on its ray, may be worked out to be as follows. [ c1 ((2, 1, 0) + (0, 2, 1) + (1, 0, 2) − (1, 1, 1)) + c2 ((1, 2, 0) + (0, 1, 2) + (2, 0, 1) − (1, 1, 1)) + c3 (1, 1, 1)]β

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where c1 = c2 = c3 . But the same argument used to show that the diagram corresponding to the first term is not a resolution may be used to show that a linear combination as above is the diagram of a resolution only if c1 = c2 = c3 is a positive integer.

It would be interesting to know if there are other extremal rays in the cone P apart from the translates of α and β .

References 1. M. Boij, J. Söderberg, Graded Betti numbers of Cohen–Macaulay modules and the multiplicity conjecture, Journal of the London Mathematical Society, 78, no. 1 (2008), p. 78–101. 2. D. Eisenbud, G. Fløystad, J. Weyman, The existence of pure free resolutions, arXiv:0709.1529, to appear in Annales de l’institut Fourier. 3. D. Eisenbud, F.-O. Schreyer, Betti numbers of graded modules and cohomology of vector bundles, Journal of the American Mathematical Society 22 (2009), p. 859–888. 4. G. Fløystad, The linear space of Betti diagrams of multigraded artinian modules, Mathematical Research Letters, 17, no. 5 (2010), p. 943–958. 5. W. Fulton, J. Harris, Representation theory, GTM 129, Springer Verlag 1991.

Koszul Cycles Winfried Bruns, Aldo Conca and Tim Römer

Abstract We prove regularity bounds for Koszul cycles holding for every ideal of dimension ≤ 1 in a polynomial ring; see Theorem 3.5. In Theorem 4.7 we generalize the “c + 1” lower bound for the Green–Lazarsfeld index of Veronese rings proved in (Bruns et al., arXiv:0902.2431) to the multihomogeneous setting. For the Koszul complex of the c-th power of the maximal ideal in a Koszul ring we prove that the cycles of homological degree t and internal degree ≥ t(c + 1) belong to the t-th power of the module of 1-cycles; see Theorem 5.2.

1 Introduction The Koszul complex and its homology are central objects in commutative algebra. Vanishing theorems for Koszul homology are the key to many open questions. The goal of the paper is the study of regularity bounds for Koszul cycles and Koszul homology of ideals in standard graded rings. Our original motivation comes from the study of the syzygies of Veronese varieties and, in particular, the conjecture of Ottaviani and Paoletti [12] on their Green–Lazarsfeld index, see [4]. In Sect. 2 we fix the notation and describe some canonical maps between modules of Koszul cycles. Given a standard graded ring R with maximal homogeneous ideal m, a homogeneous ideal I and a finitely generated graded module M, we let Winfried Bruns Universität Osnabrück, Institut für Mathematik, 49069 Osnabrück, Germany e-mail: [email protected] Aldo Conca Dipartimento di Matematica, Universitá di Genova, Via Dodecaneso 35 16146 Genova, Italy e-mail: [email protected] Tim Römer Universität Osnabrück, Institut für Mathematik, 49069 Osnabrück, Germany e-mail: [email protected] G. Fløystad et al. (eds.), Combinatorial Aspects of Commutative Algebra and Algebraic Geometry, Abel Symposia 6, DOI 10.1007/978-3-642-19492-4 2, © Springer-Verlag Berlin Heidelberg 2011

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Zt (I, M) denote the module of Koszul cycles of homological degree t. Under a mild assumption, we show in 2.4 that Zs+t (I, M) is a direct summand of Zs (I, N) where N = Zt (I, M). Section 3 is devoted to the description of (Castelnuovo–Mumford) regularity bounds for Koszul cycles and homology. We prove bounds of the following type: regR (Zt (I, M)) ≤ t(c + 1) + regR (M) + v

(1)

under assumptions on dim M/IM. Here regR (N) denotes the (relative) Castelnuovo– Mumford regularity of a finitely generated R-module N. Note that regR (N) is the ordinary Castelnuovo–Mumford regularity if R is the polynomial ring. Furthermore it is known that regR (N) is finite if R is a Koszul algebra. If R is Koszul and dim M/IM = 0, then we prove that (1) holds with v = 0 and where c is such that mc ⊂ I + Ann(M) and I is generated in degrees ≤ c, see 3.2. In 3.5 we prove that if R is a polynomial ring of characteristic 0 or big enough and dim M/IM ≤ 1 then (1) holds with v = 0 and c = regR (I). Furthermore, if R is a polynomial ring and dim M/IM = 0, then we show in 3.9 that (1) holds with c ≥ the largest degree of a generator of I and v = dim[R/I]c . We also give examples showing that the inequality regR Zt (I, M) ≤ t(regR (I) + 1) + regR (M)

(2)

cannot hold in general (i.e. without restriction on the dimension of M/IM). However (2) holds if R is a polynomial ring, M = R/J and both I and J are strongly stable monomial ideals, see 3.7 and 3.8. We leave it as an open question whether (2) holds when M = R and R is a polynomial ring. In Sect. 4 we prove that, given a vector c = (c1 , . . . , cd ) ∈ Nd+ , the Segre–Veronese ring associated to c over a field of characteristic 0 or big enough, has a Green– Lazarsfeld index larger than or equal to min(c) + 1, see 4.7. This result was announced in [4] and improves the bound of Hering, Schenck and Smith [11] by 1. In Sect. 5 we analyze the generators of the module Zt = Zt (mc , R) under the assumption that R has characteristic 0 or big enough. If R is Koszul we prove that Zt /Z1t vanishes in degrees ≥ t(c + 1), 5.2. Here Z1t denotes the image of the canonical map ∧t Z1 → Zt . This allows us to deduce that the c-th Veronese subring of a polynomial ring S satisfies the property N2c if and only if H1 (mc , S)2c = 0, see 5.3. Finally, we prove that the cycles given in [4] generate Z2 ; see 5.5.

2 Notation and Generalities In this section we collect notation and general facts about maps between modules of Koszul cycles. Let R be a ring, F be a free R-module of rank n, ϕ : F → R be an R-linear map and M be an R-module. All tensor products are over R. We n ϕ , R) = t=0 Kt (ϕ , R) = • F and K(ϕ , M) = consider the Koszul complexes K( n • F ⊗ M. The complex K(ϕ , M) can be seen as a module over t=0 Kt (ϕ , M) =

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the exterior algebra K(ϕ , R). For a ∈ K(ϕ , R) and f ∈ K(ϕ , M) the multiplication will be denoted by a. f . The differential of K(ϕ , R) and K(ϕ , M) will be denoted simply by ϕ and it satisfies

ϕ (a. f ) = ϕ (a). f + (−1)s a.ϕ ( f ) for all a ∈ Ks (ϕ , R) and f ∈ K(ϕ , M). We let Zt (ϕ , M), Bt (ϕ , M), Ht (ϕ , M) denote the cycles, the boundaries and the homology in homological degree t and set Z(ϕ , M) = ⊕Zt (ϕ , M), and so on for cycles, boundaries and homology. One knows that Z(ϕ , R) is a subalgebra of K(ϕ , R) and that B(ϕ , R) is an ideal of Z(ϕ , R) so that the homology H(ϕ , R) is itself an algebra. More generally, Z(ϕ , M) is a Z(ϕ , R)-module. We let Zs (ϕ , R)Zt (ϕ , M) denote the image of the multiplication map Zs (ϕ , R) ⊗ Zt (ϕ , M) → Zs+t (ϕ , M). Similarly, Z1 (ϕ , R)t will denote the image t of the map Z1 (ϕ , R) → Zt (ϕ , R). In the graded setting the map ϕ will be assumed to be of degree 0 and F will be a direct sum of shifted copies of R. In this way the Koszul complex K(ϕ , M) inherits a graded structure for the map ϕ and the module M. So cycles, boundaries and homology have an induced graded structure. An index on the left of a graded module always denotes the selection of the homogeneous component of that degree. If R is standard graded over a field K with maximal homogeneous ideal m all the invariants we are going to study depend actually only on the image of ϕ and not on the map itself as long as ker ϕ ⊆ mF. So, if J = Im ϕ , we will sometimes denote K(ϕ , R) simply by K(J, R) and so on. Fix a basis of the free module F, say {e1 , . . . , en }. Given I = {i1 , . . . , is } ⊂ [n] with i < i2 < · · · < is we write eI for the corresponding basis element ei1 ∧ · · · ∧ eis 1 of s F. If ϕ (ei ) = ui ∈ J we will also use the symbol [ui1 , . . . , uis ] to denote eI . For disjoint subsets A, B ⊂ [n] we set ε (A, B) = #{(a, b) ∈ A × B : a > b} and

σ (A, B) = (−1)ε (A,B) . One has

eA eB = σ (A, B)eA∪B .

For further application we record the following: Lemma 2.1. For disjoint subsets A, B,C of [n] one has

σ (A ∪ B,C)σ (B, A) = σ (B, A ∪C)σ (A,C). Proof. Just use the fact that ε (A ∪ B,C) = ε (A,C) + ε (B,C) and ε (B, A ∪ C) = ε (B, A) + ε (B,C).

Any element f ∈ s F ⊗ M can be written uniquely as f = ∑ eI ⊗ mI with mI ∈ M where the sum is over the subsets of cardinality s of [n]. If mI = 0 then we will say that eI does not appear in f . For every f ∈ Ks+t (ϕ , M) and for every I ⊂ [n] with s = #I we have a unique decomposition f = aI + eI .bI

(3)

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with aI ∈ Ks+t (ϕ , M) and bI ∈ Kt (ϕ , M), and, furthermore, eJ does not appear in aI whenever J ⊃ I and eS does not appear in bI whenever S ∩ I = 0. / With the notation above we have: Lemma 2.2. For every f ∈ Ks+t (ϕ , M) we have: (a) ∑I eI .bI = t+s s f where ∑I stands for the sum extended to all the subsets I ⊂ [n] with s = #I. (b) if f ∈ Zs+t (ϕ , M), then bI ∈ Zt (ϕ , M) for every I with s = #I. Proof. For (a) one writes f = ∑ eJ ⊗ mJ with J ⊂ [n] with #J = s + t and mJ ∈ M. Then one observes t+sthat eJ .mJ appears in eI .bI iff I ⊂ J. Hence eJ .mJ appears in ∑I eI .bI exactly s times. For (b) one applies the differential 0 = ϕ ( f ) = ϕ (aI ) + ϕ (eI ).bI +(−1)s eI .ϕ (bI ) and since eJ does not appear in ϕ (aI )+ ϕ (eI ).bI whenever

J ⊇ I then ϕ (bI ) must be 0. The multiplication Ks (ϕ , R) ⊗ Kt (ϕ , M) → Ks+t (ϕ , M) can be interpreted as a map Ks (ϕ , Kt (ϕ , M)) → Ks+t (ϕ , M) defined by a ⊗ f → a. f . Restricting the domain of the map to Ks (ϕ , Zt (ϕ , M)) we get a map Ks (ϕ , Zt (ϕ , M)) → Ks+t (ϕ , M) which is indeed a map of complexes. So it induces a map

αt : Zs (ϕ , Zt (ϕ , M)) → Zs+t (ϕ , M) defined by

∑ a ⊗ f ∈ Zs (ϕ , Zt (ϕ , M)) → ∑ a. f .

Now we define a map

γt : Ks+t (ϕ , M) → Ks (ϕ , Kt (ϕ , M)) by the formula

γt ( f ) = ∑ eI ⊗ bI I

where the sum is over the I ⊂ [n] with #I = s and bI is determined by the decomposition (3). We claim: Lemma 2.3. The map γt : K(ϕ , M) → K(ϕ , Kt (ϕ , M))[−t] is a map of complexes. Proof. Since K(ϕ , M) = K(ϕ , R) ⊗ M and we have K(ϕ , Kt (ϕ , M)) = K(ϕ , Kt (ϕ , R)) ⊗ M it is enough to prove the statement in the case M = R. Then it is enough to check

γt ◦ ϕ (eJ ) = ϕ ◦ γt (eJ ) for every J ⊂ [n] with #J = s + t. Note that

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γt (eJ ) = ∑ σ (A, B)eA ⊗ eB where the sum is over all the B such that #B = t and A = J \ B. Then

ϕ ◦ γt (eJ ) = ∑ σ (A ∪ {p}, B)σ ({p}, A)ϕ (e p )eA ⊗ eB and

γt ◦ ϕ (eJ ) = ∑ σ ({p}, A ∪ B)σ (A, B)ϕ (e p )eA ⊗ eB

where in both cases the sum is over all the partitions of J into three parts A, B, {p} with #B = t. So we have to check that

σ (A ∪ {p}, B)σ ({p}, A) = σ ({p}, A ∪ B)σ (A, B).

This is a special case of 2.1. It follows that γt gives, by restriction, a map Zs+t (ϕ , M) → Zs (ϕ , Kt (ϕ , M)).

By virtue of 2.2, its image is indeed contained in Zs (ϕ , Zt (ϕ , M)). So we have a map

βt : Zs+t (ϕ , M) → Zs (ϕ , Zt (ϕ , M)) and, by virtue of Lemma 2.2, we have t +s f αt ◦ βt ( f ) = s

for all f ∈ Zs+t (ϕ , M).

An immediate consequence: is invertible in R. Then Zs+t (ϕ , M) is a direct summand Lemma 2.4. Assume t+s s of Zs (ϕ , Zt (ϕ , M)). One can easily check that, in the graded setting, the maps described in this section are graded and of degree 0.

3 Bounds for Koszul Cycles In this section we consider a field K and a standard graded K-algebra R with maximal homogeneous ideal m. In other words, R is of the form S/J where S is a polynomial ring over K with the standard grading and J is a homogeneous of S. We will consider a finitely generated graded R-module M. Let βi,Rj (M) = dimK TorRi (M, K) j be the graded Betti numbers of M over R. We define the number tiR (M) = max{ j ∈ Z : βi,Rj (M) = 0}, whenever TorRi (M, K) = 0 and tiR (M) = −∞ otherwise. The Castelnuovo–Mumford regularity of M over R is

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regR (M) = sup{tiR (M) − i : i ∈ N}. Recall that R is a Koszul algebra if regR (K) = 0. One knows that regR (M) is finite for every finitely generated module M if R is a Koszul algebra, see Avramov and Eisenbud [2]. One says that R has the property Np if its defining ideal J is generated by quadrics and the syzygies of the quadrics are linear for p − 1 steps, that is, if tiS (R) ≤ i + 1 for i = 1, . . . , p. The Green–Lazarsfeld index of R is the largest number p such that R has the property N p , that is, index(R) = max{p : tiS (R) ≤ i + 1 for i = 1, . . . , p}. Conventions. Just to avoid endless repetitions, throughout this section ideals will be homogeneous, modules will be finitely generated and graded, linear maps will be graded of degree 0. Furthermore I will always denote an ideal and M a module of the current ring. The current ring will be denoted by S if it is the polynomial ring over a field K or by R if it is a standard graded K-algebra and m will denote its maximal homogeneous ideal. We start with a well-known fact that is easy to prove: Lemma 3.1. One has I + Ann(M) ⊆ Ann(M/IM) ⊆

I + Ann(M).

We have: Proposition 3.2. Assume R is Koszul and dim M/IM = 0. Let c be the smallest integer such that mc ⊆ I + Ann(M) and I is generated in degree ≤ c (such a number c exists by 3.1). Set Zt = Zt (I, M) and Ht = Ht (I, M). Then, for every t, regR (Zt ) ≤ t(c + 1) + regR (M) and regR (Ht ) ≤ t(c + 1) + regR (M) + c − 1. Proof. The proof is a slight generalization of the arguments given in [4, Sect. 2]. Set Bt = Bt (I, M) and Kt = Kt (I, R). Note that I + Ann(M) annihilates Ht . Hence mc Ht = 0. It follows that Ht vanishes in degrees ≥ t0R (Zt ) + c and hence regR (Ht ) ≤ t0R (Zt ) + c − 1 ≤ regR (Zt ) + c − 1. So the second formula follows from the first. The short exact sequence 0 → Bt → Zt → Ht → 0 gives regR (Bt ) ≤ max{reg(Zt ), regR (Ht ) + 1} ≤ regR (Zt ) + c and 0 → Zt+1 → Kt+1 ⊗ M → Bt → 0

Koszul Cycles

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gives regR (Zt+1 ) ≤ max{regR (Kt+1 ⊗ M), regR (Bt ) + 1} ≤ max{(t + 1)c + regR (M), regR (Zt ) + c + 1} Now the statement can be proved by induction on t, the case t = 0 being obvious

since Z0 = M. We single out a special case of 3.2: Proposition 3.3. Assume that dim S/I = 0. Set Zt = Zt (I, M) and Ht = Zt (I, M). Then, for every t, regS (Zt ) ≤ t(regS (I) + 1) + regS (M) and regS (Ht ) ≤ t(regS (I) + 1) + regS (M) + regS (I) − 1. Proof. The number c of 3.2 is ≤ regS (I).

The following remark explains why the assumption on the dimension of S/I is necessary in 3.3. Remark 3.4. The module Z1 (I, M) sits in the exact sequence: 0 → Z1 (I, M) → F ⊗ M → IM → 0. Hence regS (IM) ≤ max{regS (I) + regS (M), regS (Z1 (I, M)) − 1}. There are plenty of examples such that regS (IM) > regS (I) + regS (M) already when M = I, see Conca [7] or Sturmfels [13]. Therefore, in these examples, one has regS (Z1 (I, M)) > regS (I) + 1 + regS (M). But using a result of Caviglia [5], see also Eisenbud, Huneke and Ulrich [8], we are able to show: Theorem 3.5. Assume that dim M/IM ≤ 1. Assume also that either char K = 0 or > t. Set Zt = Zt (I, M). Then regS (Zt ) ≤ t(regS (I) + 1) + regS (M) for every t. Proof. By induction on t. For t = 1 note that, by [5], we have regS (IM) ≤ regS (I) + regS (M) and the short exact sequence of 3.4 implies that regS (Z1 ) ≤ regS (I) + 1 + regS (M). For t > 1, by virtue of 2.4 we have that Zt is a direct summand of Zt−1 (I, Z1 ). Hence regS (Zt ) ≤ regS (Zt−1 (I, Z1 )). Since Ann(Z1 ) ⊇ Ann(M) we have Ann(Z1 ) + I ⊇ Ann(M) + I and, by 3.1, dim Z1 /IZ1 ≤ M/IM ≤ 1. Hence, by induction, we have

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regS (Zt−1 (I, Z1 )) ≤ (t − 1)(regS (I) + 1) + regS (Z1 ). Since regS (Z1 ) ≤ regS (I) + 1 + regS (M) has been already established, the desired inequality follows.

Question 3.6. (1) Does the inequality in 3.5 hold over a Koszul algebra R? And is the assumption on the characteristic needed? (2) Is it true that regS (Zt (I, S)) ≤ t(regS (I) + 1) holds for every homogeneous ideal I ⊂ S? Since Z1 (I, S) is the first syzygy module of I the inequality of 3.6 is actually an equality for t = 1. An indication that the answer to 3.6(2) might be “yes” for some classes of ideals is given in 3.7 and 3.8. Recall that a monomial ideal I ⊂ S = K[x1 , . . . , xn ] is strongly stable if whenever a monomial m ∈ I is divisible by a variable xi , then mx j /xi ∈ I for every j < i. In characteristic 0 the strongly stable ideals are exactly the ideals of S which are fixed by the Borel group of the upper triangular matrices of GLn (K) acting on S. The Eliahou–Kervaire complex [9] gives the graded minimal free resolution of strongly stable ideals. For us it is important to recall that if I is strongly stable then regS (I) is the largest degree of a minimal generator of I. Proposition 3.7. Let I ⊂ S be a strongly stable ideal. Set Zt = Zt (I, S). Then Zt is generated by elements of degree ≤ t(regS (I) + 1). Proof. Set c = regS (I). The idea of the proof follows essentially the argument given in [4, Theorem 3.3]. We note first that, as we are dealing with a monomial ideal I, the modules Zt have a natural Zn -graded structure as long as we consider the free presentation F → I associated with the monomial generators of I. We do a double induction on n and on t. The case n = 1 is obvious. The case t = 1 is easy and follows from the description of the (first) syzygies of I given in [9]. By induction on t it is enough to verify that Zt /Z1 Zt−1 is generated in degree < t(c + 1). Hence it suffices to show that every Zn -graded element f ∈ Zt of total degree q ≥ t(c + 1) can be written modulo Z1 Zt−1 as a multiple of an element in Zt of total degree < q. Let α ∈ Zn be the Zn -degree of f . If αn = 0 then we can conclude by induction on n. Therefore we may assume that αn > 0. Let u ∈ I be a monomial generator of I with xn | u and [u] the corresponding free generator of F. We have the decomposition f = a + [u]b with b ∈ Zt−1 and [u] does not appear in a. Note that b has degree q − deg(u) ≥ q − c. Since Zt−1 is generated by elements of degree ≤ (t − 1)(c + 1) we may write s

b=

∑ λ jv jz j

(4)

j=1

where λ j ∈ K, z j ∈ Zt−1 are Zn graded and the v j are monomials of positive degree.

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Let λ j v j z j be a summand in (4). If xn does not divide v j , then choose i < n such that xi | v j . Since xi u/xn ∈ I, there exists a monomial generator of I, say u1 , such that u1 | xi u/xn , say u1 w = xi u/xn . Set z = xi [u] − xn w[u1 ] ∈ Z1 and subtract the element

λj

vj z z j ∈ Zt−1 Z1 xi

from f . Repeating this procedure for each λ j v j z j in (4) such that xn does not divide v j we obtain a cycle f1 ∈ Zt of degree α such that (i) f ≡ f1 mod Z1 Zt−1 ; (ii) if v[u1 , . . . , ut ] appears in f1 and u ∈ {u1 , . . . , ut }, then xn | v. We repeat the described procedure for each monomial generator u ∈ I with xn | u. We end up with an element f 2 ∈ Zt of degree α such that (iii) f ≡ f2 mod Z1 Zt−1 ; (iv) if v[u1 , . . . , ut ] appears in f2 and xn | u1 · · · ut , then xn | v. Note that if v[u1 , . . . , ut ] appears in f2 and xn u1 · · · ut , then xn | v by degree reasons. Hence for every v[u1 , . . . , ut ] appearing in f2 we have xn | v. Therefore f 2 = xn g, and

g ∈ Zt has degree < q. This completes the proof. Indeed a much stronger statement holds: Theorem 3.8. Let I, J be strongly stable ideals of S. Then regS (Zt (I, S/J)) ≤ t(regS (I) + 1) + regS (S/J) for every t. Theorem 3.8 has been proved by Satoshi Murai in collaboration with the second author and is part of an ongoing project. The following result, whose proof is surprisingly simple, generalizes Green’s theorem [10, Theorem 2.16]: Theorem 3.9. Let I ⊂ S such that dim M/IM = 0. Let c ∈ N be such that I is generated in degrees ≤ c and set v = dim[S/I]c . Set Zt = Zt (I, M) and Ht = Ht (I, M). One has regS (Zt ) ≤ t(c + 1) + regS (M) + v and regS (Ht ) ≤ t(c + 1) + regS (M) + v + c − 1 for every t. Proof. The first inequality can be deduced from the second using the standard short exact sequences relating Bt , Zt and Ht . We prove the second inequality by induction on v. If v = 0 then mc ⊂ I and the assertion has been proved in 3.2. Now let v > 0.

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Observe that Ht is annihilated by I + Ann(M). Hence (by 3.1) dim Ht = 0 and its regularity is the largest degree in which Ht does not vanish. Take f ∈ Sc \ I and set J = I + ( f ). Note that the minimal generators of I are minimal generators of J and that dim[S/J]c = v − 1. We have a short exact sequence of Koszul homology [3, 1.6.13]: Ht+1 (J, M) → Ht (−c) → Ht . By construction Ht (−c) does not vanish in degree regS (Ht ) + c while Ht vanishes in that degree. It follows that Ht+1 (J, M) does not vanish in degree regS (Ht ) + c and hence regS (Ht+1 (J, M)) ≥ regS (Ht )+c. By induction we know that regS (Ht+1 (J, M)) ≤ (t + 1)(c + 1) + regS (M) + (v − 1) + c − 1. It follows that regS (Ht ) + c ≤ (t + 1)(c + 1) + regS (M) + (v − 1) + c − 1, that is, regS (Ht ) ≤ t(c + 1) + regS (M) + v + c − 1.

Remark 3.10. (a) Let I ⊂ S be the ideal generated by a proper subspace V of forms of degree c such that Im = mc+1 . Then regS (I) = c + 1. Set Zt = Zt (I, S) and v = dim Sc /V . By virtue of 3.3 we have regS (Zt ) ≤ t(c + 2) while 3.9 gives regS (Zt ) ≤ t(c + 1) + v. So for small t the first bound is better than the second and the other way round for large t. (b) Since H0 = M/IM, for t = 0 the bound of 3.9 takes the form reg√ S (M/IM) ≤ regS (M) + v + c − 1. Even the case M = S is interesting: it says that if I = m, I is generated in degree ≤ c and v = dim[S/I]c then mc+v ⊂ I.

4 Green–Lazarsfeld Index for Segre–Veronese Rings The goal of this section is to prove a result 4.7 about the Green–Lazarsfeld index of Segre–Veronese rings which was announced in [4]. We first need to generalize some results of [4] to the multihomogeneous setting. Let d ∈ N and m = (m1 , . . . , md ) ∈ Nd and c = (c1 , . . . , cd ) ∈ Nd . We consider the polynomial ring S = K[xi j : 1 ≤ i ≤ d, 1 ≤ j ≤ mi ] with the Zd graded structure induced by assigning deg xi j = ei ∈ Zd . Consider the ideals mi = (xi j | j = 1, . . . , mi ) and d

mc = ∏ mci i . i=1

Then the module of Koszul cycles Zt (m , S) has a Zd -graded structure and also a finer Zm = Zm1 × · · · Zmd -graded structure. We have: c

Lemma 4.1. The module Zt (mc , S) is generated by elements that either have Zd degree bounded above by the vector t c +(t − 1) ∑ ei or belong to Uit for some i in {1, . . . , d} where Ui = Z1 (mc , S)c +ei .

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Proof. Set Zt = Zt (mc , S) and give it the natural Zm graded structure. The proof is a multigraded variant of the argument used above in 3.7. First note that given a monomial generator u of mc , a variable xi j |u and k such that 1 ≤ k ≤ mi and k = j, the monomial v = uxik /xi j belongs to mc and the element xik [u] − xi j [v] belongs to Ui . It is well known that these syzygies generate Z1 and that mc has a linear resolution. Now assume that f ∈ Zt is a Zm -homogeneous element of degree (α1 , . . . , αd ), αi = (αi1 , . . . , αimi ) ∈ Zmi . Assume that |αi | ≥ t(ci + 1) for some i, say for i = 1. We may also assume that α11 = 0. Using induction on t, the rewriting procedure described in the proof of 3.7 and the linear syzygies described above, we may write

f = x11 g modU1 Zt−1 . Since g ∈ Zt , the conclusion follows by induction on t. Next we note that [4, Lemma 3.4] can be extended to the present setting: Lemma 4.2. Let α ∈ Nd be a vector such that α ≤ c componentwise. Let a1 , a2 . . . , at+1 be monomials of Zd -degree equal to α and b1 , b2 . . . , bt ∈ S monomials of degree c − α . Then

∑

σ ∈St+1

(−1)σ aσ (t+1) [b1 aσ (1) , b2 aσ (2) , . . . , bt aσ (t) ]

(5)

belongs to Zt (mc , S). Now we prove a multigraded version of [4, Theorem 3.6]: Lemma 4.3. For every i = 1, . . . , d let b = c − ei and set Ui = Z1 (mc , S)c +ei . Then (ci + 1)! mbUici ⊂ mci i Zci (mc , S) + Bci (mc , S). Proof. Set u = ci and Zu = Zci (mc , S) and Bu = Bci (mc , S). The generators of Ui are of the form za (y0 , y1 ) = y0 [ay1 ] − y1 [ay0 ] where a is a monomial of Zd -degree equal to b and y0 , y1 ∈ {xi1 , . . . , ximi }. So we have to take u such elements, say za j (y0 j , y1 j ) with j = 1, . . . , u, another monomial of degree b, say au+1 , and we have to prove that u

(u + 1)!au+1 ∏ za j (y0 j , y1 j ) ∈ mui Zu + Bu .

(6)

j=1

The symmetrization argument given in the proof of [4, Theorem 3.6] works in this case as well to prove that the left hand side of (6) can be rewritten, modulo boundaries, as ∑ yi1 1 · · · yiu ,uWi where i = (i1 , . . . , iu ) ∈ {0, 1}u and Wi are cycles of the type described in 4.2.

An Nd -graded K-algebra R = α ∈Nd Rα is called standard if R0 = K and R is generated by Rei with i = 1, . . . , d. Clearly R can be presented as a quotient of an

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Nd -graded polynomial ring S = K[xi j : 1 ≤ i ≤ d, 1 ≤ j ≤ mi ] with the Nd -graded structure induced by assigning deg xi j = ei ∈ Zd . Given a vector c = (c1 , . . . , cd ) ∈ Nd , we can consider the Segre–Veronese subring of R associated to it, namely R(c) =

R jc .

j∈N

Our goal is to study the Green–Lazarsfeld index of R(c) . We note that R(c) is a quotient ring of S(c) . Furthermore one has: Lemma 4.4. (a) regS(c) (R(c) ) = 0 for c 0. (b) index(R(c) ) ≥ index(S(c) ) for c 0. More precisely, both statements hold provided one has ic ≥ α componentwise for every i ∈ Z and α ∈ Zd such that βi,Sα (R) = 0. Proof. A detailed proof of (a) is given in the bigraded setting by Conca, Herzog, Trung and Valla [6]. The same argument works as well for multigradings. Then (b) follows from (a) and [4, Lemma 2.2].

Consider the symmetric algebra T of the K-vector space Sc (i.e. a polynomial ring of Krull dimension dimK Sc ), and the natural surjection T → S(c) . The Betti numbers of S(c) as a T -module can be computed via Koszul homology. Lemma 4.5. We have

βi,Tj (S(c) ) = Hi (mc , S) jc .

Proof. One notes that S(c) is a direct summand of S and then proceeds as in [4, Lemma 4.1]

So we may reinterpret 4.1 in terms of syzygies of S(c) , obtaining: Corollary 4.6. One has βi,Tj (S(c) ) = 0 provided ( j − i − 1) min(c) ≥ i. Proof. Since mc annihilates Hi (mc , S), it follows from 4.1 that Hi (mc , S)α = 0 for every α ≥ i(c + ∑ ei ) + c componentwise. Replacing α with j c we have that βi,Tj (S(c) ) = 0 if j c ≥ i(c + ∑ ei ) + c which is equivalent to ( j − i − 1) min(c) ≥ i.

In [11] Hering, Schenck and Smith proved that index(S(c) ) ≥ min(c). We improve the bound by one: Theorem 4.7. One has min(c) ≤ index(S(c) ). Moreover, min(c) + 1 ≤ index(S(c) ) if char K = 0 or char K > 1 + min(c). Proof. The first statement is an immediate consequence of 4.6. In fact, if i ≤ min(c) then ( j − i − 1) min(c) ≥ i for every j > i + 1 and hence, by 4.6, tiT (S(c) ) = i + 1. Set u = min(c). For the second statement, we have to show that Hu+1 (mc , S) j c = 0 for every j > u + 2. By virtue of 4.1 we know that Zu+1 (mc , S) is generated by:

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(1) elements of degree ≤ (u + 1) c +u ∑s es and (2) elements of Uiu+1 where Ui = Z1 (mc , S)c +ei and i = 1, . . . , d; they have degree (u + 1) c +(u + 1)ei . So an element f ∈ Zu+1 (mc , S) j c can come from a generator of type (1) by multiplication of elements of degree α ∈ Zd such that

α ≥ j c −(u + 1) c +u ∑ es . s

Since j > u + 2, we have

α ≥ 2 c +u ∑ es ≥ c . s

∈ mc Zu+1 (mc , S)

So f and f = 0 in homology. Alternatively, f ∈ Zu+1 (mc , S) j c can come from a generator of type (2) by multiplication of elements of degree α ∈ Zd such that α = j c −(u + 1) c −(u + 1)ei ≥ 2 c −(u + 1)ei . If ci > u then α ≥ c, and we conclude as above that f = 0 in homology. If, instead, ci = u, then α ≥ 2 c −(ci + 1)ei . We have that f ∈ m2 c −(ci +1)ei Uici +1 = mc −ci ei mc −ei Uici Ui . But, assuming K has either characteristic 0 or > u + 1, 4.3 implies: mc −ei Uici ⊂ mci i Zci (mc , S) + Bci (mc , S). Hence f ∈ mc Zu (mc , S)Ui + Bci (mc , S)Ui ⊂ Bci +1 (mc , S) and we conclude that f = 0 in homology.

5 Generating Koszul Cycles In this section we consider the Koszul cycles Zt (I, R) where R is standard graded and I is a homogeneous ideal. For simplicity, in this section we let Zt denote the cycles Zt (I, R), and similarly write Bt , Ht and Kt for boundary, homology and components of the Koszul complex K(I, R). We consider the multiplication map Zs ⊗ Zt → Zs+t

(7)

and we want to understand in which degrees it is surjective. Note that the map (7) has a factorization us,t αt Zs ⊗ Zt −→ Zs (I, Zt ) −→ Zs+t where the first map us,t is the canonical one and the second is the map αt described in Sect. 2.

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Proposition 5.1. Suppose R has characteristic 0 or larger than s + t. Then: (1) The multiplication map Zs ⊗ Zt → Zs+t is surjective in degree j if the module TorR1 (Ks−1 /Bs−1 , Zt ) vanishes in degree j. (2) If R = K[x1 , . . . , xn ] and dim R/I = 0 then the multiplication map Zs ⊗ Zt → Zs+t is surjective in degree j for every j ≥ regR Zs + regR Zt . In particular, the map Zs ⊗ Zt → Zs+t is surjective in degree j for every j ≥ (s + t)(regR (I) + 1). Proof. To prove (1) we note that, as αt is surjective, we may as well consider the map us,t : Zs ⊗ Zt → Zs (ϕ , Zt ). Tensoring 0 → Zs → Ks → Bs−1 → 0 and 0 → Bs−1 → Ks−1 → Ks−1 /Bs−1 → 0 with Zt , we have exact sequences f

Zs ⊗ Zt → Ks ⊗ Zt → Bs−1 ⊗ Zt → 0 and

g

TorR1 (Ks−1 /Bs−1 , Zt ) → Bs−1 ⊗ Zt → Ks−1 ⊗ Zt . The composition g ◦ f is the map of the Koszul complex Ks ⊗ Zt → Ks−1 ⊗ Zt . So Zs (ϕ , Zt ) = ker(g ◦ f ) and the image of us,t is ker f . It follows that us,t is surjective in degree j iff g is injective in degree j, that is TorR1 (Ks−1 /Bs−1 , Zt ) vanishes in degree j. √ To prove (2) we first observe, since I = m, one has that (Zt )P is free for every prime ideal P = m. Hence TorRi (M, Zt ) has Krull dimension 0 for every finitely generated R-module M and every t ≥ 0 and i > 0. Then we may apply [8, Corollary 3.1] and have that regR TorRi (M, Zt ) ≤ regR M + regR Zt + i and in particular regR TorR1 (Ks−1 /Bs−1 , Zt ) ≤ regR Ks−1 /Bs−1 + regR Zt + 1. But regR Ks−1 /Bs−1 = regR Zs − 2 and hence regR TorR1 (Ks−1 /Bs−1 , Zt ) ≤ regR Zs + regR Zt − 1 In other words, TorR1 (Ks−1 /Bs−1 , Zt ) vanishes in degrees ≥ regR Zs + regR Zt . Together with (1) this concludes the proof of (2).

Theorem 5.2. Assume that R is Koszul and K has characteristic 0 or > t and take I = mc . Then for every t the module Zt /Z1t vanishes in degree ≥ t(c + 1) and Z1t has an R-linear resolution.

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Proof. We prove the first assertion by induction on t. It is enough to prove that the multiplication map Z1 ⊗ Zt−1 → Zt is surjective in degrees j ≥ tc + t. By virtue of 5.1(1), it is enough to prove that TorR1 (R/mc , Zt−1 ) vanishes for j ≥ tc +t. But since R/mc vanishes in degree ≥ c, it is easy to see that TorR1 (R/mc , Zt−1 ) vanishes in degrees ≥ t1R (Zt−1 ) + c. We know [4, Proposition 2.4] that regR (Zi ) ≤ ic + i for every i (here we use the fact that R is Koszul). So we have t1R (Zt−1 )−1 ≤ (t −1)c+(t −1), i.e. t1R (Zt−1 ) ≤ (t − 1)c + t. Then we have t1 (Zt−1 ) + c ≤ (t − 1)c + t + c = tc + t. This proves the first assertion. For the second, one just notes that regR (Zt ) ≤ tc + t and that Z1t coincides with Zt truncated in degree t(c + 1). Therefore Z1t must have an R-linear resolution.

We have the following consequence: Corollary 5.3. Let S = K[x1 , . . . , xn ] with char K = 0 or > 2c. One has H1 (mc , S)2c = 0 iff index(S(c) ) ≥ 2c, i.e. S(c) has the N2c -property. Proof. By virtue of 5.2 Z2c (mc , S) coincides with Z1 (mc , S)2c in degrees ≥ 2c(c + 1). Hence, by assumption, H2c (mc , S) vanishes in degrees ≥ 2c(c + 1). This implies T (S(c) ) = 0 if jc ≥ 2c(c + 1), that is, j ≥ 2c + 2. In other words, t T (S(c) ) ≤ that β2c, j 2c 2c + 1. Since S(c) is Cohen–Macaulay, one can conclude that tiT (S(c) ) ≤ i + 1 for i = 1, . . . , 2c, that is, index(S(c) ) ≥ 2c.

Remark 5.4. The interesting aspect of Corollary 5.3 is that we know explicitly the generators of Z1 (mc , S) and hence the inclusion Z1 (mc , S)2c ⊂ B2c boils down to a quite concrete statement. Unfortunately we have not been able to settle it. Note also that Ottaviani and Paoletti conjectured that index(S(c) ) = 3c − 3 apart from few known exceptions and at least in characteristic 0, see [12] or [4] for the precise statements. In [4] we have proved that index(S(c) ) ≥ c + 1. As we mentioned in [4] there are computational evidences that the cycles of [4, Lemma 3.4] generate Zt (mc , S). We show below that this is the case for t = 2 and any c. To this end we recall that for every monomial b of degree c − 1 and for variables x j , xk we have an element zb (x j , xk ) = x j [bxk ] − xk [bx j ] ∈ Z1 (mc , S). It is well-know and easy to see that the elements zb (x j , xk ) generate Z1 (mc , S). For a monomial a we set max(a) = max{i : xi |a} and min(a) = min{i : xi |a}. More precisely, the elements zb (x j , xk ) with j < k and max(b) ≤ k form a Gröbner basis of Z1 (mc , S) with respect to any term order selecting x j [bxk ] as leading term of zb (x j , xk ). We have: Proposition 5.5. If K has characteristic = 2 then the module Z2 (mc , S) is generated by two types of elements: (1) The elements of [4, Lemma 3.4] of degree 2c + 1, (2) and by the elements of Z1 (mc , S)2 of degree 2c + 2, that is, the elements of the form za (xi , x j )zb (xh , xk ). Proof. Consider the map

α1 : Z1 (mc , Z1 (mc , S)) → Z2 (mc , S)

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of Sect. 2. We know that Z2 (mc , S) has regularity ≤ 2c + 2 and the only generators of degree 2c + 2 are the elements of (2). So we only need to deal with the elements of degree 2c + 1. To this end we look at the component of degree 2c + 1 of α1 . Let a, b be monomials of degree c − 1. The element [axi ] ⊗ zb (x j , xk ) + [axk ] ⊗ zb (xi , x j ) + [ax j ] ⊗ zb (xk , xi )

(8)

belong to Z1 (mc , Z1 (mc , S)) and has degree 2c + 1. The image under α1 of the elements in (8) are exactly the cycles of [4, Lemma 3.4] in Z2 (mc , S). Since α1 is surjective, to complete the proof it is enough to prove the following statement: Claim. The cycles described in (8) generate Z1 (mc , Z1 (mc , S)) in degree 2c + 1. Let F ∈ Z1 (mc , Z1 (mc , S)) be an element of degree 2c + 1. So F is a sum of elements of the form [u] ⊗ f with u a monomial of degree c and f ∈ Z1 (mc , S) with deg( f ) = c + 1. Choose [u] to be the largest in the lexicographic order induced by x1 > · · · > xn and look at the coefficient f of [u] in F, i.e. F = [u] ⊗ f + sum of terms [v] ⊗ g with v < u. Let x j [bxk ] be the leading term of f with j < k and max(b) ≤ k. If min(u) < j then we may add a suitable scalar multiple of (8) to “kill” the leading term of F and we are done. If instead min(u) ≥ j, then, since 0 = ϕ (F) = u f + sum of terms vg with v < u we have that x j u[bxk ] must cancel, and so x j u = xs v for some v < u in the lex-order. But this is impossible.

References 1. L.L. Avramov, A. Conca, S.B. Iyengar, Free resolutions over commutative Koszul algebras. Math. Res. Lett. 17, no. 2 (2010), 197–210 2. L.L. Avramov and D. Eisenbud, Regularity of modules over a Koszul algebra. J. Algebra 153 (1992), 85–90. 3. W. Bruns and J. Herzog, Cohen–Macaulay rings. Rev. ed. Cambridge Studies in Advanced Mathematics 39, Cambridge University Press 1998. 4. W. Bruns, A. Conca and T. Römer, Koszul homology and syzygies of Veronese subalgebras. arXiv:0902.2431, to appear in Math. Ann. 5. G. Caviglia, Bounds on the Castelnuovo–Mumford regularity of tensor products. Proc. Amer. Math. Soc. 135 (2007), 1949–1957. 6. A. Conca, J. Herzog, N.V. Trung and G. Valla, Diagonal subalgebras of bigraded algebras and embeddings of blow-ups of projective spaces. Amer. J. Math. 119 (1997), 859–901. 7. A. Conca, Regularity jumps for powers of ideals. In: Commutative algebra. Lect. Notes Pure Appl. Math., 244, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 21–32. 8. D. Eisenbud, C. Huneke, and B. Ulrich, The regularity of Tor and graded Betti numbers. Amer. J. Math. 128 (2006), 573–605. 9. S. Eliahou and M. Kervaire, Minimal resolutions of some monomial ideals. J. Algebra 129 (1990), 1–25.

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10. M.L. Green, Koszul cohomology and the geometry of projective varieties. II. J. Differ. Geom. 20 (1984), 279–289. 11. M. Hering, H. Schenck and G.G. Smith, Syzygies, multigraded regularity and toric varieties. Compos. Math. 142 (2006), 1499–1506. 12. G. Ottaviani and R. Paoletti, Syzygies of Veronese embeddings. Compos. Math. 125 (2001), 31–37. 13. B. Sturmfels, Four counterexamples in combinatorial algebraic geometry. J. Algebra 230 (2000), 282–294.

Boij–Söderberg Theory David Eisenbud and Frank-Olaf Schreyer

Abstract In this article we start with a survey of the recent breakthroughs concerning Betti table of graded modules over the polynomial ring and cohomology tables coherent sheaves on projective space. We then ask how this theory can be extended to a more general setting. Our first new result concerns cohomology tables of very ample polarized varieties. We prove a necessary and sufficient condition that the Boij–Söderberg cone of cohomology tables of coherent sheaves on a variety X of dimension d with polarization OX (1) coincides with the corresponding cone of (Pd , O(1)), and conjecture that our condition is always satisfied. The last section concerns cohomology tables of vector bundles with respect to more than one line bundle, where we start with the simplest case P1 × P1 . We identify some extremal rays in the Boij–Söderberg cone of vector bundles on P1 × P1 , and conjecture that these are all.

1 Introduction The Hilbert polynomial of a graded module over a polynomial ring is an important invariant that is refined by the graded Betti numbers of the module, which are usually encoded together as the Betti Table of the module (see below for precise definitions). If we think of the module as representing a sheaf on projective space, then the Hilbert polynomial appears as the Euler characteristic of twists of the sheaf, and a more natural refinement is, perhaps, the table of dimensions of the cohomology spaces of different twists, the Cohomology Table of the sheaf. David Eisenbud Dept. of Mathematics, University of California, Berkeley, Berkeley CA 94720, USA e-mail: [email protected] Frank-Olaf Schreyer Facultät für Mathematik und Informatik, Campus E 2 4, Universität des Saarlandes, D-66123 Saarbrücken, Germany e-mail: [email protected] G. Fløystad et al. (eds.), Combinatorial Aspects of Commutative Algebra and Algebraic Geometry, Abel Symposia 6, DOI 10.1007/978-3-642-19492-4 3, © Springer-Verlag Berlin Heidelberg 2011

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In late 2006 Mats Boij and Jonas Söderberg, made a remarkable conjecture specifying the possible Betti Tables of modules of finite length, up to a rational multiple [2]. It is easy to see that the Betti tables form a semigroup—the direct sum of modules corresponds to the addition of Betti tables. Allowing multiplication by positive rational numbers instead of just positive integers, we get a rational convex cone. What Boij and Söderberg did was to conjecture that the Betti tables of what are called pure resolutions define the extremal rays of this cone. They also conjectured the existence of such pure resolutions—very few were known—and thus that the cone was closed, as well. We proved these conjectures by showing, on the one hand, the existence of the necessary pure resolutions and, on the other, identifying the facets of the cone as coming from the extremal rays in the cone of cohomology tables of vector bundles on projective space. We also identified the extremal rays in the corresponding cone of cohomology tables of vector bundles, and showed that vector bundles with such extremal cohomology tables actually exist. A flurry of other papers and preprints including [3, 6, 9, 10, 11] and [12] have added to the basic picture and its applications; in particular, the whole picture now extends in some form to arbitrary finitely generated graded modules over a polynomial ring, and to coherent sheaves on a projective space. The first two sections of this note survey some of what we now know about these cases. One can imagine many extensions of the basic ideas in this theory. For example, one might ask about the cohomology tables of vector bundles or sheaves whose support is restricted to a subvariety of projective space. Somewhat surprisingly, the possibilities are often identical to those for a projective space of the same dimension. We will show in Sect. 4 that this depends on the existence of a single “Ulrich” sheaf on X. We have conjectured elsewhere that Ulrich sheaves exist on every projective variety. A different sort of extension would be to pack more data into the cones. For example we could look at multigraded modules, and ask about the cone of multigraded Betti numbers. More generally, we could look at modules equivariant for the action of a reductive group, and ask about the representations in the resolution instead of the degrees. This direction has been pursued by Sam and Weyman [10]. As a first step, one might try to determine—or at least guess!—the extremal tables. We can ask similar questions about the cohomology of vector bundles or coherent sheaves. The last section of this paper takes up one aspect of this. We study vector bundles F on P1 × P1 and ask about the rational cone of bigraded cohomology tables {hi (F (a, b)}. We give a conjectural description of the cone in terms of extremal rays.

2 Betti Tables Let S = k[x1 , . . . , xn ] be a polynomial ring over a field k, graded with each xi of degree 1, and let M be a finitely generated graded S-module. As usual, we write S(− j) for the graded free module of rank 1 with generator in degree j. By the

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Hilbert Syzygy theorem there exist finite free resolutions of M; that is, sequences of graded free modules and degree 0 homomorphisms F:

ϕ

ϕm

1 · · · ←− Fm ←− 0 F0 ←−

∼ M. Such a resolution is said to be that are exact at Fi for i > 0, while coker ϕ1 = minimal if no proper summand of Fi maps onto the kernel of ϕi−1 . Minimal resolutions are unique up to isomorphism, and have length m ≤ n. In particular, if F is a minimal free resolution, and we write Fi = ⊕ j S(− j)βi, j (M) , then the graded Betti numbers βi, j (M) = dim((Fi ⊗S k) j ) are invariants of M alone. We define the Betti table of M to be this collection of numbers {βi, j (M)}. We may regard the Betti table of M as an element of an (infinite-dimensional) rational vector space, B :=

∞

Qn+1

−∞

with coordinates βi, j (M). Since βi, j (M ⊕ N) = βi, j (M) + βi, j (N) the Betti tables of finitely generated modules form a sub-semigroup of this vector space. The following Theorem, conjectured by Boij and Söderberg, specifies the cone of positive rational linear combinations of Betti tables of finitely generated modules precisely. One way of specifying a cone is to give it’s extremal rays—the half-lines in the cone that are not in the convex hull of the remaining elements of the cone. In the case of the cone of Betti tables, the extremal rays will turn out to be the pure modules: Definition 2.1. A finitely generated graded S-module M is called pure of type d := (d0 , . . . , dm ) if (a) In a minimal free resolution of M as above, the free module Fi generated by elements of degree di ; that is, βi, j = 0 when j = di . (b) M is Cohen–Macaulay of codimension m; that is, Fi = 0 for i > m and the annihilator of M is an ideal of codimension m. It is easy to see that if there is a pure module of type d, then d0 < · · · < dm . Much more is true: if M is a pure module, then a result of Herzog and Kühl [15] shows that the Betti table of M is determined by d up to a rational multiple: that is, there is a constant r = r(M) depending on M such that

βi,di (M) =

r . ∏t=i |dt − di |

Thus the pure modules of type d define a single ray in the cone of Betti tables. One more preparation is necessary: we order the strictly increasing sequences d: we say that d = (d0 , . . . , dm ) ≤ d = (d0 , . . . , dm ) if m ≥ m and di ≤ di for i = 1, . . . , m . (One can think of this as the termwise order if one simply extends each sequence d = (d0 , . . . , dm ) to d = (d0 , . . . , dm , ∞, ∞, . . . ).) We can now state the main result of the theory concerning the cone of Betti tables:

38


Theorem 2.2 ([3, 8]). Let S = k[x1 , . . . , xn ] be as above. (a) For every strictly increasing sequence of integers d = (d0 , . . . , dm ) with m ≤ n, there exist pure finitely generated graded S-modules of type d. (b) The Betti table of any finitely generated graded S-module may be written uniquely as a positive rational linear combination of the Betti tables of a set of pure finitely generated modules whose types form a totally ordered sequence. The second statement of the Theorem has two nice interpretations, which may help clarify its meaning. First, geometrically, it really says that the cone of Betti tables is a simplicial fan, that is, it is the union of simplicial cones, meeting along facets, with each simplicial cone spanned by the rays corresponding to a set of pure Betti tables whose types form a totally ordered set. These simplices and cones are of course infinite dimensional; but one can easily reduce to the finite-dimensional case by specifying that one wants to work with resolutions where the free modules are generated in a given bounded range of degrees. Second, algorithmically, the Theorem implies that there is a greedy algorithm that gives the decomposition. Rather than trying to specify this formally, we give an Example. For this purpose, we write the Betti table of a module M as an array whose entries in the i-th column are the βi, j —that is, the i-th column corresponds to the free module Fi for reasons of efficiency and tradition, we put βi, j in the ( j − i)-th row. For our example we take n = 3, and let M = S/(x2 , xy, xz2 ). The minimal free resolution of M has the form S ←− S(−2)2 ⊕ S(−3) ←− S(−3) ⊕ S(−4)2 ←− S(−5) ←− 0 and is represented by an array ⎛ ⎞ 1 β (M) = ⎝ 2 1 ⎠ 121 where all the entries not shown are equal to zero. To write this as a positive rational linear combination of pure diagrams, we first consider the “top row”, corresponding to the generators of lowest degree in the free modules of the resolution. These are in positions ⎛ ⎞ ∗ ⎝ ∗∗ ⎠ ∗ corresponding to the degree sequence (0, 2, 3, 5). There is in fact a pure module M1 = S/I1 with resolution ⎛ ⎞ 1 β (M1 ) = ⎝ 5 5 ⎠ . 1


39

The greedy algorithm now instructs us to subtract the largest possible multiple q1 of β (M1 ) that will leave the resulting table β (M) − q1 β (M1 ) having only non-negative terms. We see at once that q1 = 1/5. We now repeat this process starting from β (M) − q1 β (M1 ); the Theorem guarantees that there will always be a pure resolution whose degree sequence matches the top row of the successive remainders. In this case we arrive at the expression ⎞ ⎛ ⎞ ⎛ ⎛ ⎞ ⎞ ⎛ ⎛ ⎞ 1 3 1 1 1 1 1 1 1 ⎠+ ⎝ ⎠. ⎠+ ⎝ β (M)=⎝ 2 1 ⎠ = ⎝ 5 5 ⎠+ ⎝ 10 5 10 6 3 43 1 1 15 8 121 All the fractions and diagrams that occur are of course invariants—apparently new invariants—of M.

3 Facets of the Cone and Cohomology Tables We next focus on the facets of the cone of Betti tables. According to the simplicial structure of the cone [2], an outer facet corresponds to a sequence of three degree sequences which differ in at most two consecutive positions. For example the degree sequences to the following Betti tables form such a chain. 386 24 1 < < 1 42 683 The facet equation is defined by the vanishing on all Betti tables of pure resolutions corresponding to degree sequences that are < (0, 1, 3, 4) and all Betti table of pure resolutions corresponding to degree sequences > (0, 1, 3, 4). This allows to compute the coefficients of the facet equation recursively using zero coefficients on the support of the right hand table as start values. ⎛ ⎞ .. .. .. .. . .⎟ ⎜. . ⎜21 −12 5 0 ⎟ ⎜ ⎟ ⎜12 −5 0 3⎟ ⎜ ⎟ ⎜ 5 0 −3 4 ⎟ ⎜ ⎟ ⎜ 0 3 −4 3 ⎟ ⎜ ⎟ ⎜ 0 0 0 0⎟ ⎜ ⎟ ⎜ 0 0 0 0⎟ ⎝ ⎠ .. .. .. .. . . . . What we have to prove is that this linear form is non-negative on the Betti table of any minimal free resolution. Our key observation is that the numbers appearing are dimensions of cohomology groups of what we call supernatural vector bundles.

40


Definition 3.1. A vector bundle E on Pm has natural cohomology [13], if for each k at most one of the groups H i (E (k)) = 0. It has supernatural cohomology, if in addition the Hilbert polynomial

χ (E (k)) =

rank E m!

m

∏ (k − z j ) j=1

has m distinct integral roots z1 < z2 < . . . < zm . For a coherent sheaf E on Pm we denote by

γ (E ) = (γ j,k ) ∈

∞

∏

Qm+1 with γ j,k = h j (E (k))

k=−∞

its cohomology table. Analogous to the Theorem on free resolutions we have Theorem 3.2 ([8]). The extremal rays of the rational cone of cohomology tables of vector bundles are generated by cohomology tables of supernatural vector bundles. The crucial new concept is the following pairing between Betti tables of modules and cohomology table of coherent sheaves. We define β , γ for a Betti table β = (βi,k ) and a cohomology table γ = (γ j,k ) by β , γ = ∑ (−1)i− j ∑ βi,k γ j,−k i≥ j

k

Theorem 3.3 (Positivity 1, [8, 9]). For F any free resolution of a finitely generated graded K[x0 , . . . , xm ]-module and E any coherent sheaf on Pm we have β (F), γ (E ) ≥ 0. For example the facet equation above, is obtained from the vector bundles E on P2 , which is the kernel of a general map O 5 (−1) → O 3 . The coefficients of the functional −, γ (E ) are ⎛ ⎞ .. .. .. .. . . . . ⎜ ⎟ ⎜21 −12 5 0 ⎟ ⎜ ⎟ ⎜12 −5 0 3 ⎟ ⎜ ⎟ ⎜ 5 0 −3 4 ⎟ ⎜ ⎟ ⎜ 0 3 −4 3 ⎟ ⎜ ⎟ ⎜ 0 4 −3 0 ⎟ ⎜ ⎟ ⎜ 0 3 0 −5 ⎟ ⎜ ⎟ ⎜ 0 0 5 −12⎟ ⎜ ⎟ ⎜ 0 0 12 −21⎟ ⎝ ⎠ .. .. .. .. . . . .


41

This is not quite what we wanted. We define truncate functionals −, γ τ ,c by putting zero coefficients in the appropriate spots. Theorem 3.4 (Positivity 2, [8, 9]). For F any minimal free resolution of a finitely generated graded K[x0 , . . . , xm ]-module and E any coherent sheaf on Pm we have β (F), γ (E ) τ ,c ≥ 0. To prove the positivity theorems we consider the tensor product of F with the ˇ Cech resolution C : . . . → C p (E) =

∑

0≤i0 0 and all j ∈ {1, . . . , n} we have Hi (z1 , . . . , z j ; grm (I k )) = 0 for all 0. Then under the assumption that M (m; I k ) is acyclic, this sequence is a d-sequence with respect to I k . Suppose we can choose a K-basis z of S1 such that ((z1 , . . . , z j ) grm (I k ) : z j+1 )/(z1 , . . . , z j ) grm (I k )

(2)

has finite length for all k and all j, then, since reg(grm (I k )) = 0, [8, Proposition 20.20] implies that (((z1 , . . . , z j ) grm (I k ) : z j+1 )/(z1 , . . . , z j ) grm (I k )) = 0 for all > 0.

(3)

Observe that the regularity of grm (I k ) is indeed zero, because grm (I k ) is generated in degree 0 and the approximation complex M (m; I k ) provides a linear resolution of grm (I k ). For a sequence satisfying (3) for all k it follows that Hi (z1 , . . . , z j ; grm (I k )) = 0 for all i > 0, all k and > 0, and so z is a d-sequence with respect to all I k , as explained before. This is equivalent to saying that z is a d-sequence with respect to R(I). But how can we find a sequence z such that property (2) (and consequently property (3)) is satisfied for all powers of I?

Powers of Componentwise Linear Ideals

53

Suppose we have already chosen z1 , . . . , z j satisfying (2). We apply the graded version of [3, Proposition 2.5] to the standard graded S/(z1 , . . . , z j )-algebra A=

grm (I k )(z1 , . . . , z j ) grm (I k )

k≥0

to conclude that A =

Ass(grm (I k )(z1 , . . . , z j ) grm (I k ))

k≥0

is a finite set. Hence since K is infinite we may choose a linear form z j+1 ∈ S/(z1 , . . . , z j ) which is contained in no prime ideal of A different from the graded maximal ideal of S/(z1 , . . . , z j ). This is the desired next linear form in our sequence. Hence in each step of the constructing of z we have an open choice. In other words, generic sequences will be d-sequences with respect to R(I). Corollary 2.5. Let K be an infinite field, I ⊂ S a graded ideal and z a generic K-basis of S1 . The following conditions are equivalent: (a) All powers of I are componentwise linear; (b) z is a d-sequence with respect to R(I). Let us analyze what it means that z is a d-sequence with respect to R(I). After a change of coordinates we may assume that our given sequence is the sequence x1 , . . . , xn . Let f1 , . . . , f m a homogeneous system of generators of I, and let T = S[y1 , . . . , ym ] be the polynomial ring over S in the variables y1 , . . . , ym . We consider the surjective S-algebra homomorphism T → R(I) with y j → f j for j = 1, . . . , m. Then R(I) ∼ = T /J where J is the kernel of the map T → R(I). Identifying R(I) with T /J, we see that x1 , . . . , xn is a d-sequence with respect to R(I) if and only if (x1 , . . . , xi−1 ) + J : xi ∩ ((x1 , . . . , xn ) + J) = (x1 , . . . , xi−1 ) + J

for i = 1, . . . , n. (4)

This condition can be easily checked by any computer algebra system. For example, if we let I be the ideal of 2-minors of the generic symmetric matrix ⎛ ⎞ abc ⎝ b d e ⎠, ce f one easily finds with CoCoA that a, b, e, d + f , c, f is a d-sequence on R(I), and hence all powers of I have a linear resolution. Passing to the initial ideal of J with respect to some term order we can deduce a sufficient condition for the property that all powers of an ideal are componentwise linear. We denote by Ji the ideal which is the image of J under the canonical epimorphism T → T /(x1 , . . . , xi ). Then (4) implies that x1 , . . . , xn is a d-sequence on R(I), if and only if Ji−1 : xi ∩ (xi , . . . , xn ) + Ji−1 = Ji−1

for i = 1, . . . , n.

(5)

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Let < be a monomial order. We denote by in(I) the initial ideal with respect to this order. Lemma 2.6. Suppose there exists a monomial order < on T /(x1 , . . . , xi−1 ) such that in(Ji−1 ) : xi ∩ ((xi , . . . , xn ) + in(Ji−1 )) = in(Ji−1 ).

(6)

Then (5) holds for this integer i. Proof. Let f ∈ Ji−1 : xi ∩ ((xi , . . . , xn ) + Ji−1 ). Then xi f ∈ Ji−1 , and so xi in( f ) ∈ in(Ji−1 ). Therefore, in( f ) ∈ in(Ji−1 ) : xi . On the other hand, since f ∈ (xi , . . . , xn ) + Ji−1 , it follows that in( f ) ∈ in((x1 , . . . , xi ) + Ji−1 ) ⊂ (x1 , . . . , xi ) + in(Ji−1 ). Thus our hypothesis implies that in( f ) ∈ in(Ji−1 ). Hence there exists g ∈ Ji−1 such that in(g) = in( f ). We may assume that the leading coefficients of f and g are 1, and set h = f − g. Then xi h = xi f − xi g ∈ J j−1 and h ∈ (xi , . . . , xn ) + Ji−1 . Since in(h) < in( f ), we may assume by induction that h ∈ Ji−1 . This then implies that f ∈ Ji−1 , as desired. For a monomial ideal L we denote as usual by G(L) the unique minimal set of monomial generators of L. We also set yb = yb11 · · · ybmm for any integer vector b = (b1 , . . . , bm ). Now the preceding considerations yield Proposition 2.7. The sequence x1 , . . . , xn is d-sequence on R(I), and hence all powers of I are componentwise linear, if for each i there exists a monomial order on T /(x1 , . . . , xi−1 ) such that whenever xi divides u ∈ G(in(Ji−1 )), then u is the form u = xi yb and x j yb ∈ in(Ji−1 ) for all j ≥ i. In order to prove this proposition we just have to see that (6) holds for all i. This will follow from Lemma 2.8. Let I ⊂ T be a monomial ideal. Then the following conditions are equivalent: (a) I : x1 ∩ (mT + I) = I. (b) If x1 divides u ∈ G(I), then u is of the form u = x1 yb and x j yb ∈ I for all j = 1, . . . , n. Proof. (a) ⇒ (b): Let u ∈ G(I) such that x1 divides u. Say, u = x1a u with a ≥ 1 and x1 does not divide u . Suppose a > 1; then x1a−1 u ∈ I. But x1a−1 u ∈ I : x1 ∩ (mT + I), a contradiction. Hence we see that u = x1 u . Suppose some xi divides u for some i > 1. Then u ∈ I : x1 ∩ (mT + I), but u ∈ I, again a contradiction. Thus u = x1 yb for some exponent vector b. The monomial xi yb belongs to I : x1 ∩ (mT + I) for all i, and hence xi yb ∈ I for all i. (b) ⇒ (a): We only need to prove the inclusion I : x1 ∩ (mT + I) ⊂ I. Our assumptions imply that G(I) can be written as {x1 yb1 , . . . , x1 ybk , uk+1 , . . . , um } with some integer vectors bi , and with monomials u j ∈ G(I) which are not divisible by x1 for j = k + 1, . . . , m. It follows that I : x1 = (yb1 , . . . , ybk , uk+1 , . . . , um ).


55

Thus if u ∈ I : x1 ∩ mT + I, then u is either a multiple of one of the u j , in which case u ∈ I, or u = vybi for some i, and x j divides v for some j. Write v = x j v ; then u = v x j ybi ∈ I, according to the assumptions made in (b). We demonstrate the use of Proposition 2.7 with a simple example. Example 2.9. Let I be the ideal of 2-minors of x1 a b . x2 b c We have I = (−x2 a + x1 b, −x2 b + x1 c, −b2 + ac) and J = (−by1 + ay2 − xy3 , cy1 − by2 + x2 y3 ). We want to show that x1 , x2 , a, c, b is a d-sequence. We apply Proposition 2.7 by using the reverse lexicographic order induced by x1 > x2 > a > b > c > y1 > y2 > y3 . Then in(J0 ) = (cy1 , by1 , b2 y2 ), in(J1 ) = (cy1 , by1 , b2 y2 ), in(J3 ) = (cy1 , by1 , b2 y2 ) and in(J4 ) = (by1 , by2 ).

in(J2 ) = (cy1 , by1 , b2 y2 ),

Thus we see that the conditions of Proposition 2.7 are satisfied, and hence x1 , x2 , a, c, b is indeed a d-sequence on R(I). Example 2.9 naturally leads us to pose the following question. Let I be the ideal of 2-minors of x a1 a2 · · · an−1 an . y a2 a3 · · · an an+1 This ideal defines a rational scroll. Is it true that the sequence x, y, a1 , an+1 , an , . . . , a2 is a d-sequence on R(I)? This is the case for n ≤ 6.

3 Chordal Graphs Let [n] = {1, 2, . . . , n} denote the vertex set and G a finite graph on [n] with no loop and no multiple edge. We write E(G) for the set of edges of G. The edge ideal of G is the ideal I(G) of S = K[x1 , . . . , xn ] which is generated by those monomials xi x j with {i, j} ∈ E(G). A subset C of [n] is called vertex cover of G if, for each edge {i, j} of G, one has either i ∈ C or j ∈ C. A minimal vertex cover of G is a vertex cover C with the property that any proper subset of C cannot be a vertex cover of G. We say that G is unmixed if all minimal vertex covers have the same cardinality. A mixed graph is a graph which is not unmixed. The vertex cover ideal of G is the ideal of S which is generated by those squarefree monomials ∏i∈C xi such that C is a minimal vertex cover of G. The vertex cover ideal of G is the Alexander dual of I(G); in other words, the ideal

(xi , x j ). IG := I(G)∨ = {i, j}∈E(G)

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Recall that a finite graph G is called chordal if each cycle of G of length > 3 possesses a chord. In the paper [9, Theorem 3.2] by Francisco and Van Tuyl it is shown that every vertex cover ideal of a chordal graph is componentwise linear. Example 3.1. (a) Let G be the line of length 3. Thus E(G) = {{1, 2}, {2, 3}, {3, 4}}. Then G is unmixed and IG = (x1 x3 , x2 x3 , x2 x4 ). Since x1 , x3 , x2 + x 4 , x2 is a d-sequence with respect to R(IG ), all powers of IG are (componentwise) linear. (b) Let G be the line of length 4. Thus E(G) = {{1, 2}, {2, 3}, {3, 4}, {4, 5}}. Then G is mixed and IG = (x1 x3 x4 , x1 x3 x5 , x2 x4 , x2 x3 x5 ). Since x1 , x3 , x5 , x2 + x4 , x2 is a d-sequence with respect to R(IG ), all powers of IG are componentwise linear. A complete graph on [n] is a finite graph such that {i, j} ∈ E(G) for all 1 ≤ i < j ≤ n. Let Gn denote the complete graph on [n]. Its vertex cover ideal IGn is generated by all squarefree monomials of S of degree n − 1; in other words, IGn is an ideal of Veronese type ([11]). In particular, IGn is a polymatroidal ideal ([5]). Hence all powers of IGn have linear resolutions. Theorem 2.4 then guarantees that a generic K-basis z = z1 , . . . , zn of S1 is a d-sequence with respect to R(IGn ). Let m ≥ 1 and Γ a connected graph on [n + m] such that the induced subgraph of G on [n] is Gn and that {i, j} ∈ E(G) if n < i < j ≤ n + m. Such a graph is called a star graph based on Gn . A star graph based on Gn is a chordal graph. Example 3.2. Let n = 3 and m = 3. Let G be the star graph based on G3 with the edges {1, 4}, {2, 4}, {2, 5}, {3, 5}, {1, 6}, {3, 6} together with all edges of G3 . Since both {1, 2, 3} and {2, 3, 4, 6} are minimal vertex covers of G, it follows that G is mixed. Let K be a field of characteristic 0. Since the K-basis x4 , x5 , x6 , x1 + x2 + x3 , 2x1 + 3x2 + 5x3 , 7x1 + 11x2 + 13x3 of S1 is a d-sequence with respect to R(IG ), all powers of IG are componentwise linear. The simple computational observation done in Example 3.2 now grows the following Theorem 3.3. All powers of the vertex cover ideal of a star graph based on Gn are componentwise linear. Proof. Let G be a star graph on the vertex set {x1 , . . . , xn ,t1 , . . . ,tm } based on Gn with vertex set {x1 , . . . , xn }. Let T = S[t1 , . . . ,tm ] denote the polynomial ring in n + m variables over K and IG ⊂ T the vertex cover ideal of G. First, we observe that IG is generated by the following monomials:

Powers of Componentwise Linear Ideals n

ui = ∏ x j j=1 j=i

57

∏

tk ,

i = 1, . . . , n,

k {xi ,tk }∈E(G)

and

n

un+1 = ∏ x j . j=1

It is clear that all these monomials belong to IG . Conversely, suppose that C is an arbitrary vertex cover of G and uC its corresponding monomial. If {x1 , . . . , xn } ⊂ C, then un+1 divides uC . Thus assume that xi ∈ C for some i. Then x j ∈ C for all j = i, because if x j ∈ C for some j = i, then the edge {xi , x j } is not covered by C. It follows that {x1 , . . . , xi−1 , xi+1 , . . . , xn } ⊂ C. Moreover since xi ∈ C we must have tk ∈ C for all k with {xi ,tk } ∈ E(G). This shows that ui divides uC . Notice that un+1 is not always part of the minimal monomial set G(IG ) of generators of IG . In fact, un+1 ∈ G(IG ) if and only if for each i = 1, . . . , n there exists k such that {xi , yk } ∈ E(G). For the following considerations, however, it is not important whether un+1 belongs to G(IG ) or not. Second, we write the Rees algebra of IG as the factor ring R(IG ) = T [y1 , . . . , yn+1 ]/J, where J is the kernel of the T -algebra homomorphism given by y j → u j for j = 1, . . . , n + 1. We claim that J is generated by the binomials fi = xi yi − ∏ tk yn+1 , i = 1, . . . n, k

where for each i the product ∏k tk in fi is taken over all k with {xi ,tk } ∈ E(G). To see why this is true, we write L for the ideal generated by the binomials f1 , . . . , fn , and have to show that L = J. We first observe that f1 , . . . , f n is a regular sequence. Indeed, let < be a lexicographic monomial order with yi > yn+1 > x j ,tk for all i, j, k and i = n + 1. Then in< ( fi ) = xi yi for i = 1, . . . , n. Since gcd(in< ( fi ), in< ( f j )) = 1 for all i = j, it follows that f1 , . . . , f n is a Gröbner basis of L, and that f1 , . . . , f n is a regular sequence. Now it follows that dim T [y1 , . . . , yn+1 ]/L = dim T + 1 = dim R(IG ). Hence, if L is a prime ideal, then one has L = J. To show that L is a prime ideal, we consider the ideal (L, x1 y1 ) = (vyn+1 , x1 y1 , f2 , . . . , f n ), where v = ∏k tk and where the product is taken over all k with {x1 ,tk } ∈ E(G). The initial terms of the generators of the ideal (L, x1 y1 ) are vyn+1 , x1 y1 , . . . , xn yn . Since these initial terms form a regular sequence of monomials, it follows as above that f1 , . . . , fn , x1 y1 is a regular sequence. In particular the variable x1 is a nonzero divisor on T [y1 , . . . , yn+1 ]/L. By the same reason all xi are nonzero divisors on T [y1 , . . . , yn+1 ]/L. Let x = ∏ni=1 xi ; then the natural homomorphism T [y1 , . . . , yn+1 ]/L → (T [y1 , . . . , yn+1 ]/L)x is a monomorphism. In the localized polynomial ring T [y1 , . . . , yn+1 ]x the ideal Lx is generated by the elements xi−1 fi = yi − xi−1 (∏k tk )yn+1 . Thus we see that (T [y1 , . . . , yn+1 ]/L)x ∼ = T [yn+1 ]x , which is a domain. It follows that T [y1 , . . . , yn+1 ]/L is a domain. In other words, one has L = J, as desired.

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Now we have that in< (J) = (x1 y1 , . . . , xn yn ). Hence t1 , . . . ,tm is a regular sequence on T [y1 , . . . , yn+1 ]/in< (J), and consequently is a regular sequence on the Rees algebra R(IG ) = T [y1 , . . . , yn+1 ]/J. Since any regular sequence is a d-sequence, it remains to be shown that we can find a basis z1 , . . . , zn of S1 which is a d-sequence on R(IG ) = R(IG )/(t1 , . . . ,tm ). Note that R(IG ) ∼ = S[y1 , . . . , yn ]/(x1 y1 , . . . , xn yn ).

Thus the theorem follows form the following Lemma 3.4.

Lemma 3.4. Let R = S[y1 , . . . , yn ]/(x1 y1 , . . . , xn yn ), and z1 , . . . , zn a generic K-basis of S1 . Then z1 , . . . , zn is a d-sequence on R. Proof. We will apply Theorem 2.3 and have to show that the approximation complex M (m; R) is acyclic, where m = (x1 , . . . , xn ). Recall that M (m; R)i = Hi (m; R) ⊗ A(−i), where A = R[e1 , . . . , en ] is a polynomial ring over R in the variables e1 , . . . , en , and where Hi (m; R) is the ith Koszul homology of the sequence x1 , . . . , xn with values in R. Since R is a complete intersection, it follows from [2, Theorem 2.3.9] that H(m; R) is the ith exterior algebra of H1 (m; R). Since H1 (m; R) is a free K[y1 , . . . , yn ]-modules whose basis is given by the homology classes [yi ei ] of the cycles yi ei , we see that Hi (m; R) is free (R/m)[y1 , . . . , yn ]-module with basis {bJ : J = { j1 , j2 , . . . , ji } 1 ≤ j1 < j2 < . . . < ji ≤ n}, where bJ = [y j1 y j2 · · · y ji e j1 ∧ e j2 ∧ · · · ∧ e ji ]. Thus M (m; R)i can be identified with the free B-module

J,|J|=i bJ B,

where

B = K[y1 , . . . , yn , e1 , . . . , en ]. After this identification the differential of M (m; R) is given by

∂ (bJ ) =

i

∑ (−1)k+1 y jk e jk bJ\{ jk }

for J = { j1 < j2 < . . . < ji }

k=1

Hence M (m; R) is isomorphic to the Koszul complex K(y1 e1 , . . . , yn en ; B). Since the sequence y1 e1 , . . . , yn en is a regular sequence, it follows then that M (m; R) is acyclic, as desired. Theorem 3.3 and computational evidence, including Example 3.6 leads us to present the following Conjecture 3.5. All powers of the vertex cover ideal of a chordal graph are componentwise linear. Example 3.6. Let G be the chordal graph drawn below. The graph G is no longer a star graph.


59

Then G is mixed and IG is generated by x1 x3 x5 x6 , x2 x3 x4 x5 x6 , x1 x2 x4 x6 x7 , x1 x2 x3 x4 x6 , x2 x3 x4 x5 x7 , x1 x2 x3 x5 x7 , x1 x2 x5 x6 x7 , x2 x4 x5 x6 x7 . We can easily find a d-sequence which is a “generic” K-basis of S1 created by CoCoA with “Randomized.” Hence all powers of the vertex cover ideal of G are componentwise linear. In [13] the Cohen–Macaulay chordal graphs are classified. Let G be a finite graph on [n], and Δ (G) its clique complex, that is to say, the simplicial complex whose faces are the cliques (i.e. complete subgraphs) of G. It is shown in [13] that if G is chordal, then G is Cohen–Macaulay if and only if [n] is the disjoint union of those facets of Δ (G) which have a free vertex. A vertex of a facet is free, if it belongs to no other facet. In support of Conjecture 3.5 we have the following result. Theorem 3.7. Let G be a graph on [n], and suppose that [n] is the disjoint union of those facets of the clique complex of G with a free vertex. Then all powers of IG have a linear resolution. Proof. Let F1 , . . . , Fm be the facets of Δ (G) which have a free vertex. Since [n] = F1 ∪ F2 ∪ · · · ∪ Fs is a disjoint union, we may assume that if i ∈ Fp , j ∈ Fq and p < q, then i < j. In particular, 1 ∈ F1 and n ∈ Fs . Moreover, we may assume that if i1 , i2 ∈ Fi where i1 is a nonfree vertex and i2 is a free vertex, then i1 < i2 . Observe that any minimal vertex cover of G is of the following form: (F1 \ {a1 }) ∪ (F2 \ {a2 }) ∪ · · · ∪ (Fs \ {as }),

where a j ∈ Fj .

In particular, G is unmixed and all generators of IG have degree n − s. Now let R(IG ) be the Rees algebra of vertex cover ideal of G. Suppose u1 , . . . , um is the minimal set of monomial generators of IG . Then there is a surjective K-algebra homomorphism K[x1 , . . . , xn , y1 , . . . , ym ] −→ R(IG ) xi → xi

and y j → u j ,

whose kernel J is a binomial ideal. Let < be the lexicographic order induced by the ordering x1 > x2 > · · · > xn > y1 > · · · > ym . We are going to show that the generators of in> (J) are at most of degree 1 in the xi . This is the so-called x-condition and it implies that all powers of IG have a linear resolution, see [12]. Suppose that

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xi1 xi2 · · · xi p y j1 y j2 · · · y jq with i1 ≤ i2 ≤ . . . ≤ i p is a minimal generator of in< (J). Then xi1 xi2 · · · xi p y j1 y j2 · · · y jq − xk1 xk2 · · · xk p y1 y2 · · · yq ∈ J.

(7)

It follows that i1 < min{k1 , . . . , k p }, and there exists an index jr such that xi1 does not divide u jr . Say, i1 ∈ Fc . Then (7) implies that there exists d ∈ [p] with kd ∈ Fc . In particular, i1 = max{i : i ∈ Fd }. Let i0 = max{i : i ∈ Fd }. Since i0 is a free vertex, it follows that xi1 (u jr /xi0 ) is a minimal generator of IG , say, ug . Therefore, f = xi1 y jr − xi0 yg ∈ J and in( f ) = xi1 y jr divides xi1 xi2 · · · xi p y j1 y j2 · · · y jq , as desired.

References 1. K. Akin, D. Buchsbaum and J. Weyman, Resolution of determinantal ideals: the submaximal minors, Adv. Math. 39 (1980), 1–30. 2. W. Bruns and J. Herzog, Cohen–Macaulay rings (Revised), Cambridge University Press (1998). 3. M. Brodmann, Asymptotic depth and connectedness of schemes, Proc. Amer. Math. Soc. 108 (1990), 573–581. 4. A. Conca, Regularity jumps for powers of ideals, Commutative algebra, Lect. Notes Pure Appl. Math. 244, Chapman & Hall/CRC, Boca Raton, FL, (2006), 21–32. 5. A. Conca and J. Herzog, Castelnuovo–Mumford regularity of products of ideals, Collect. Math. 54 (2003), 137–152. 6. A. Conca, J. Herzog and T. Hibi, Rigid resolutions and big Betti numbers, Comm. Math. Helv. 79 (2004), 826–838. 7. A. Conca, J. Herzog and G. Valla, Sagbi bases with applications to blow-up algebras, J. Reine Angew. Math. 474 (1996), 113–138. 8. D. Eisenbud, Commutative Algebra; with a view towards algebraic geometry, Springer (1995). 9. C. A. Francisco and A. Van Tuyl, Sequentially Cohen–Macaulay edge ideals, Proc. Amer. Math. Soc. 135 (2007), 2327–2337. 10. J. Herzog and T. Hibi, Componentwise linear ideals, Nagoya Math. J. 153 (1999), 141–153. 11. J. Herzog and T. Hibi, Cohen–Macaulay polymatroidal ideals, European J. Combin. 27 (2006), 513–517. 12. J. Herzog, T. Hibi and X. Zheng, Monomial ideals whose powers have a linear resolution, Math. Scand. 95 (2004), 23–32. 13. J. Herzog, T. Hibi and X. Zheng, Cohen–Macaulay Chordal Graphs, J. Combin. Theory Ser. A 113 (2006), 911–916. 14. J. Herzog, A. Simis and W. Vasconcelos, Approximation complexes and blowing-up rings, J. of Alg. 74 (1982), 466–493. 15. J. Herzog, A. Simis and W. Vasconcelos, Approximation complexes and blowing-up rings II, J. of Alg. 82 (1983), 53–83. 16. S. Iyengar and T. Römer, Linearity defects of modules over commutative rings, J. of Alg. 322 (2009), 3212–3237. 17. C. Huneke, The theory of d-sequences and powers of ideals, Adv. Math. 46 (1982), 249–279. 18. T. Römer, On minimal graded free resolutions, Dissertation, Universität Duisburg-Essen, 2001. 19. T. Römer, Homological properties of bigraded algebras, Illinois J. Math. 45 (2001), 1361– 1376. 20. K. Yanagawa, Alexander Duality for Stanley–Reisner Rings and Squarefree N n -Graded Modules, J. of Alg. 225 (2000), 630–645.

Modules with 1-Dimensional Socle and Components of Lusztig Quiver Varieties in Type A Joel Kamnitzer and Chandrika Sadanand

Abstract We study modules with 1-dimensional socle for preprojective algebras for type A quivers. In particular, we classify such modules, determine all homomorphisms between them, and then explain how they may be used to describe the components of Lusztig quiver varieties.

1 Introduction For any simply-laced Kac–Moody Lie algebra g, Lusztig [L] has constructed canonical bases for its representations using the geometry of quiver varieties. In particular, Lusztig considered the variety Rep(w)v of representations of the preprojective algebra Λ on a fixed vector space of dimension v and having dimension of socle bounded by w. The irreducible components of this variety index Lusztig’s canonical basis for a particular weight space of a highest weight representation of g. The components of Rep(w)v are also in natural bijection with the components of Nakajima’s Lagrangian quiver varieties. This is shown in the work of Saito [Sai, Sect. 4.6], who also studied a crystal structure on these components jointly with Kashiwara [KS]. Because the components of Rep(w)v index the canonical basis, it would be interesting to describe them in an explicit fashion using known combinatorics. In certain special cases (including g = sln ), this has been done by Savage [Sav], using ad-hoc methods. In a forthcoming paper [BK], Pierre Baumann and the first author will use module-theoretic means to give a uniform description of the components using Joel Kamnitzer Department of Mathematics, University of Toronto, Toronto, Canada e-mail: [email protected] Chandrika Sadanand Department of Mathematics, Stony Brook University, Stony Brook, USA e-mail: [email protected] G. Fløystad et al. (eds.), Combinatorial Aspects of Commutative Algebra and Algebraic Geometry, Abel Symposia 6, DOI 10.1007/978-3-642-19492-4 5, © Springer-Verlag Berlin Heidelberg 2011

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the theory of MV polytopes [K]. In our description, a key role is played by certain Λ -modules with one dimensional socle. In the current paper, we focus on the case g = sln . Using elementary means, we classify Λ -modules with one dimensional socle and explain how these modules can be used to describe components of Rep(w)v . Similar results (and more) will be formulated and proved for general g in [BK]. More specifically in Sect. 3, we classify Λ -modules with one dimensional socle by showing that they are all isomorphic to certain Maya modules introduced by Savage [Sav]. These Maya modules are in bijection with subsets of {1, . . . , n} (other than {1, . . . , i}). Next, we compute the space of homomorphisms between two such modules, obtaining an explicit combinatorial formula. We show that this formula is related to a truncated permutahedron, which is the MV polytope for this situation. In Sect. 4, we show how Maya modules can be used to describe the components of Rep(w)v . We begin by computing the space of homomorphisms between certain Maya modules and modules associated to tableaux by Savage [Sav]. We use this to rephrase Savage’s description of the components in a module-theoretic fashion.

1.1 Acknowledgements We would like to thank Pierre Baumann for allowing us to use some ideas from our forthcoming joint work. We would also thank Alistair Savage for his generous explanations. We thank Bernard Leclerc and Bernhard Keller for helpful conversations. Finally, we thank Lucy Zhang and the referee for their comments on this paper. Part of this work was conducted by C.S. as an NSERC undergraduate summer research project.

2 Background 2.1 Notation Let Q denote the root lattice of SLn . So Q = (x1 , . . . , xn ) ∈ Zn : ∑ xi = 0 . For i = 1, . . . , n − 1, let αi = (. . . , 0, 1, −1, 0, . . . ) denote the simple roots (the 1 is in the ith position). Let Q+ be the subset of Q given by non-negative sums of the αi . Let ωi = (1, . . . , 1, 0, . . . , 0) denote the fundamental weights (the first i entries are 1s). If A and B are i element subsets of {1, . . . , n}, then we define A − B := 1A − 1B ∈ Q

Modules with 1-Dimensional Socle . . .

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where 1A is the n-tuple which is 1 in positions indexed by numbers in A and 0 in the other positions. We write A ≥ B if A − B ∈ Q+ .

2.2 The Preprojective Algebra Let Ω be a simply-laced Dynkin quiver (that is a Dynkin diagram with orientation) with edge set Ω and vertex set I. Let Λ denote the preprojective algebra of the quiver Ω . By definition Λ is the quotient P(Ω ⊕ Ω )/ ∑ ττ − ττ τ ∈Ω

of the path algebra of the doubled quiver Ω ⊕ Ω by the preprojective relation. For this paper, we will work exclusively with the type An−1 quiver with the leftward orientation.

For this quiver we have vertex set I = {1, . . . , n − 1} and edge sets

Ω = {2 → 1, 3 → 2, . . . , n − 1 → n − 2} Ω = {1 → 2, 2 → 3, . . . , n − 2 → n − 1} So a Λ -module M consists of an I-graded vector space M = ⊕i∈I Mi with linear maps (i → i + 1) : Mi → Mi+1

(i → i − 1) : Mi → Mi−1

such that the preprojective relations (i + 1 → i)(i → i + 1) = (i − 1 → i)(i → i − 1) for i = 1, . . . , n − 1 are satisfied. Here and later, we adopt the convention that (1 → 0) : M1 → 0 and (n − 1 → n) : Mn−1 → 0 are 0. If M is a Λ -module, then it has a dimension vector v = (vi )i∈I ∈ NI , where vi = dim(Mi ). It will be convenient to encode this as an element of Q+ as αv = ∑i vi αi .

2.3 Socle of Modules The only simple Λ -modules are the one-dimensional modules Si , which have dimension 1 in the ith slot and 0 elsewhere. If M is any Λ -module, then the socle of M is defined to be the maximal semisimple submodule of M. The Si th isotypic component of the socle of M is called the i-socle of M and is denoted soci (M).

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More explicitly, soc(M) is the submodule of M whose ith graded piece is soci (M) = {w ∈ Mi : (i → i + 1)(w) = 0 and (i → i − 1)(w) = 0} All arrows act by 0 in soc(M).

2.4 Lusztig Quiver Varieties If v ∈ NI , then we may consider the variety Repv of representations of Λ on a fixed I-graded vector space of dimension v. By the work of Lusztig [L], the irreducible components of Repv index the canonical basis for (Un)αv , where Un denotes the universal enveloping algebra of the upper triangular subalgebra of g. In particular, the number of irreducible components of Repv is equal to the Kostant partition function of αv . For each point x ∈ Repv , we can consider the corresponding abstract Λ -module Mx . For w ∈ NI , we consider the variety Rep(w)v of consisting of those points x ∈ Repv with dim soci (Mx ) ≤ wi for all i. Under Lusztig’s construction the components of these varieties are related to the irreducible representations as follows. Let λ = λw := ∑i wi ωi and μ = λw − αv (here ωi are the fundamental weights). The irreducible components of Rep(w)v index the canonical basis for the μ weight space of the irreducible representation V (λ ) of SLn .

3 Modules with One-Dimensional Socle 3.1 The Maya Modules Let A be a proper subset of {1, . . . , n} of size i, other than {1, . . . , i}. The Maya module N(A) has the following description. If A = {a1 < · · · < ai }, then N(A) has basis w1,1 , . . . , wa1 −1,1 , . . . , wk,k , . . . , wak −1,k , . . . , wi,i , . . . , wai −1,i where w j,k ∈ N(A) j . We define

( j → j − 1)(w j,k ) = w j−1,k ( j → j + 1)(w j,k ) = w j+1,k+1

(1)

Note that N(A) has a 1-dimensional socle Si , spanned by wi,i . Let us call the span of wk,k , . . . , wak −1,k the kth “row” of N(A) and let us call N(A) j the jth “column”. So the kth row starts at column k and extends to column ak − 1. This can be seen in the following picture of the module N({3, 6, 7}).


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Lemma 3.1. Let v = dim(N(A)). Then αv = {1, . . . , i} − A. Proof. Since we have an explicit basis for N(A) it is easy to see that dim N(A) j = |{r ∈ {1, . . . , i} : r ≤ j < ar }| From this, the desired result follows immediately.

3.2 The Uniqueness Theorem We will now show that every Λ -module with 1-dimensional socle is isomorphic to a Maya module. We start by characterizing the dimension vectors of modules with 1-dimensional socle. If v is a dimension vector, we will extend v by defining v0 = 0 = vn (this will eliminate some special cases below). Lemma 3.2. Let M be a module with socle Si . Let v = dim(M). Then v j = v j+1 or v j + 1 = v j+1 , for all j < i v j−1 = v j or v j−1 = v j + 1, for all j > i

(2) (3)

Proof. Suppose that (2) does not hold for some j < i. Then either v j > v j+1 or v j + a = v j+1 for some a > 1. Suppose that v j > v j+1 . Then dim ker( j → j + 1) > 0. Consider a non-zero element w ∈ ker( j → j + 1). Then ( j + 1 → j) ◦ ( j → j + 1)(w) = 0 ⇒ ( j − 1 → j) ◦ ( j → j − 1)(w) = 0 But, ( j → j − 1)(w) = 0, since M has no j-socle. Hence dim ker( j − 1 → j) > 0. Continuing in this manner we see that dim ker(1 → 2) > 0. This means that M has 1-socle, a contradiction. Now, suppose that v j + a = v j+1 for some a > 1. Assume j + 1 < i. In this case, dim ker( j + 1 → j) ≥ a > 1

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Let w ∈ ker( j + 1 → j), then ( j → j + 1) ◦ ( j + 1 → j)(w) = 0 ⇒ ( j + 2 → j + 1) ◦ ( j + 1 → j + 2)(w) = 0 But ( j + 1 → j + 2)(w) = 0, since M has no j + 1-socle. Therefore ( j + 1 → j + 2) gives us an injective map from ker( j + 1 → j) to ker( j + 2 → j + 1) and so dim ker( j + 2 → j + 1) ≥ a Continuing in this manner, when we reach i, we see that since the socle is onedimensional, it must be the case that dim ker (i → i + 1)|ker(i→i−1) = 1 and hence we find dim ker(i + 1 → i) ≥ a − 1 > 0 Again, we consider an element w ∈ ker(i + 1 → i). Then (i → i + 1) ◦ (i + 1 → i)(w) = 0 ⇒ (i + 2 → i + 1) ◦ (i + 1 → i + 2)(w) = 0 Again, since M does not have i + 1-socle, dim ker(i + 2 → i + 1) > 0 Continuing in this manner, we see that dim ker(n − 1 → n − 2) > 0, which implies that M has (n − 1)-socle. This is a contradiction. The proof of (3) follows similarly. Lemma 3.3. Suppose that v ∈ NI satisfies the condition (2) and (3). Then αv = {1, . . . , i} − A for some i element subset of {1, . . . , n}, A = {1, . . . , i}. Proof. Let αv = ∑n−1 j=1 v j α j , and let x j be the jth coordinate of αv . Then for each j = 1, . . . , n, we have x j = 1⇐⇒v j = v j−1 + 1, x j = −1⇐⇒v j + 1 = v j−1 , x j = 0⇐⇒v j = v j−1 . Also note that x j = 1 ⇒ j ≤ i and x j = −1 ⇒ j > i. So define A := { j ≤ i : x j = 0} ∪ { j > i : x j = 1} and then it is easily seen that A has the desired properties.

Now we formulate and prove the uniqueness statement. Theorem 3.4. Let M be a module with socle Si and dimension v. Let A be such that αv = {1, . . . , i} − A. Then M ∼ = N(A). This result is well-known to experts. For example, it follows from the fact that certain Nakajima quiver varieties are 0-dimensional. It can also be proved using the


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crystal structure on components of quiver varieties (due to Kashiwara–Saito [KS]). Here we prefer to give an elementary argument. Proof. Our goal is to find a basis for M whose module structure matches the Maya module structure (1). Let A = {a1 < · · · < ai }. Let wi,i ∈ Mi be a basis for the socle of M. Assume that ai > i + 1. We claim that there exists wi+1,i ∈ Mi+1 such that (i + 1 → i)(wi+1,i ) = wi,i . Suppose that no such wi+1,i exists. From the proof of Lemma 3.2, we see that wi,i spans the kernel of (i → i + 1). Hence if wi,i is not in the image of (i + 1 → i), then (i → i+1)◦(i+1 → i) is an isomorphism. By the preprojective relations, this means that (i+2 → i+1)◦(i+1 → i+2) is an isomorphism. From the proof of Lemma 3.2, we know that (i+2 → i+1) is injective, so both (i+2 → i+1) and (i+1 → i+2) are isomorphisms. Hence (i + 1 → i + 2) ◦ (i + 2 → i + 1) is an isomorphism. Continuing in this fashion, we find that all ( j → j + 1) are isomorphisms for j ≥ i and so we see that vi+1 = · · · = vn = 0. This contradicts ai > i + 1. By a similar argument, there exist wi+2,i , . . . , wai −1,i such that i+1→i

i+2→i+1

a −1→a −2

i i wi,i ←−−−− wi+1,i ←−−−−− . . . ←− −−−− −− wai −1,i .

Since (i → i − 1)(wi,i ) = 0, from the preprojective relations we find that (k → k + 1)(wk,i ) = 0 for all k. Thus wi,i , . . . , wai −1,i spans a submodule which we denote by N. Note that N ∼ = N({1, . . . , i − 1, ai }). If M = N, then we are done. Suppose that N = 0 and consider the quotient module M/N. Since dim M/N = dim M − dim N, we see that if v = dim M/N, then

αv = {1, . . . , i − 1} − {a1 , . . . , ai−1 }. We claim that soc(M/N) = Si−1 . As above, there exists w ∈ Mi−1 such that (i − 1 → i)(w) = wi,i and as above (i − 1 → i − 2)(w) = 0. Hence [w] ∈ soc(M/N). To see that there is no other socle, note that if [u] ∈ soc(M/N) j , then ( j → j − 1)(u) ∈ N and ( j → j + 1)(u) ∈ N. Suppose that j < i − 1, then ( j → j + 1)(u) ∈ N implies that ( j → j + 1)(u) = 0 which implies that u = 0 since ( j → j + 1) is injective (as in the proof of Lemma 3.2). Suppose that j = i − 1, then the injectivity of ( j → j + 1) forces u = w. If j = i, then the [u] = 0, since the kernel of (i → i − 1) is spanned by wi,i . Similarly if j > i, then [u] = 0, since ( j → j − 1) is injective (as in the proof of Lemma 3.2), so u must be a multiple of w j,i . Thus, we have shown that soc(M/N) = Si−1 . Thus by the induction hypothesis, we see that M/N ∼ = N({a1 , . . . , ai−1 }) and we obtain a short exact sequence of Λ modules 0 → N → M → N({a1 , . . . , ai−1 }) → 0. Let us pick a vector space splitting. Thus combining the standard basis of N({a1 , . . . , ai−1 }) with the above basis of N, we obtain a basis wk,l for M with l = 1, . . . , i and k = l, . . . , al − 1. This module structure with respect to this basis does not match (1), since extra terms involving the basis for N may enter into the result of applying

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quiver arrows to the basis elements of N({a1 , . . . , ai−1 }). Hence we will now adjust our basis. In particular, for each l = 1, . . . , i − 1 and k = i + 1, . . . al − 1, we see that there is a scalar ck,l such that (k → k − 1)(wk,l ) = wk−1,l + ck,l wk−1,i We may eliminate this scalar by setting wk,l = wk,l − ck,l wk,i for these (k, l) and wk,l = wk,l otherwise. Next, note that (i − 1 → i)(wi−1,i−1 ) = 0 in N({a1 , . . . , ai−1 }) and thus (i − 1 → i)(wi−1,i−1 ) = cwi,i in M for some scalar c. Since M has no i − 1 socle, c is non-zero. Scaling all wk,l by 1/c (for l < i), we may assume that c = 1. It then follows from the preprojective relations that (k → k + 1)(wk,i−1 ) = wk+1,i for all k = i, . . . , ai−1 − 1. Now consider some wk,l for l < i − 1 and k ≥ i − 1. Then (k → k + 1)(wk,l ) = wk+1,l+1 + cl wk+1,i for some scalar cl . By the preprojective relations cl depends only on l. Then we make the adjustment wk,l = wk,l − cl wk,i−1 for all k = i − 1, . . . , al−1 − 1 and wk,l = wk,l for all other (k, l). After all these adjustments, we see that wk,l satisfy the Maya module structure (1). Thus M ∼ = N(A) as desired.

3.3 Computation of Hom Spaces Now we compute the space of homomorphisms between Maya modules. Theorem 3.5. Let A, B be i, j element subsets respectively. Then we have dim Hom(N(A), N(B)) = # of r ∈ {1, . . . , i}, such that r ≤ j < ar , and ar−l ≤ b j−l for l = 0, . . . , r − 1 where A = {a1 < · · · < ai } and B = {b1 < · · · < b j }. Proof. Let R := r ∈ {1, . . . , i} : r ≤ j < ar , and ar−l ≤ b j−l for l = 0, . . . , r − 1 We construct a map ϕ : R → Hom(N(A), N(B)), and then show that it gives a bijection between R and a basis for Hom(N(A), N(B)). This will yield the desired result.


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For simplicity of notation, we will use wk,l for the basis for N(A) and wk,l for the basis for N(B). For each r ∈ R, let us define ϕ (r) = ϕr to be the homomorphism which takes the rth row of N(A) to the bottom row of N(B) and then extended to higher rows in the obvious way. More explicitly, we define ϕr by

wk, j−l , if l ≥ 0, and k ≥ j − l ϕr (wk,r−l ) = 0, otherwise Such a wk, j−l will always exist since j − l ≤ k < ar−l ≤ b j−l . A simple check using the structure of Maya modules (1) shows that ϕr is a homomorphism. Now, suppose that ψ in any element of Hom(N(A), N(B)). Since we have explicit bases for N(A) and N(B) we may consider the matrix coefficients involving wj, j , the generator of the socle of N(B). ψ takes N(A) j to N(B) j so for each r ∈ {1, . . . , i} such that r ≤ j < ar , we get a matrix coefficient sr , such that

ψ (w j,r ) = sr wj, j + · · · . Note that if all the sr are zero, then ψ = 0. This is because every submodule of N(B) must contain wj, j (since wj, j spans the socle of N(B)) and so any non-zero homomorphism from N(A) to N(B) must hit wj, j . Thus, the collection sr completely determines ψ . Also note that if r ∈ / R, then ar−l > b j−l for some l. This means that we can find some non-zero w ∈ N(A) and p ∈ Λ such that pw = w j,r but ψ (w) = 0 (in fact we can choose w = war−l −1,r−l ). Hence for r ∈ / R, we see that sr = 0. Combining these observations, we see that ψ = ∑r∈R sr ϕr . Thus the ϕr span Hom(N(A), N(B)). These ϕr are linearly independent since ϕr vanishes on a j,r for r > r . Thus the ϕr form a basis for Hom(N(A), N(B)) as desired.

3.4 Connection with MV Polytopes We now make the connection between Theorem 3.5 and MV polytopes. For each subset B of {1, . . . , n} of size i, we may consider the truncated permutahedron P(B) which is defined as P(B) := conv({1C − 1{1,..., j} : C is a subset of {1, . . . , n} of size j and C ≤ B }) These polytopes P(B) are relevant since Naito–Sagaki [NS] have shown that these are the MV polytopes associated to the vertices of the crystal corresponding to the minuscule representation Λ i Cn . These vertices are precisely labeled by subsets B of size i. Corollary 3.6. For each subset B of {1, . . . , n}, the max value of 1A , on the polytope P(B) is given by dim Hom(N(A), N(B)).

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Proof. Assume for simplicity that i ≤ j. A similar proof holds in the i > j case. By Theorem 3.5, dim Hom(N(A), N(B)) = r − s where r is the maximal element of {1, . . . , i} such that ar−l ≤ b j−l for j = 0, . . . , r − 1 and s = |{1, . . . , j} ∩ A|. Now, we claim that r = maxC≤B |C ∩ A|. First note that if we choose C to be the smallest possible j element subset of {1, . . . , n} such that {a1 , . . . , ar } ⊂ C, then C ≤ B and |C ∩ A| ≥ r. On the other hand, for any C ≤ B, we claim that |C ∩ A| ≤ r. To see why this is the case, note that by the definition of r, not all the inequalities ar+1 ≤ b j , ar ≤ b j−1 , . . . , a1 ≤ b j−r

(4)

can hold. So now suppose that C ≤ B and C ∩ A contains at least r + 1 elements. Let us choose r + 1 of these elements and order them ai1 < · · · < air+1 . Then since C ≤ B, we find that air+1 ≤ b j , . . . , ai1 ≤ b j−r . But since ail ≥ al for all l, this implies that all the inequalities (4) hold—a contradiction. Hence we conclude that r = maxC≤B |C ∩ A|. Thus dim Hom(N(A), N(B)) = r − s = max |C ∩ A| − |{1, . . . , j} ∩ A| C≤B

= max1A , 1C − 1{1,..., j} C≤B

as desired.

4 Description of Irreducible Components 4.1 Savage’s Description of the Components Alistair Savage has given a description of the components of Rep(w)v in terms of tableaux. We would like to reformulate his description in terms of Hom spaces. Let Tabμ (λ ) denote the set of semistandard Young tableaux (SSYT) of shape λ and content μ . If X is a box in a SSYT T , then we will write r(X) for the row of X and c(X) for the content of X. For each T ∈ Tab(λ )μ , Savage has identified a component CT of Rep(w)v . Let T ∈ Tab(λ )μ . A Λ -module is said to be of type T if there exists a basis for M with the following properties. For each box X in T , there are vectors wXr(X) , . . . , wXc(X)−1 ∈ M and the collection of all these vectors (over all boxes X) forms a basis for M. Moreover, we have ( j → j − 1)(wXj ) = wXj−1 ,

( j → j + 1)(wXj ) = ∑ dYX wYj+1 Y

(5)


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for some scalars dYX , where the sum varies over all those boxes Y such that r(Y ) < r(X) ≤ c(Y ) < c(X). Let CT = {x ∈ Rep(w)v : Mx is of type T } denote the closure of the locus of those modules of type T . Theorem 4.1 ([Sav, Sect. 5]). CT is a component of Rep(w)v and this provides a bijection between the components of Rep(w)v and Tab(λ )μ .

4.2 Description of the Components by Hom Spaces We would like to reformulate Savage’s description. The key will be the following generalization of Theorem 3.5. A connected subset of {1, . . . , n} is one of the form {t − i + 1,t − i + 2, . . . ,t}. Theorem 4.2. Let M be a module of type T and let A = {t − i + 1, . . . ,t} be a connected subset of {1, . . . , n}. Then dim Hom(M, N(A)) = # of boxes X in T , such that r(X) ≤ i < c(X) ≤ t. Proof. The idea is similar to the proof of Theorem 3.5. To each box X of T in the above set, we can define a homomorphism ϕX : M → N(A) by taking the row indexed by X to the bottom row of N(A). We then extend to all of M. More explicitly, we define ϕX (wXk ) = wk,i for k ≥ i (note that such wk,i exists since i ≤ k < c(X) ≤ t = ai ). We define ϕX (wXk ) = 0 for k < i. We also define ϕX (wYk ) = 0 for all Y = X with r(Y ) ≥ r(X) or c(Y ) ≥ c(X). Now we proceed to define ϕX (wYk ) for those boxes Y with r(Y ) < r(X) and c(Y ) < c(X). We do so by an inductive procedure on r(Y ). Suppose that Y is a box with r(Y ) = r(X) − 1 and c(Y ) < c(X). Then we define

ϕX (wYk ) = dYX wk,i−1 . Note that such wk,i−1 exists since k < c(Y ) < c(X) ≤ t and so k < t − 1 = ai−1 . Next, suppose that Y is a box with r(Y ) = r(X) − 2 and c(Y ) < c(X). Then we define ϕX (wYk ) = dYX wk,i−1 + ∑ dYZ dZX wk,i−2 , Z

where the sum ranges over all those boxes Z such that r(Z) = r(X) − 1 and c(Y ) < c(Z) < c(X). Continuing in this fashion, we define ϕX on all of M. The structure of the module as given in (5) ensures that ϕX is a Λ -module homomorphism.

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The fact that these ϕX form a basis for Hom(M, N(A)) follows along the same lines as in the proof of Theorem 3.5. We now combine this result and Savage’s theorem. For each A = {t −i+1, . . . ,t}, we define a constructible function fA : Rep(w)v → N x → dim(Hom(Mx , N(A)) Since this is a constructible function it takes a constant value on a constructible dense subset of each component of Rep(w)v . For each component Z ⊂ Rep(w)v , let f A (Z) denote this constant value. Also, for each T ∈ Tab(λ )μ , let gA (T ) denote the number of boxes X in T such that r(X) ≤ i < c(X) ≤ t. Note that the collection {gA (T )} (where A ranges over all connected subsets) determines T . Theorem 4.3. For each component Z ⊂ Rep(w)v , there exists a tableau T ∈ Tab(λ )μ such that fA (Z) = gA (T ) for all connected subsets A ⊂ {1, . . . , n}. This provides a bijection between the components of Rep(w)v and the SSYT of shape λ and filling μ . Proof. Theorem 4.2 shows that if M is of type T , then fA (M) = gA (T ). Theorem 4.1 shows that for each component Z, there exists a unique tableau T such that there is a dense subset of Z consisting of modules of type T . Combining these two results, we obtain the desired result.

References [BK] P. Baumann and J. Kamnitzer, Preprojective algebras and MV polytopes; arXiv:1009.2469. [K] J. Kamnitzer, Crystal structure on Mirković–Vilonen polytopes, Adv. in Math. 215, no. 1 (2007) 66–93; math.QA/0505398. [KS] M. Kashiwara and Y. Saito, Geometric construction of crystal bases, Duke Math. J. 89, no. 1 (1997) 9–36. [L] G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, JAMS 4 (1991) 365–421. [NS] S. Naito and D. Sagaki, Mirković–Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal, Adv. Math. 221, no. 6 (2009) 1804–1842; arXiv:0806.3112. [Sai] Y. Saito, Crystal bases and quiver varieties, Math. Ann. 324, no. 4 (2002) 675–688. [Sav] A. Savage, Geometric and combinatorial realizations of quiver varieties, Algebr. Represent. Theory 9, no. 2 (2006) 161–199; math.RT/0310314.

Realization Spaces for Tropical Fans Eric Katz and Sam Payne

Abstract We introduce a moduli functor for varieties whose tropicalization realizes a given weighted fan and show that this functor is an algebraic space in general, and is represented by a scheme when the associated toric variety is quasiprojective. We study the geometry of these tropical realization spaces for the matroid fans studied by Ardila and Klivans, and show that the tropical realization space of a matroid fan is a torus torsor over the realization space of the matroid. As a consequence, we deduce that these tropical realization spaces satisfy Murphy’s Law.

1 Introduction The tropicalization of a d-dimensional subvariety of a torus with respect to the trivial valuation is the underlying set of a pure d-dimensional fan with a locally constant positive integer weight function on its smooth points. The question of which subvarieties of a torus, if any, have a given tropicalization has generated many discussions, but the literature treats only the special cases of existence of realizations for curves of genus zero or one [Spe07] and tropical linear spaces [Mik08], in characteristic zero. Here we use Hilbert schemes and Tevelev’s approach to compactifications of subvarieties of tori [Tev07] to address the general case, constructing a fine moduli scheme over the integers that parametrizes varieties whose tropicalization realizes a given weighted simplicial fan. Eric Katz Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA e-mail: [email protected] Sam Payne Stanford University, Mathematics, Bldg. 380, 450 Serra Mall, Stanford, CA 94305, USA e-mail: [email protected] Sam Payne is supported by the Clay Mathematics Institute. G. Fløystad et al. (eds.), Combinatorial Aspects of Commutative Algebra and Algebraic Geometry, Abel Symposia 6, DOI 10.1007/978-3-642-19492-4 6, © Springer-Verlag Berlin Heidelberg 2011

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Let T be the torus whose lattice of one parameter subgroups is N ∼ = Zn , and let NR = N ⊗Z R. Fix a pure d-dimensional simplicial fan Δ in NR with a weight function w that assigns a positive integer w(σ ) to each d-dimensional cone σ in Δ . Say that a tropical realization of (Δ , w) is a subvariety Y ◦ of T whose tropicalization is the underlying set |Δ |, with weight function given by w(σ ) on the relative interior of σ . In Sect. 2, we show that a subvariety Y ◦ of T is a tropical realization of (Δ , w) if and only if its closure Y in the toric variety X(Δ ) is proper and the intersection number (V (σ ) ·Y ) is equal to w(σ ) for every d-dimensional cone σ in Δ . The existence of Hilbert schemes makes it convenient to work with these proper subvarieties of toric varieties that appear as closures of tropical realizations rather than with the tropical realizations themselves, especially when working in families. Definition 1.1. Let S be a scheme. A family of tropical realizations of (Δ , w) over S is a subscheme Y of X(Δ ) × S, flat and proper over S, such that every geometric fiber is reduced, irreducible, and has intersection number w(σ ) with V (σ ), for every d-dimensional cone σ in Δ . The pullback of a family of tropical realizations of Δ under a morphism S → S is a family of tropical realizations over S , so the map RΔ ,w taking a scheme S to the set of all families of tropical realizations of Δ over S is a contravariant functor from schemes to sets. In Sect. 3 we show that this functor is an algebraic space, in the sense of [Knu71]. Recall that the toric variety X(Δ ) is quasiprojective if and only if the fan Δ can be extended to a regular subdivision of NR . Theorem 1.2. If the toric variety X(Δ ) is quasiprojective then the tropical realization functor RΔ ,w is represented by a scheme of finite type. In other words, there is a scheme RΔ ,w of finite type and a universal family of tropical realizations of (Δ , w) over RΔ ,w such that any family of tropical realizations of Δ over an arbitrary scheme S is the pullback of this universal family under a unique morphism S → RΔ ,w . We call RΔ ,w the tropical realization space of (Δ , w). Theorem 1.2 has been applied by the first author to construct nonarchimedean analytic moduli spaces for realizations of tropical polyhedral complexes over valued fields [Kat10]. Remark 1.3. If (Δ , w ) is a refinement of (Δ , w) then a subvariety of T is a tropical realization of (Δ , w ) if and only if it is a tropical realization of (Δ , w). So there is a natural bijection between the k-points of RΔ ,w and of RΔ ,w , for every field k, although these schemes need not be isomorphic. In Sect. 4, we study the tropical realization spaces of a special class of fans associated to matroids. The matroid fan ΔM associated to a matroid M was introduced by Ardila and Klivans [AK06], who called it the fine subdivision of the Bergman fan of the matroid. Theorem 1.4. Let M be a matroid. The tropical realization space of the matroid fan ΔM is naturally a torus torsor of rank #M − c(M) over the realization space of M.

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Here #M is the number of elements in the underlying set of the matroid and c(M) is the number of connected components of M. Corollary 1.5. The matroid fan Δ M is tropically realizable over a field k if and only if the matroid M is realizable over k. For instance, the matroid associated to the configuration of all seven lines in the projective plane P2 (F2 ) over the field with two elements is realizable only over fields of characteristic two. The associated matroid fan is a two-dimensional fan in a six-dimensional real vector space with twenty-one maximal cones corresponding to the twenty-one full flags in P2 (F2 ), and it is the tropicalization of a two-dimensional subvariety of a six-dimensional torus only over fields of characteristic two. Remark 1.6. Corollary 1.5 has been known as a folklore result in the tropical geometry community, and a version of this result over the complex numbers has been reported by Mikhalkin, based on joint work with Sturmfels and Ziegler [Mik08], but there is no proof in the literature. The geometric tropicalization theory of Hacking, Keel, and Tevelev [HKT09] implies that Δ M is the tropicalization of the complement of a hyperplane arrangement over a field k if and only if that hyperplane arrangement realizes the matroid M. In Sect. 4, we show that every tropical realization of ΔM is the complement of a hyperplane arrangement, which proves the corollary. For those interested only in the corollary, this is simpler than deducing it from Theorem 1.4, whose proof is more technical. Combining Theorem 1.4 and Mnëv’s Universality Theorem, we deduce that tropical realization spaces can have arbitrary singularity types, even when the tropical realizations themselves are all smooth. Following Vakil [Vak06], we say that a singularity type is an equivalence class of pointed schemes (Y, y), with the equivalence relation generated by setting (Y, y) ∼ (Y , y ) if there is a smooth morphism from Y to Y that takes y to y . A collection of schemes satisfies Murphy’s Law if every singularity type occurs on some scheme in the collection. We say that a weighted fan (Δ , w) is multiplicity free if w(σ ) is equal to one for all σ . In this case, we write RΔ for the tropical realization space RΔ ,w . Theorem 1.7. Every singularity type of finite type over Spec Z appears in the tropical realization space of a multiplicity free two-dimensional fan Δ such that the universal family is smooth over RΔ . In other words, tropical realization spaces of multiplicity free two-dimensional fans satisfy Murphy’s Law. Mnëv’s Universality Theorem also implies that deciding whether an arbitrary matroid is realizable over a given field k that is not algebraically closed is as difficult as the generalized Hilbert’s Tenth Problem of deciding whether an arbitrary system of polynomials with integer coefficients has a solution over k [Stu87]. This problem is known to be undecidable over many function fields, such as Fq (t), R(t), and C(t, u); decidability over the rational numbers is an open problem [Poo08]. Conventions. Throughout this paper, all tropicalizations are with respect to the trivial valuation and all schemes are locally of finite type.

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Acknowledgements. We are grateful to D. Eisenbud, E. Feichtner, D. Helm, and R. Vakil for helpful discussions related to this work and to the referee for several useful comments and suggestions.

2 Realization and Compactification As in the introduction, Δ is a pure d-dimensional fan in NR and X = X(Δ ) is the associated toric variety with dense torus T . The weight function w assigns a positive integer w(σ ) to each d-dimensional cone σ in Δ . Here we discuss how subvarieties of T that are tropical realizations of (Δ , w) can be characterized in terms of their closures in X. Let Y ◦ be a d-dimensional subvariety of T . The tropicalization Trop(Y ◦ ) is the set of vectors v ∈ NR such that the initial degeneration inv (Y ◦ ) in T is nonempty. It is the underlying set of a rational fan of pure dimension d, and comes with a tropical multiplicity function m on its subset of smooth points, defined as follows. We say that a point in Trop(Y ◦ ) is smooth if it has an open neighborhood homeomorphic to Rd . There is an open dense set of smooth points v in Trop(Y ◦ ) such that a d-dimensional subtorus Tv acts freely on inv Y ◦ . For such v, the tropical multiplicity m(v) is defined to be the length of the zero dimensional quotient inv Y ◦ /Tv . The balancing condition implies that the tropical multiplicity extends to a locally constant function on the set of all smooth points of Trop(Y ◦ ). In particular, the tropical multiplicity is constant on the relative interior of each maximal cone of Δ . The balancing condition then says that these integers associated to maximal cones are a Minkowski weight, in the sense of [FS97]. Since the balancing condition is necessary for a weighted fan to be tropically realizable, nothing is lost by assuming that all weighted fans in this paper are balanced. We now state the conventional definition of a tropical realization over a field k. In Proposition 2.4 below, we show that the tropical realizations in this naive sense are exactly those subvarieties of T whose closure in X(Δ ) is a family of tropical realizations over Spec k. To avoid any possibility of confusion, we always write Y ◦ for a subvariety of the torus T , and Y for its closure in X(Δ ). Definition 2.1. A tropical realization of (Δ , w) is a subvariety Y ◦ of T such that the underlying set of Trop(Y ◦ ) is |Δ | and the tropical multiplicity m(v) is equal to w(σ ) for v in the relative interior of σ . Lemma 2.2. Let Y ◦ be a d-dimensional subvariety of T . The underlying set of Trop(Y ◦ ) is equal to |Δ | if and only if its closure Y in X(Δ ) is proper and meets every T -invariant subvariety. Proof. This follows from Lemma 2.2 of [Tev07] in the case where X is smooth, since Y and |Δ | both have pure dimension d. For the general case, and for further references, see Sect. 8 of [Kat09].


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Since Δ is simplicial, the toric variety X(Δ ) is the coarse moduli space of a smooth toric stack [Iwa09]. It follows that there is a natural intersection product on the rational Chow groups A∗ (X)Q [Vis89] with all of the refined properties of the intersection product on Chow groups of smooth varieties, as presented in Chap. 8 of [Ful98]. In particular, if σ is a d-dimensional cone in Δ , then V (σ ) ·Y is a welldefined class in A0 (V (σ ) ∩ Y )Q . If Y is proper then this class has a well-defined degree, which is a rational number denoted (V (σ ) ·Y ). Lemma 2.3. If Trop(Y ◦ ) = |Δ | then the intersection number (V (σ ) ·Y ) is equal to the tropical multiplicity m(v) for v in the relative interior of σ . Proof. If Δ is sufficiently fine, then Y is Cohen–Macaulay and the scheme theoretic intersection V (σ ) ∩ Y is canonically identified with the zero-dimensional quotient inv (Y ◦ )/Tv , for a generic v in the relative interior of σ . The general case follows by choosing a suitable refinement and applying the projection formula. See Sect. 9 of [Kat09] for details. We now prove the following proposition, which was mentioned in the introduction. Proposition 2.4. A subvariety Y ◦ of the torus T is a tropical realization of Δ if and only if its closure Y in X(Δ ) is proper, and (V (σ ) ·Y ) = w(σ ), for every d-dimensional cone σ ∈ Δ . Proof. If Y ◦ is a tropical realization of (Δ , w) then Y is proper and (V (σ ) · Y ) = w(σ ) by the two preceding lemmas. We now show the converse. Suppose Y is proper and (V (σ ) · Y ) = w(σ ) for every d-dimensional cone σ in Δ . Since Y is proper, and both Y and Δ are d-dimensional, Trop(Y ◦ ) is a union of d-dimensional cones of Δ . For any d-dimensional cone σ in Δ , the intersection number (V (σ ) · Y ) is positive, so Y must intersect V (σ ). It follows that Trop(Y ◦ ) contains σ , and since this holds for all maximal cones of Δ , Trop(Y ◦ ) = |Δ |. By the preceding lemma, m(v) is equal to (V (σ ) ·Y ) for v in the relative interior of σ , and the proposition follows. The proposition motivates our definitions of a family of tropical realizations of (Δ , w) and the tropical realization functor RΔ ,w in the introduction. Corollary 2.5. Let k be a field. Then RΔ ,w (k) is the set of closures in X(Δ ) of the tropical realizations of (Δ , w) over k. Proof. The corollary follows from Proposition 2.4, since RΔ ,w (k) is the set of proper k-subvarieties of X(Δ ) whose intersection number with V (σ ) is w(σ ) for every d-dimensional cone σ in Δ .

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3 Representability In Sect. 2 we showed that a subvariety Y ◦ of T is a tropical realization of (Δ , w) if and only if its closure Y in X(Δ ) is proper and (V (σ ) · Y ) = w(σ ) for every ddimensional cone σ in Δ . Proposition 3.1. The tropical realization functor RΔ ,w is an open algebraic subspace of the Hilbert functor of any toric compactification of X(Δ ). Proof. Let Σ be a complete simplicial fan that contains Δ as a subfan; for a construction of such a fan, see Theorem III.2.8 of [Ewa96]. Then X(Σ ) is a proper simplicial toric variety that contains X(Δ ) as a T -invariant open subvariety. The Hilbert functor of X(Σ ) is an algebraic space [Art69], and there is an open subfunctor of the Hilbert functor that parametrizes reduced and irreducible d-dimensional subschemes that are contained in X(Δ ). Since intersection numbers are locally constant in flat families, it follows that RΔ ,w is also an open subfunctor of the Hilbert functor of X(Σ ), and hence is an algebraic space. We now prove Theorem 1.2, which says that the tropical realization functor RΔ ,w is represented by a scheme of finite type over the integers when X(Δ ) is quasiprojective. Recall that toric varieties are canonically defined over the integers, and X(Δ ) is quasiprojective if and only if Δ can be extended to a regular subdivision of NR , which means that X(Δ ) can be embedded as a T -invariant open subvariety of a projective toric variety. Theorem 3.2. If X(Δ ) is quasiprojective then the tropical realization space RΔ ,w is represented by a scheme of finite type. Proof. Fix a projective toric variety X(Σ ) ⊂ Pr that contains X(Δ ) as a T -invariant open subvariety. By Proposition 3.1, the tropical realization functor RΔ ,w is represented by an open subscheme RΔ ,w of the Hilbert scheme of X(Σ ). We must show that RΔ ,w is of finite type. Suppose Y is the closure of a tropical realization of (Δ , w). We will show that there are only finitely many possibilities for the Hilbert polynomial of Y , and the theorem follows because the Hilbert scheme parametrizing subschemes of X(Σ ) with fixed Hilbert polynomial is of finite type. First, we claim that the degree of Y in Pr is determined by w. Now the Chow homology class [Y ] in Ad (X(Σ ))Q is determined by the intersection numbers (V (σ )· Y ) for d-dimensional cones σ in Σ [FS97], and Y is disjoint from V (σ ) if σ is not contained in Δ . Therefore (V (σ ) ·Y ) is w(σ ) if σ is in Δ , and zero otherwise. This determines [Y ] and the claim follows, since the degree of Y is given by the push forward of [Y ] under the given embedding of X(Σ ) in Pr . Next, the Castelnuovo–Mumford regularity of Y is bounded in terms of its degree and r [BM93]. So there is an integer N0 , depending only on (Δ , w), such that the Hilbert function of Y evaluated at N agrees with the Hilbert polynomial for N ≥ N0 . Therefore, the Hilbert polynomial of Y is determined by the value of the Hilbert function at N0 , . . . , N0 + d.


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Finally, the Hilbert function of a d-dimensional subvariety of Pr is bounded uniformly in terms of d, r, and its degree; for the construction of such a bound, see Exercise 3.28 in [Kol96]. It follows that there are only finitely many possible values for the Hilbert function of Y at N0 , . . . , N0 +d, and hence there are only finitely many possibilities for the Hilbert polynomial of Y , as required. Since any fan Δ has a refinement Δ that can be extended to a regular subdivision of NR , Theorem 3.2 implies that, for an arbitrary weighted simplicial fan (Δ , w), the tropical realizations of (Δ , w) over a field k are in natural bijection with the k-points of a scheme of finite type. Remark 3.3. These tropical realization spaces can also be used to construct more general moduli, such as for tropical realizations of pairs. If Δ is a subfan of positive codimension in Δ , then inside the product RΔ ,w × RΔ ,w there is the closed algebraic subspace parametrizing pairs (Y ,Y ) such that Y is contained in Y . The fibers of this moduli space of pairs under second projection parametrize tropical realizations of (Δ , w ) that are contained in a fixed subvariety Y of X(Δ ).

4 Tropical Realization of Matroid Fans Let M be a loop free matroid on a finite set E, and let e0 , . . . , en be the elements of E. Let N be the lattice N = ZE /e0 + · · · + en . The matroid fan ΔM is a simplicial fan in NR that encodes the lattice of flats of M. Ardila and Klivans introduced this fan in [AK06] and called it the fine subdivision of the Bergman fan of the matroid; the fan is defined as follows. For a subset I ⊂ E, let eI be the vector eI = ∑ ei ei ∈I

in NR . The rays of the matroid fan Δ M correspond to the proper flats F E of the matroid, and the ray ρF corresponding to a flat F is spanned by eF . More generally, the k-dimensional cones of the matroid fan correspond to the k-step flags of proper flats. If F is a flag of flats F1 ⊂ · · · ⊂ Fk then the corresponding cone σF is spanned by {eF1 , . . . , eFk }. Since each cone σF in ΔM is spanned by a subset of a basis for the lattice N, the toric variety X(Δ M ) is smooth. Furthermore, since every flag of flats in a matroid can be extended to a full flag, the matroid fan Δ M is of pure dimension equal to the rank of M minus one.

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Example 4.1. Let Un be the uniform matroid on {0, . . . , n}, the matroid in which every subset is a flat. Then the matroid fan ΔUn in Rn+1 /(1, . . . , 1) is the first barycentric subdivision of the fan corresponding to Pn , and X(Δ Un ) is obtained from Pn by a sequence of blowups X(Δ Un ) = Xn−1 → · · · → X1 → X0 = Pn , where Xi+1 → Xi is the blowup along the strict transforms of the i-dimensional T invariant subvarieties of Pn . The matroid fan ΔUn can also be realized as the normal fan of the n-dimensional permutahedron. Note that the labeling E = {e0 , . . . , en } of the underlying set of the matroid M induces an inclusion of the matroid fan Δ M as a subfan of Δ Un . In particular, the toric variety X(Δ M ) is quasiprojective, so the tropical realization functor RΔM is represented by a scheme of finite type. Furthermore, the dense torus T in X(Δ M ) is naturally identified with the dense torus in Pn . Proposition 4.2. Let Y ◦ be a tropical realization of Δ M . Then the closure of Y ◦ in Pn is a d-dimensional linear subspace. We will prove the proposition using a projection to Pd and the following lemmas. After possibly renumbering the elements of E, we may assume {e0 , . . . , ed } is a basis for M. Let U be the uniform matroid Ud . Consider the projection from NR to Rd+1 /(1, . . . , 1) taking ei to the image of the ith standard basis vector for i ≤ d, and taking e j to zero for j > d. Every cone in ΔM projects into some cone of ΔU , inducing a map of fans ΔM → ΔU . Composing the induced map of toric varieties X(ΔM ) → X(ΔU ) with the birational projection X(ΔU ) → Pd described in Example 4.1 gives a natural map p : X(ΔM ) → Pd . This map can also be factored as a birational morphism followed by a linear projection, as follows. Let Di be the ith coordinate hyperplane in Pn . Then p factors as p1 p2 Pn (D0 ∩ · · · ∩ Dd ) −→ Pd , X(Δ M ) −→ where p1 is the birational morphism induced by the open immersion of X(ΔM ) in the iterated blowup X(ΔUn ) of Pn , and p2 is the linear projection away from D0 ∩ · · · ∩ Dd . Let Y be the closure in X(Δ M ) of a tropical realization Y ◦ of Δ M . Lemma 4.3. The map p takes Y birationally onto Pd . Proof. We show that the map from ΔM to ΔU is of tropical degree one, and it follows that the map Y → X(Δ U ) is birational, by Theorem 1.1 of [ST08]. The lemma follows by composing this map with the birational projection from X(ΔU ) to Pd . Let v be a point in the interior of the maximal cone σ in ΔU corresponding to a full flag S1 ⊂ · · · ⊂ Sd of proper subsets of {0, . . . , d}. Let F be the flag F1 ⊂ · · · ⊂ Fd in which Fi is the flat spanned by {e j | j ∈ Si }. Then σF is the unique cone in Δ M whose image meets the relative interior of σ , and it projects bijectively onto σ .


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Therefore, the preimage of v in |Δ M | is a single point in the relative interior of σF . Since the lattice N ∩ span(σF ) maps isomorphically onto Zd+1 /(1, . . . , 1), the map of fans ΔM → Δ U is of tropical degree one, as required. Since p is birational and Pd is normal, the complement of the open set over which p is an isomorphism has codimension at least two. In particular, for any irreducible in Y , the strict transform of divisor D in Pd there is a unique irreducible divisor D D, that maps onto D, and the map from D to D is birational. Let x0 , . . . , xd be the standard homogeneous coordinates on Pd , and let H be the its strict transform in Y . For 1 ≤ i ≤ n, coordinate hyperplane cut out by x0 , with H let χi be the character of T , viewed as a rational function on X(ΔM ), corresponding to the unique lattice point ui ∈ Hom(N, Z) such that ⎧ ⎨ −1 if i = 0, ui , e j = 1 if i = j, ⎩ 0 otherwise. The character χi is invertible on the dense open subset Y ◦ of Y , and hence restricts to a nonzero rational function χi |Y on Y . and all its other Lemma 4.4. The rational function χi |Y has a simple pole along H, poles are contracted by p, for 1 ≤ i ≤ n. Proof. On X(Δ M ), the rational function χi has simple poles along those divisors DρF corresponding to flats F that contain e0 but not ei , and no other poles. In particular, p maps each pole of χi into H, and every T -invariant divisor of X(ΔM ) that maps onto are contracted H is a simple pole of χi . It follows that all poles of χi |Y other than H by p, and the multiplicity of the pole along H is independent of i. It will therefore suffice to show that χ1 |Y has a simple pole along H. Now χ1 |Y is the pullback of the rational function x1 /x0 on Pd under the birational map p. Since x1 /x0 has a simple pole along H, its pullback has a simple pole along and the lemma follows. the strict transform H, Proof of Proposition 4.2. To prove the proposition, we must show that the rational function χi |Y is a linear combination of 1, χ1 |Y , . . . , χd |Y , for i > d. Since p : Y → Pd is proper and birational, χi |Y induces a rational function fi on Pd whose poles are the push forward of the poles of χi |Y . By Lemma 4.4, fi has a simple pole along H and no other poles, and therefore may be expressed as a linear combination fi = a0 + a1 x1 /x0 + · · · + ad xd /x0 of the T -eigensections of O(H). Since χ j |Y is the pull back of x j /x0 for 1 ≤ j ≤ d, pulling back the expression above to Y gives χi |Y as a linear combination of 1, χ1 |Y , . . . , χd |Y , as required. Let Y ◦ be a tropical realization of ΔM , and let Y be its closure in X(Δ M ). By Proposition 4.2, the image of Y in Pn is a copy of Pd embedded linearly, so Y ◦ is exactly the complement in Pd of the arrangement of n + 1 hyperplanes given by

82


intersecting with the coordinate hyperplanes in Pn . In other words, when we factor p as a birational morphism followed by a linear projection, as described above, then Y ◦ is the complement of the hyperplane arrangement H0 , . . . , Hn , where Hi is the projection of p1 (Y ) ∩ Di , where Di is the ith coordinate hyperplane in Pn . We now show that this hyperplane arrangement realizes the matroid M, in the following standard sense. In terms of flats, a collection of hyperplanes H0 , . . . , Hn realizes M if the codimension of any intersection Hi0 ∩ · · · ∩ Hir is equal to the rank of the flat spanned by ei0 , . . . , eir . This is equivalent to the condition that the intersection of any d + 1 projective hyperplanes H j0 ∩ · · · ∩ H jd is empty if and only if {e j0 , . . . , e jd } is a basis for M. See [Oxl92] for this and other standard facts from matroid theory. Proposition 4.5. Let Hi be the hyperplane given by intersecting Pd with the ith coordinate hyperplane in Pn . Then {H0 , . . . , Hn } realizes M. Proof. Let Di be the ith coordinate hyperplane in Pn , so Hi is the intersection of Di with π (Y ) ∼ . For any I ⊂ {0, . . . , n}, the preimage in X(Δ M ) of the coordinate = Pd linear subspace i∈I Di is the union of the T -invariant divisors DρF for proper flats F that contain I. Therefore, if {ei0 , . . . , eid } is a basis for M then the preimage of Di0 ∩ · · · ∩ Din in X(Δ M ) is empty. It follows that the preimage of Hi0 ∩ · · · ∩ Hid in Y is empty, and since π : Y → Pd is surjective, it follows that Hi0 ∩ · · · ∩ Hid is empty as well. Conversely, if {ei0 , . . . , eid } is not a basis for M, then it is a subset of a proper flat F. Now Y meets the divisor DρF , and any point in Y ∩ DρF projects into Hi0 ∩ · · · ∩ Hid , so this intersection is nonempty, as required. The torus acts naturally on the set of tropical realizations of any fan; if Y ◦ is a tropical realization of (Δ , w) then the translation t · Y ◦ is also a tropical realization of (Δ , w), for t in T . We say that two tropical realizations of ΔM are isomorphic if they differ by translation by an element of T and that two realizations of M are isomorphic if they differ by an automorphism of Pd . Corollary 4.6. The map taking a tropical realization Y ◦ of Δ M to the hyperplane arrangement H0 , . . . , Hn in Pd gives a natural bijection between the isomorphism classes of tropical realizations of Δ M over a field k and the isomorphism classes of realizations of M over k. Proof. First, we show that the map from isomorphism classes of tropical realizations of ΔM to realizations of M is injective. Suppose Y1◦ and Y2◦ are tropical realizations of ΔM . Since there is a unique automorphism of Pd taking Hi to the ith coordinate hyperplane, for 0 ≤ i ≤ d, the projected images p(Y1◦ ) and p(Y2◦ ) in Pd must be equal. Then the two embeddings of this variety in T are determined by the push forwards under p of the rational functions χi |Y j◦ , for 0 ≤ i ≤ n, which are rational functions on Pd whose divisors of zeros and poles are independent of all choices, as in the proof of Lemma 4.4. Therefore χi |Y1◦ is a nonzero scalar multiple of χi |Y2◦ . It follows that Y1◦ is the translation of Y2◦ by an element of T , so the map from isomorphism classes of tropical realizations of Δ M to isomorphism classes of


83

realizations of M is injective. It remains to show that every realization of M comes from a tropical realization of ΔM in this way. Given a hyperplane arrangement realizing M, there is a natural embedding of the complement of this arrangement in T , well-defined up to translation, given by the space of global invertible functions. The tropicalization of the image is exactly Δ M , by the geometric tropicalization theory in Sect. 2 of [HKT09]. In particular, there is a natural surjection from the set of k-points of RΔM to the set of isomorphism classes of realizations of M over k. In the following concluding section we show that this surjection is induced by a smooth and surjective morphism of schemes, and deduce that tropical realization spaces satisfy Murphy’s Law.

5 Murphy’s Law for Tropical Realization Spaces Recall that a hyperplane in Pd is cut out by a nonzero section of the line bundle O(1) and a family of hyperplanes over a scheme S is cut out by a section of O(1) on the relative projective space PdS that is nonzero on the fiber over each point in S. The functor taking a scheme S to the set of such sections is represented by a scheme Γ that is isomorphic to Ad+1 0, and there is a locally closed subscheme ΓM of Γ n+1 parametrizing tuples of sections (s0 , . . . , sn ) cutting out families of realizations of the matroid M. These are exactly the tuples such that {si0 , . . . , sid } generates O(1) if and only if {ei0 , . . . , eid } is a basis for M. acts by coNote that PGLd+1 acts on ΓM by automorphisms of Pd and Gn+1 m ordinatewise scaling of the sections si . These actions commute, and the product acts freely on ΓM . Furthermore, two tuples of sections cut out the PGLd+1 ×Gn+1 m same family of hyperplane arrangements if and only if they differ by an element of Gn+1 m and two families of hyperplane arrangements are isomorphic, by definition, if they differ by an element of PGLd+1 . The quotient RM = ΓM /(PGLd+1 ×Gn+1 m ) is the usual realization space of the matroid M and represents the functor taking a scheme S to the set of isomorphism classes of families of hyperplane arrangements realizing M over S. Theorem 1.4 says that RΔM is naturally a torus torsor over RM . To prove this, we will describe a quotient of T that acts freely on RΔM , with quotient RM . Recall that a circuit in a matroid is a minimal set that is not contained in a basis, and a matroid is connected if any two elements are contained in a circuit. The matroid M has a unique decomposition into connected components M = F0 · · · Fc , which are disjoint flats such that every circuit of M is contained in some Fi , and the restriction of M to each Fi is connected. Let NM be the sublattice of M generated by

84


vF0 , . . . , vFc . Note that this sublattice has rank exactly c and is saturated in N; the sum vF0 + · · · + vFc is zero. Let TM be the subtorus of T whose lattice of one-parameter subgroups is NM . Proposition 5.1. The subtorus TM acts trivially on the tropical realization space RΔM , and the quotient T /TM acts freely. To prove the proposition, we use the following lemma characterizing subvarieties of T that are invariant under a subtorus. Let T be a subtorus of T whose lattice of one-parameter subgroups is N . Lemma 5.2. Let Y ◦ be a subvariety of T . Then Y ◦ is invariant under the action of T if and only if Trop(Y ◦ ) is invariant under translation by NR . Proof. Suppose Y ◦ is invariant under the action of T . Then base change to an extension field K with a valuation that surjects onto R, such as the field of generalized power series k((t R )). Since tropicalization commutes with base change, by Proposition 6.1 of [Pay09], Trop(Y ◦ ) is the image of Y ◦ (K) under the valuation. Then the action of T (K) on Y (K) induces an action of NR on Trop(Y ◦ ) by translations, as required. Conversely, suppose Trop(Y ◦ ) is invariant under translation by NR . Then the image of Trop(Y ◦ ) in NR /NR has dimension dimY ◦ − dim NR . This image is exactly the tropicalization of the closure of the image of Y ◦ in T /T . Therefore, the image of Y ◦ in T /T has dimension dimY ◦ − dim T , and hence Y ◦ is invariant under the action of T . Proof of Proposition 5.1. We begin by showing that the matroid fan Δ M is preserved under translation by (NM )R . The support of the matroid fan |Δ M | is the image in NR of the set of vectors (v0 , . . . , vn ) in Rn+1 such that, for every circuit {ei0 , . . . , eik } of M, the minimum of the set of coordinates {vi0 , . . . , vik } occurs at least twice [FS05]. Now, let F be a connected component of M. Since every circuit of M is either contained in F or disjoint from F, it follows that |Δ M | is invariant under translation by vF . Since this holds for each connected component of M, |Δ M | is invariant under translation by (NM )R . Next we show that (NM )R is the intersection of the affine spans of the maximal cones in ΔM , and hence every vector in NR that acts on Δ M by translations is in (NM )R . Suppose the image v in NR of (v0 , . . . , vn ) ∈ Rn+1 is in the span of every maximal cone of Δ M . To show that v is in (NM )R , we must show that v j is equal to vk whenever e j and ek are in the same connected component of M. Now, if e j and ek are in the same connected component then they are contained in a circuit {ei1 , . . . , eis , e j , ek }. Then there is a full flag of proper flats F = F1 ⊂ · · · ⊂ Fd such that Fr is the span of {ei1 , . . . , eir } for 1 ≤ r ≤ s, and Fs+1 is the span of the full circuit. The maximal cone σF is spanned by {vF1 , . . . , vFd }, and since each Fi contains either none or both of e j and ek , and v is in the span of the vectors vFi , it follows that the coordinates v j and vk are equal, as required.


85

By Lemma 5.2, since (NM )R acts on |ΔM | by translations, the subtorus TM preserves each tropical realization of Δ M , so the action of TM on the tropical realization space RΔM is trivial. It remains to show that T /TM acts freely on RΔM . By Fulton’s criterion for intersection multiplicity one, the closure of each tropical realization of ΔM meets the orbit Oσ corresponding to a maximal cone transversally in a single point. Therefore, to show that T /TM acts freely on RΔM it will suffice to show that T /TM acts freely on the union of these orbits Oσ . Since the pointwise stabilizer of Oσ is exactly the subtorus of T corresponding to the span of σ , and the intersection of these subtori is TM , it follows that T /TM acts freely on RΔM , as required. We will now prove the following precise version of Theorem 1.4. Theorem 5.3. The matroid realization space RM is naturally isomorphic to the quotient of the tropical realization space RΔM by the free action of T /TM . Proof. Recall that M is a fixed matroid on a finite set E = {e0 , . . . , en } whose elements are numbered so that e0 , . . . , ed is a basis for M. Let Di be the ith coordinate hyperplane in Pn . We have defined a natural morphism p : X(Δ ) → Pd that factors as a birational morphism p1 to Pn followed by the linear projection p2 from D0 ∩ · · · ∩ Dd . If Y is the closure in X(ΔM ) of a tropical realization of Δ M then p1 (Y ) is a d-dimensional linear subspace of Pn that is disjoint from D0 ∩ · · · ∩ Dd , and p2 maps p1 (Y ) ∩ Di to a hyperplane Hi in Pd . In Sect. 4 we showed that the hyperplane arrangement H0 , . . . , Hn realizes M. Now, suppose that YS is a family of tropical realizations of ΔM over a scheme S. By the Hilbert polynomial criterion for flatness, Hi = p2 (p1 (YS ) ∩ Di ) is a flat family of hyperplanes. Let si be a section of O(1) on PdS cutting out Hi . By Nakayama’s Lemma, si0 , . . . , sid generate O(1) if and only if they generate the fiber at every point. Therefore it follows from Proposition 4.5 that H0 , . . . , Hn is a family of realizations of M, and hence is the pullback under a unique morphism from S to RM . In particular, the universal family of tropical realizations over RΔM determines a natural morphism ϕ : RΔM → RM . We will show that ϕ is the quotient morphism for the free action of T /TM on RΔM given by Proposition 5.1; to prove this, we first show that the quotient map from ΓM to RM factors through ϕ . Recall that the closure Y in X(ΔM ) of any tropical realization of the matroid fan ΔM over a field is the strict transform of a d-dimensional linear subspace of Pn under a sequence of iterated blowups of strict transforms of coordinate linear subspaces. We construct a family of tropical realizations of ΔM from a tuple of sections of O(1) cutting out a realization of M by a similar sequence of blowups, giving a natural morphism ψ : ΓM → RΔM , as follows. Let S be a scheme, and let s0 , . . . , sn be sections of O(1) on PdS cutting out a family of realizations of M. For each flat F of M, let VF be the family of linear subspaces

86


VF =

Hi

ei ∈F

obtained by intersecting the corresponding families of hyperplanes. We construct a scheme YS from a sequence of blowups of PdS YS = Yd → Yd−1 → · · · → Y1 → Y0 ∼ = PdS , where the map from Y j+1 to Y j is the blowup along the strict transform of the flats VF of relative dimension j over S. By construction, YS is flat and proper over S, and we now construct an embedding of YS in X(ΔM ), making YS a family of tropical realizations of ΔM . By [Cox95], to give a morphism from YS to X(Δ M ) we must give a line bundle LF for every ray ρF in Δ M together with a collection of compatible isomorphisms u,v ∼ ϕi : ⊗F LF F − → OYS for u ∈ Hom(N, Z). Since the lattice points ui considered in Sect. 4, for 1 ≤ i ≤ n, characterized by the pairings ⎧ ⎨ −1 if i = 0, ui , e j = 1 if i = j, ⎩ 0 otherwise, form a basis for Hom(N, Z), it will suffice to give the isomorphisms ϕui . Then all other ϕu are given uniquely as a composition of the ϕui , and the compatibilu ,v ity condition is automatically satisfied. Now, the line bundle ⊗F LF i F is exactly O(Hi ) ⊗ O(H0 )∨ , so there are natural isomorphisms given by the sections s0 /si of Hom(O(Hi ) ⊗ O(H0 )∨ , OYS ). From the case where S is Spec k, treated in Sect. 4, it follows that the resulting map to X(ΔM ) × S is an embedding and that the image is a family of tropical realizations of ΔM over S. By the universal property of the tropical realizations space, this family is pulled back under a unique morphism from S to RΔM . In particular, the universal family of tuples of sections over ΓM determines a natural morphism ψ from ΓM to RΔM , taking a tuple of sections (s0 , . . . , sn ) to the tropical realization of ΔM given by embedding the complement of the vanishing loci of the si in the dense torus T in Pn by homogeneous coordinates [s0 : · · · : sn ]. The ϕ ◦ ψ takes a tuple of sections s0 , . . . , sn to the realization of M cut out by the si , and is the quotient map from ΓM to RM , as claimed. By construction, ψ is invariant under the action of PGLd+1 and the diagonal subtorus Gm in Gn+1 m , and descends to a T -equivariant map

ψ : ΓM /(PGLd+1 ×Gm ) → RΔM . The composition ϕ ◦ ψ is then the quotient map for a free T -action. Since the subtorus TM acts trivially on RΔM , and the quotient T /TM acts freely, ψ must be the quotient by the action of TM and ϕ is the quotient by the action of T /TM , making RΔM a T /TM torsor over RM , as required. We conclude by using Theorem 1.4 to show that tropical realization spaces satisfy Murphy’s Law.


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Proof of Theorem 1.7. Let (Y, y) be a singularity of finite type over Spec Z. By the scheme theoretic version of Mnëv’s Universality Theorem [Mnë85], as presented by Lafforgue in Sect. 1.8 of [Laf03], the singularity type of (Y, y) occurs at some point of the realization space of a rank three matroid M. Since the tropical realization space RΔM is smooth and surjective over RM , by Theorem 1.4, it follows that this singularity type also appears on RΔM . It remains to show that the universal family is smooth over RΔM , and since the universal family is flat it is enough to show that the fibers are smooth. Now the constructions of Sect. 4 show that each fiber is isomorphic to the blowup of P2 at finitely many distinct points, and is therefore smooth, as required.

References [AK06]

F. Ardila and C. Klivans, The Bergman complex of a matroid and phylogenetic trees, J. Combin. Theory Ser. B 96 (2006), no. 1, 38–49. [Art69] M. Artin, Algebraization of formal moduli. I, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 21–71. [BM93] D. Bayer and D. Mumford, What can be computed in algebraic geometry?, Computational algebraic geometry and commutative algebra (Cortona, 1991), Sympos. Math., XXXIV, Cambridge Univ. Press, Cambridge, 1993, pp. 1–48. [Cox95] D. Cox, The functor of a smooth toric variety, Tohoku Math. J. (2) 47 (1995), no. 2, 251–262. [Ewa96] G. Ewald, Combinatorial convexity and algebraic geometry, Graduate Texts in Mathematics, vol. 168, Springer-Verlag, New York, 1996. [FS97] W. Fulton and B. Sturmfels, Intersection theory on toric varieties, Topology 36 (1997), no. 2, 335–353. [FS05] E. Feichtner and B. Sturmfels, Matroid polytopes, nested sets and Bergman fans, Port. Math. (N.S.) 62 (2005), no. 4, 437–468. [Ful98] W. Fulton, Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 2, Springer-Verlag, Berlin, 1998. [HKT09] P. Hacking, S. Keel and J. Tevelev, Stable pair, tropical, and log canonical compact moduli of Del Pezzo surfaces, Invent. Math. 178 (2009), no. 1, 173–228. [Iwa09] I. Iwanari, The category of toric stacks, Compos. Math. 145 (2009), no. 3, 718–746. [Kat09] E. Katz, A tropical toolkit, Expo. Math. 27 (2009), no. 1, 1–36. [Kat10] E. Katz, Tropical realization spaces and tropical approximations, preprint, arXiv: 1008.1836v1, 2010. [Knu71] D. Knutson, Algebraic spaces, Lecture Notes in Mathematics, vol. 203, SpringerVerlag, Berlin, 1971. [Kol96] J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 32, SpringerVerlag, Berlin, 1996. [Laf03] L. Lafforgue, Chirurgie des grassmanniennes, CRM Monograph Series, vol. 19, American Mathematical Society, Providence, RI, 2003. [Mik08] G. Mikhalkin, What are tropical counterparts of algebraic varieties?, Oberwolfach Reports 5 (2008), no. 2, 1460–1462. [Mnë85] N. Mnëv, Varieties of combinatorial types of projective configurations and convex polyhedra, Dokl. Akad. Nauk SSSR 283 (1985), no. 6, 1312–1314. [Oxl92] J. Oxley, Matroid theory, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 1992.

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E. Katz and S. Payne S. Payne, Analytification is the limit of all tropicalizations, Math. Res. Lett. 16 (2009), no. 3, 543–556. B. Poonen, Undecidability in number theory, Notices Amer. Math. Soc. 55 (2008), no. 3, 344–350. D. Speyer, Uniformizing tropical curves I: genus zero and one, preprint, arXiv: 0711.2677v1, 2007. B. Sturmfels and J. Tevelev, Elimination theory for tropical varieties, Math. Res. Lett. 15 (2008), no. 3, 543–562. B. Sturmfels, On the decidability of Diophantine problems in combinatorial geometry, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 121–124. J. Tevelev, Compactifications of subvarieties of tori, Amer. J. Math. 129 (2007), no. 4, 1087–1104. R. Vakil, Murphy’s law in algebraic geometry: Badly-behaved deformation spaces, Invent. Math. 164 (2006), no. 3, 569–590. A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670.

A Relation Between Symmetric Polynomials and the Algebra of Classes, Motivated by Equivariant Schubert Calculus Dan Laksov

Abstract In previous work we developed a general formalism for equivariant Schubert calculus of grassmannians consisting of a basis theorem, a Pieri formula and a Giambelli formula. Part of the work consists in interpreting the results in a ring that can be considered as the formal generalized analog of localized equivariant cohomology of infinite grassmannians. Here we present an extract of the theory containing the essential features of this ring. In particular we emphasize the importance of the GKM condition. Our formalism and methods are influenced by the combinatorial formalism given by A. Knutson and T. Tao for equivariant cohomology of grassmannians, and of the use of factorial Schur polynomials in the work of L.C. Mihalcea.

1 Introduction In previous work [L1, L2] we developed a general formalism for equivariant Schubert calculus of grassmannians consisting of a basis theorem, a Pieri formula and a Giambelli formula. Part of the work consists in interpreting the results in a ring that can be considered as the formal generalized analog of localized equivariant cohomology of infinite grassmannians. More precisely the ring can be obtained as the graded limit of the equivariant cohomologies of the grassmannians of d-planes in n space, as n goes to infinity. Here we present an extract of the theory containing the essential features of this ring. In particular we emphasize the importance of the conditions made explicit by M. Goresky, R. Kottwitz, and R. MacPherson and called GKM conditions (see [GKM]). Our formalism and methods are influenced by the combinatorial formalism given by A. Knutson and T. Tau for equivariant cohomology of grassmannians and of the

Dan Laksov Department of Mathematics, KTH, S-100 44, Stockholm, Sweden e-mail: [email protected] G. Fløystad et al. (eds.), Combinatorial Aspects of Commutative Algebra and Algebraic Geometry, Abel Symposia 6, DOI 10.1007/978-3-642-19492-4 7, © Springer-Verlag Berlin Heidelberg 2011

89

90

D. Laksov

use of factorial Schur polynomials in the work of L.C. Mihalcea on the equivariant quantum cohomology ring of grassmannians [M1] and [M2]. We like to thank W. Fulton for valuable comments.

2 Partitions We first give the terminology of partitions used in the following. Notation 1. Let P(d) be the set of all partitions λ : λ1 ≥ · · · ≥ λd ≥ 0. We consider P(d) as a lattice with inequality λ ≥ λ if λi ≥ λi for i = 1, . . . , d. An inversion on a strict partition μ : μ1 > · · · > μd > 0 is a pair (i, j = μ p ) with i < j and i ∈ / {μ p+1 , . . . , μd }. The set of inversions on μ we denote by inv(μ ). The length of μ is the number of inversions (μ ) = ∑di=1 μi − d + i − 1. For every partition λ ∈ P(d) we write μ (λ ) : λ1 + d − 1 + 1 > · · · > λd + d − d + 1 > 0, that is, μ (λ )i = λi + d − i + 1 for i = 1, . . . , d. We call μ (λ ) the strict partition corresponding to λ . Note that d

(μ (λ )) = ∑ λi . i=1

3 Factorial Schur Polynomials In this section we define factorial Schur polynomials and give their main properties. Notation 2. Let A be a commutative ring with unit and let y1 , y2 , . . . be elements of A. We denote by A[T ] and A[T1 , . . . , Td ] the ring of polynomials in the variable T , respectively in the independent variables T1 , . . . , Td , over A. The A-algebra of symmetric polynomials we denote by A[T1 , . . . , Tn ]sym . We call the polynomials (T |y)i = (T − y1 ) · · · (T − yi ) in A[T ] factorial powers. For all collections of polynomials f 1 , . . . , fd in A[T ] we let f ( f i (T j )) =

1 (T1 )

.. .

... f1 (Td )

..

.

.. .

.

f d (T1 ) ... fd (Td )

We denote the Vandermonde determinant by V (T1 , . . . , Td ) = det (T jd−i ) = det ((T j |y)d−i ) =

∏

1≤i< j≤d

(Ti − T j ).

A Relation Between Symmetric Polynomials and the Algebra of Classes

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Let λ ∈ P(d) be a partition, and let μ = μ (λ ) be the corresponding strict partition. We write det ((T j |y)μi −1 ) sλ (T1 , . . . , Td |y) = det ((T j |y)d−i ) and we call sλ (T1 , . . . , Td |y) a factorial Schur polynomial. It is clear that the polynomial sλ (T1 , . . . , Td |y) is symmetric in T1 , . . . , Td and that, for all positive integers i1 , . . . , id , we have that sλ (yi1 , . . . , yid |y) is a homogeneous expression in y1 , y2 , . . . of degree (μ ) = ∑di=1 μi − d + i − 1 = ∑di=1 λi with coefficients in Z. The following result is fundamental (see also [MS] and [L2]). Theorem 3.1. Let λ and λ be partitions in P(d) and let μ = μ (λ ) and μ = μ (λ ) be the corresponding strict partitions. Assume that λ is not greater than or equal to λ . Then (1) sλ (yμ1 , . . . , yμd |y) = ∏(i, j)∈inv(μ ) (y j − yi ). (2) sλ (yμ , . . . , yμ |y) = 0. 1 d

Proof. We can assume that A = Z[y1 , y2 , . . . ] where y1 , y2 , . . . are independent variables over Z, because we obtain the general situation by specializing y1 , y2 , . . . over Z to arbitrary elements in A. Let M = ((yμ j |y)μi −1 ) and M = ((yμ j |y)μi −1 ). (1) If i < j we have (yμ j |y)μi −1 = 0, and if i = j we have (yμ j |y)μi −1 =

μi −1

∏ (yμi − yk ).

k=1

Consequently, M is lower triangular and the product of the elements on the diagoμi −1 (yμi − yk ). We have V (yμ1 , . . . , yμd )sλ (yμ1 , . . . , yμd |y) = det (M) = nal is ∏di=1 ∏k=1 μ −1 d i ∏i=1 ∏k=1 (yμi − yk ). Since V (yμ1 , . . . , yμd ) = ∏1≤i< j≤d (yμi − yμ j ) and this is not a zero divisor it follows that sλ (yμ1 , . . . , yμd |y) = ∏(i, j)∈inv(μ ) (y j − yi ) as we wanted to prove. (2) By assumption there is an index p such that λ p < λ p , that is, μ p < μ p . For i ≤ p ≤ j we obtain μ j ≤ μ p < μ p ≤ μi , and thus (yμ j |y)μi −1 = 0. Consequently the (p× (d − p+1))-matrix in the upper right corner of M is zero, and thus det (M ) = 0. We obtain that V (yμ , . . . , yμ )sλ (yμ , . . . , yμ |y) = det (M ) = 0. Since V (yμ , . . . , yμ ) = 1 1 1 d d d ∏1≤i< j≤d (yμi − yμ j ) is not a zero divisor we obtain that sλ (yμ1 , . . . , yμd |y) = 0, as we wanted to prove. Notation 3. Denote by ∏λ ∈P(d) A the collection of all lists of elements of A indexed by P(d). The λ -coordinate of a list α we denote by α |λ . A list α satisfies the GKM (Goresky–Kottwitz–MacPherson) condition with respect to y1 , y2 , . . . if, for all pairs of partitions λ , λ in P(d) such that the corresponding strict partitions μ = μ (λ )

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and μ = μ (λ ) differ only by i = μ p = μq = j for exactly one pair p and q, we have that yi − y j divides α |λ − α |λ . We say that a polynomial f in A[T1 , . . . , Td ] satisfies the GKM condition when the list α f defined by α f |λ = f (yμ1 , . . . , yμd ) with μ = μ (λ ) does so. Lemma 3.2. Let α in ∏λ ∈P(d) A be a list that satisfies the GKM condition, and let λ ∈ P(d) be minimal with the property that α |λ = 0. Then yi − y j divides α |λ for all pairs (i, j) in inv(μ (λ )). Proof. Let μ = μ (λ ) : μ1 > · · · > μd > 0 be the strict partition associated with λ . If (i, j = μ p ) is an inversion in μ we define q by μq < i < μq−1 . Then μq < i < j = μ p , and hence q > p and μ p+1 ≥ μq > μq−1 . We let

μ : μ1 > · · · > μ p−1 > μ p+1 > · · · > μq−1 > i > μq > · · · > μd > 0 when q − 1 > p and

μ : μ1 > · · · > μ p−1 > i > μq > · · · > μd > 0 = i = j = μ p when q − 1 > p, when q − 1 = p. Thus μ differs from μ only for μq−1 and for μ p = i = j = μ p when q − 1 = p. In both cases μ > μ . Since α satisfies the GKM condition we have that yi − y j divides α |λ − α |λ where μ = μ (λ ). However, it follows from the minimality of μ and from the inequality μ > μ that α |λ = 0. Thus yi − y j divides α |λ as asserted.

Proposition 3.3. The symmetric polynomials in A[T1 , . . . , Td ] satisfy the GKM condition. In particular, if f is a symmetric polynomial and μ : μ1 > · · · > μd > 0 is a strict partition that is minimal with the property that f (yμ1 , . . . , yμd ) = 0, then yi − y j divides f (yμ1 , . . . , yμd ) for all pairs (i, j) in inv(μ ). Proof. Let μ : μ1 > · · · > μd > 0 and μ : μ1 > · · · > μd > 0 be strict partitions that differ only in that i = μ p = μq = j. If f is symmetric we have that f (yμ1 , . . . , yμd ) − f (yμ , . . . , yμ ) 1

d

= f (yi , yμ1 , . . . , yμ p−1 , yμ p+1 , . . . , yμd ) − f (y j , yμ , . . . , yμ , yμ , . . . , yμ ). 1

q−1

q+1

d

Since the partitions

μ1 > · · · > μ p−1 > μ p+1 > · · · > μd > 0 and

μ1 > · · · > μq−1 > μq+1 > · · · > μd > 0

are equal the first part of the proposition follows. The second part we obtain from Lemma 3.2.


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Proposition 3.4. Let f in A[T1 , . . . , Td ] be a polynomial that can be written in the form f (T1 , . . . , Td ) = ∑ aλ sλ (T1 , . . . , Td |y) λ ∈I

with aλ in A for a finite subset I of P(d). Assume that for all μ = μ (λ ) with λ ∈ I we have (1) f (yμ1 , . . . , yμd ) = 0. (2) ∏(i, j)∈inv(μ ) (y j − yi ) is not a zero divisor in A, Then aλ = 0 for all λ ∈ I . Proof. Assume that the proposition does not hold and let λ be minimal in I with the property that aλ = 0. Let μ = μ (λ ) be the corresponding strict partition. For all λ = λ such that aλ = 0 we have that λ is not greater than or equal to λ . From Theorem 3.1 with λ and λ exchanged it follows that sλ (yμ1 , . . . , yμd |y) = 0. Consequently we have that f (yμ1 , . . . , yμd ) = aλ sλ (yμ1 , . . . , yμd |y). It follows from Theorem 3.1 that we then have f (yμ1 , . . . , yμd ) = aλ ∏(i, j)∈inv(μ ) (y j − yi ). The assumptions (1) and (2) thus contradict that aλ = 0 and we have proved the proposition. Remark 3.5. We have chosen the assumption on f in Proposition 3.4 to emphasize that we do not need to know that the factorial Schur polynomials generate the Amodule of symmetric polynomials, on the contrary, this follows from our results.

4 Factorial Schur Polynomials over Polynomial Rings We show that the properties of Schur polynomials of Sect. 2 characterize Schur polynomials among symmetric polynomials when the base ring is a polynomial ring over the integers in countably many variables. Proposition 4.1. Let A = Z[y1 , y2 , . . . ] where y1 , y2 , . . . are independent variables over Z. Then assertions (1) and (2) in Theorem 3.1 characterize Schur polynomials among symmetric polynomials that are homogeneous in y1 , y2 , . . . . More precisely: Let λ be in P(d) and let f be a symmetric polynomial in A[T1 , . . . , Td ]. Then f (T1 , . . . , Td ) = sλ (T1 , . . . , Td |y) if the following three conditions hold: (1) f (yμ1 , . . . yμd ) = ∏(i, j)∈inv(μ ) (y j − yi ) where μ = μ (λ ). (2) f (yμ , . . . , yμ ) = 0 when μ = μ (λ ) and λ is not greater than or equal to λ . 1

d

(3) f (yμ , . . . , yμ ) is homogeneous of degree ∑di=1 λi in y1 , y2 , . . . for all strict par1 d titions μ : μ1 > · · · > μd > 0. Proof. Assume that f satisfies the three conditions of the proposition, but that the proposition does not hold. Let λ ∈ P(d) be minimal such that f (yμ , . . . , yμ ) = sλ (yμ , . . . , yμ |y) 1

d

1

d

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with μ = μ (λ ). Then λ ≥ λ because otherwise we have f (yμ , . . . , yμ ) = 0 by 1 d assumption (2) and sλ (yμ , . . . , yμ |y) = 0 by Theorem 3.1 (2). Moreover we must 1 d have λ > λ since f (yμ1 , . . . , yμd ) =

∏

(y j − yi ) = sλ (yμ1 , . . . , yμd |y)

(i, j)∈inv(μ )

by assumption (1) and Theorem 3.1 (1). Since the strict partition μ is minimal with the property that the polynomial g = f (yμ , . . . , yμ ) − sλ (yμ , . . . , yμ |y) is non-zero, it follows from Proposition 3.3, 1 1 d d and from the assumption that A = Z[y1 , y2 , . . . ] that ∏(i, j)∈inv(μ ) (y j − yi ) divides g. However g is homogeneous in y1 , y2 , . . . of degree (μ ) = ∑di=1 λi by assumption (3). Since the polynomial ∏(i, j)∈inv(μ ) (y j − yi ) is of degree d

( μ ) = ∑ λi > (μ ) i=1

it can not divide the non-zero polynomial g and we obtain a contradiction to the assumption that the proposition does not hold. We have thus proved the proposition. Remark 4.2. A similar reasoning to that in the proof of Proposition 4.1 gives a slightly stronger version of Proposition 4.1 that we used in [L1]. Let P(d)k be the partitions λ ∈ P(d) satisfying λ1 ≤ k, where k is a natural number or ∞. The conclusion of Proposition 4.1 then holds for λ ∈ P(d)k when the conditions (1), (2), and (3) hold with λ in P(d)k .

5 Factorial Schur Polynomials and Schubert Classes Here we introduce the ring alluded to in the introduction and give a natural isomorphism between this ring and the ring of symmetric polynomials. In this section we let A = Z[y1 , y2 , . . . ] where y1 , y2 , . . . are independent variables over Z. Notation 4. It is clear that the collection of lists ∏λ ∈P(d) A is a ring with componentwise addition and multiplication and with unit 1 where 1|λ = 1A for all λ ∈ P(d). We have that ∏λ ∈P(d) A is an A-algebra via the homomorphism that maps a ∈ A to the list αa with αa |λ = a for all λ ∈ P(d). Let σ : A[T1 , . . . , Td ]sym → ∏ A λ ∈P(d)

be the A-algebra homomorphism defined by σ ( f (T1 , . . . , Td )) = α f where

α f |λ = f (yμ1 , . . . , yμd )


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for all λ with μ = μ (λ ). We write Sλ = σ (sλ (T1 , . . . , Td |y)). Definition 5.1. A list α in ∏λ ∈P(d) A is a class if the following two conditions are satisfied: (1) α satisfies the GKM condition. (2) There is an integer N(α ) such that total degree of α |λ in the variables y1 , y2 , . . . is at most N(α ) for all λ ∈ P(d). The classes in ∏λ ∈P(d) A form an A-subalgebra denoted by H(d). A class α ∈ H(d) is called a Schubert class corresponding to λ ∈ P(d) when the following three conditions are fulfilled: (1) α |λ = ∏(i, j)∈inv(μ ) (y j − yi ) with μ = μ (λ ). (2) α |λ = 0 when λ ∈ P(d) is not greater than or equal to λ . (3) α |λ is homogeneous of degree λ1 + · · · + λd in the variables y1 , y2 , . . . for all λ ∈ P(d). Lemma 5.2. For a list α in ∏λ ∈P(d) A we write supp(α ) = {λ : α |λ = 0} and m(α ) = {λ : λ is not greater than or equal to an element in supp(α )}. Let λ be minimal in supp(α ) and write β = σ (sλ (T1 , . . . , Td |y)). Moreover, let a be an element in A and write α = α − aβ . When α |λ = 0 we have m(α ) ∪ {λ } ⊆ m(α ). / m(α ). Then there is a λ ∈ supp(α ) Proof. Let λ ∈ m(α ) and assume that λ ∈ such that λ ≤ λ . In particular λ ∈ m(α ) and thus α |λ = 0. We can not have λ ≥ λ because then λ ≥ λ and thus λ ∈ m(α ) contradicting the assumption λ ∈ supp(α ). Thus it follows from Theorem 3.1 (2) that β |λ = 0. Consequently α |λ = α |λ − aβ |λ = 0, that contradicts that λ ∈ supp(α ). Hence we have proved that λ ∈ m(α ). Assume next that λ ∈ / m(α ). Then there is λ ∈ supp(α ) such that λ ≤ λ . / supp(α ) by assumption we have λ = λ . It thus follows from the miniSince λ ∈ mality of λ that α |λ = 0. Moreover, we obtain from Theorem 3.1 (2) that β |λ = 0. Consequently α |λ = α |λ − aβ |λ = 0 that contradicts that λ ∈ supp(α ). We have thus proved that λ ∈ m(α ). Theorem 5.3. Let A = Z[y1 , y2 , . . . ] where y1 , y2 , . . . are independent variables over Z.

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(1) The homomorphism σ induces an isomorphism of A-algebras

τ : A[T1 , . . . , Td ]sym → H(d) given by τ ( f (T1 , . . . , Td )) = α f where α f |λ = f (yμ1 , . . . , yμd ) for all λ ∈ P(d) where μ = μ (λ ). (2) Sλ = τ (sλ (T1 , . . . , Td |y)) is the unique Schubert class that corresponds to λ . (3) The elements Sλ for λ ∈ P(d) form a basis for the A-module H(d). Proof. (1) Clearly the set of elements α f |λ = f (yμ1 , . . . , yμd ) for all λ ∈ P(d) with μ = μ (λ ) have bounded total degree in y1 , y2 , . . . . When f is symmetric in the variables T1 , . . . , Td it follows from Proposition 3.3 that α f satisfies the GKM condition. Hence σ induces an A-algebra homomorphism

τ : A[T1 , . . . , Td ]sym → H(d). This homomorphism is injective because, if f = 0, the polynomial f (yμ1 , . . . , yμd ) in the variables y1 , y2 , . . . is non-zero when the variables yμ1 , . . . , yμd do not appear in the coefficients of f (T1 , . . . , Td ). We now prove that τ is surjective. Let α be a non-zero class and N an integer such that the total degree of α |λ is at most N for all λ ∈ P(d), and let λ be minimal in supp(α ). It follows from Theorem 3.1 (1) that

∏

(i, j)∈inv(μ )

(y j − yi ) = sλ (yμ , . . . , yμ |y) 1

d

with μ = μ (λ ), and since α satisfies the GKM condition it follows from Lemma 3.2 that sλ (yμ , . . . , yμ |y) divides α |λ . In particular (μ ) ≤ N. 1 d Let α |λ = asλ (yμ , . . . , yμ |y) with a ∈ A and let α = α − aτ (sλ (T1 , . . . , Td |y)). 1 d If α = 0 we have proved that α is in the image of τ . If α = 0 we have that the total degree of α |λ is at most N for all λ ∈ P(d) and it follows from Lemma 5.2 that m(α ) ∪ {λ } ⊆ m(α ), Thus when α = 0 we can repeat the procedure. We continue the process and see that we, by subtracting from elements β appearing in the process an element of the form bτ (sλ (T1 , . . . , Td |y)) with b ∈ A and ( μ (λ )) ≤ N, we can successively expand the set m(β ) by taking away minimal elements of supp(β ). Since there is only a finite number of partitions λ in P(d) with (μ (λ )) ≤ N the process must ultimately stop by giving zero. We have thus proved that H(d) is generated as an A-module by the elements Sλ for λ ∈ P(d). In particular, τ is surjective. (2) It follows from Theorem 3.1 and from the definition of the homomorphism τ that Sλ = τ (sλ (T1 , . . . , Td |y)) is a Schubert class corresponding to λ . That it is unique follows from the injectivity of τ . (3) It remains to prove that the elements Sλ for λ ∈ P(d) are linearly independent over A. This follows from Proposition 3.4. Remark 5.4. The injectivity of τ in Theorem 5.3 can also be deduced from Proposition 3.4, and the well known result that the polynomials sλ (T1 , . . . , Td |y) for all


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λ ∈ P(d) generate the A-module of symmetric polynomials A[T1 , . . . , Td ]sym ([M], Chap. 1, Sect. 5, Example 20, p.54, or [L1] Proposition 1.5). One reason for using such a proof may be that it also works in the case when the index set is the finite set P(d)k of Remark 4.2.

References [GKM] M. Goresky, R. Kottwitz, & R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), 25–83. [KT] A. Knutson & T. Tao, Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J. 119 (2003), 221–260. [L1] D. Laksov, Schubert calculus and equivariant cohomology of Grassmannians, Advances in Math. 217 (2008), 1869–1888. [L2] D. Laksov, A formalism for equivariant Schubert calculus, Algebra Number Theory 3 (2009), 711–727. [M] I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs. Oxford Science Publications. Second edition. With contributions by A. Zelevinsky, The Clarendon Press, Oxford University Press, Springer, Oxford (1995). [M1] L.C. Mihalcea, Equivariant quantum Schubert calculus, Adv. Math. 203 (2006), 1–33. [M2] L.C. Mihalcea, Giambelli Formulae for the equivariant quantum cohomology of the Grassmannian, Trans. Amer. Math. Soc. 360 (2008), 2285–2301. [MS] A. Molev & B.E. Sagan, A Littlewood–Richardson rule for factorial Schur functions, Trans. Amer. Math. 351 (1999), 4429–4443.

Theory and Applications of Lattice Point Methods for Binomial Ideals Ezra Miller

Abstract This survey of methods surrounding lattice point methods for binomial ideals begins with a leisurely treatment of the geometric combinatorics of binomial primary decomposition. It then proceeds to three independent applications whose motivations come from outside of commutative algebra: hypergeometric systems, combinatorial game theory, and chemical dynamics. The exposition is aimed at students and researchers in algebra; it includes many examples, open problems, and elementary introductions to the motivations and background from outside of algebra. Keywords binomial ideal, primary decomposition, polynomial ring, affine semigroup, commutative monoid, lattice point, convex polyhedron, monomial ideal, combinatorial game, lattice game, rational strategy, misère quotient, Horn hypergeometric system, mass-action kinetics

1 Introduction Binomial ideals in polynomial rings over algebraically closed fields admit binomial primary decompositions: expressions as intersections of primary binomial ideals. The algebra of these decompositions is governed by the geometry of lattice points in polyhedra and related lattice-point combinatorics arising from congruences on commutative monoids. The treatment of this geometric combinatorics is terse at the source [DMM10]. Therefore, a primary goal of this exposition is to provide a more leisurely tour through the relevant phenomena; this is the concern of Sects. 2, 3, and 4. That the geometry of congruences should govern binomial primary decomposition was a realization made in the context of classical multivariate hypergeometric Ezra Miller Mathematics Department, Duke University, Durham, NC 27708, USA e-mail: [email protected] G. Fløystad et al. (eds.), Combinatorial Aspects of Commutative Algebra and Algebraic Geometry, Abel Symposia 6, DOI 10.1007/978-3-642-19492-4 8, © Springer-Verlag Berlin Heidelberg 2011

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series, going back to Horn, treated in Sect. 5. Lattice-point combinatorics related to monoids and congruences has recently been shown relevant to the theory of combinatorial games, and is sure to play a key role in algorithms for computing rational strategies and misère quotients, as discussed in Sect. 6. Finally, binomial commutative algebra is central to a long-standing conjecture on the dynamics of chemical reactions under mass-action kinetics. The specifics of this connection are briefly outlined in Sect. 7, along with the potential relevance of combinatorial methods for binomial primary decomposition. Limitations of time and space prevented the inclusion of algebraic statistics in this survey; for an exposition of binomial aspects of Markov bases and conditional independence models, such as graphical models, as well as applications to phylogenetics, see [DSS09] and the references therein. Sections 2, 3, and 4 are complete in the sense that statements are made in full generality, and precise references are provided for the details of any argument that is only sketched. In contrast, Sects. 5, 6, and 7 are more expository. The results there are sometimes stated in less than full generality—but still mathematically precisely—to ease the exposition. In addition, Sects. 5, 6, and 7 are independent of one another, and to a large extent independent of Sects. 2, 3, and 4, as well; readers interested in the applications should proceed to the relevant sections and refer back as necessary. Acknowledgements. I am profoundly grateful to the organizers and participants of the International School on Combinatorics at Sevilla, Spain in January 2010, where these notes were presented as five lectures. That course was based on my Abel Symposium talk at Voss, Norway in June 2009; I am similarly indebted to that meeting’s organizers. Thanks also go to my coauthors, from whom I learned so much while working on various projects mentioned in this survey. Funding was provided by NSF CAREER grant DMS-0449102 = DMS-1014112 and NSF grant DMS-1001437.

Theory 2 Affine Semigroups and Prime Binomial Ideals 2.1 Affine Semigroups Let Zd ⊂ Rd denote the integer points in a real vector space of dimension d. Any integer point configuration ⎡ ⎤ | | A = {a1 , . . . , an } ⊂ Zd ←→ A = ⎣ a1 · · · an ⎦ ∈ Zd×n | | can be identified with a d × n integer matrix.

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Definition 2.1. A monoid is a set with an associative binary operation and an identity element. An affine semigroup is a monoid that is isomorphic to NA = N{a1 , . . . , an } = {c1 a1 + · · · + cn an | c1 , . . . , cn ∈ N} for some lattice point configuration A ⊂ Zd . Thus a monoid is a group without inverses. Although a semigroup is generally not required to have an identity element, standard terminology from the literature dictates that an affine semigroup is a monoid, and in particular (isomorphic to) a finitely generated submonoid of an integer lattice Zd for some d. Example 2.2. The configuration in the plane Z2 drawn as solid dots in the following diagram generates the affine semigroup comprising all lattice points in the real cone bounded by the thick horizontal ray and the diagonal ray.

This example will henceforth be referred to as “ 0123 ”. Example 2.3. A point configuration is allowed to have repeated elements, such as

in Z2 , which generates the affine semigroup NA = N2 of all lattice points in the nonnegative quadrant. This example will henceforth be referred to as “ 1100 0111 ”.

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Example 2.4. It will be helpful, later on, to have a three-dimensional example ready. Consider a square at height 1 parallel to the horizontal plane. It can be represented as a matrix and point configuration in Z3 as follows:

The affine semigroup NA comprises all of the lattice points in the real cone generated by the vertices of the square. In all of these examples, the affine semigroups are normal: each one equals the set of all lattice points from a rational polyhedral cone. General affine semigroups need not be normal, though they always comprise “most” of the lattice points in a cone. Example 2.5. In Example 2.2, the outer columns of the matrix, namely 10 and 13 , correspond to the extremal rays; they therefore generate the rational polyhedral cone whose lattice points constitute the 0123 affine semigroup. Consequently, the config uration 10 12 13 generates the same rational polyhedral cone, but the affine semigroup it generates is different—and not normal—because the point 11 does not lie in it, even though the configuration still generates Z2 as a group. The geometry of binomial primary decomposition is based on the sort of geometry that arises from the projection determined by A. To be more precise, A determines a monoid morphism Zd ← Nn . This morphism can be expressed as the restriction of the group homomorphism (the linear map) Zd ← Zn induced by A. The diagram is as follows:


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The kernel L of the homomorphism Zd ← Zn induced by A is a saturated lattice in Zn , meaning that Zn /L is torsion-free, or equivalently that L = RL ∩ Zn , where RL = R ⊗Z L is the real subspace of Rn generated by L. It makes little sense to say that the monoid morphism Zd ← Nn has a kernel: it is often the case that L ∩ Nn = {0}, even when the monoid morphism is far from injective. However, when Zd ← Nn fails to be injective, the fibers admit clean geometric descriptions, inherited from the fact that the fibers of the vector space map Rd ← Rn are the cosets of RL in Rn . Definition 2.6. A polyhedron in a real vector space is an intersection of finitely many closed real half-spaces. This survey assumes basic knowledge of polyhedra. Readers for whom Definition 2.6 is not familiar are urged to consult [Zie95, Chaps. 0, 1, and 2]. A

Lemma 2.7. The fiber of the monoid morphism Zd ← Nn over a given lattice point α ∈ Zd is the set Fα = Nn ∩ (u + L) = Nn ∩ Pα of lattice points in the polyhedron Pα = (u + RL) ∩ Rn≥0 for any vector u ∈ Zd satisfying Au = α , where L = ker A. This description is made particularly satisfying by the fact that the polyhedra for various α ∈ Zd are all related to one another. Example 2.8. When A = [1 1 1] is the “coordinate-sum” map N ← N3 , the polyhedra Pα for α ∈ N are equilateral triangles, the lattice points in them corresponding to the monomials of total degree α in three variables:

The polyhedra in Example 2.8 are all scalar multiples of one another, but this phenomenon is special to codimension 1. In general, when n − d > 1, the polyhedra Pα in the family indexed by α ∈ NA have facet normals chosen from the same fixed set of possibilities—namely, the images in L∗ of the dual basis vectors of (Zn )∗ —so their shapes feel roughly similar, but faces can shrink or disappear.

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Example 2.9. In the case of 0123, a basis for the kernel ker A can be chosen so that the inclusion Zn ← ker A is given by the matrix

8 7 and β = 12 so that, for example, the depicted polytopes Pα and Pβ for α = 12 both have outer normal vectors that are the negatives of the rows of B. Moving to 6 would shrink the bottom edge entirely. The corresponding fibers Fα , Pγ for γ = 12 Fβ , and Fγ comprise the lattice points in these polytopes.

2.2 Affine Semigroup Rings Definition 2.10. The affine semigroup ring of NA over a field k is

k[NA] =

k · tα ,

α ∈NA

a subring of the Laurent polynomial ring k[Zd ] = k[t1±1 , . . . ,td±1 ], in which

α

tα = t1α1 · · ·td d

and

tα + β = tα t β .

The definition could be made with k an arbitrary commutative ring, but in fact the case we care about most is k = C, the field of complex numbers. The reason is that the characteristic zero and algebraically closed hypotheses enter at key points; these notes intend to be precise about which hypotheses are needed where. Definition 2.11. Denote by πA the surjection πA

k[NA] k[x] t ai ← xi onto the affine semigroup ring k[NA] from the polynomial ring k[x] := k[x1 . . . , xn ]. The next goal is to calculate the kernel IA = ker(πA ). To do this it helps to note that both k[NA] and the polynomial ring are graded, in the appropriate sense.


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Definition 2.12. Let A ∈ Zd×n , so ZA ⊆ Zd is a subgroup. A ring R is A-graded if R is a direct sum of homogeneous components R=

α ∈ZA

Rα ,

such that Rα Rβ ⊆ Rα +β .

An ideal in an A-graded ring is A-graded if it is generated by homogeneous elements. Example 2.13. The affine semigroup ring k[NA] is A-graded, with k[NA]α = k · tα if α ∈ NA, and k[NA]α = 0 otherwise. The polynomial ring k[x] is also A-graded, with k[x]α = k · {Fα }, the vector space spanned by the fiber Fα from Lemma 2.7. Proposition 2.14. The kernel of the surjection πA from Definition 2.11 is IA = xu − xv | u, v ∈ Nn and Au = Av = xu − xv | u, v ∈ Nn and u − v ∈ ker A . Proof. The “⊇” containment follows simply because πA (xu ) = xAu . The reverse containment uses the A-grading: in the ring R = k[x]/xu −xv | Au = Av , the dimension of the image of k[x]α as a vector space over k is either 0 or 1 since Au = Av = α if u and v lie in the same fiber Fα . On the other hand, R k[NA]α maps surjectively onto the affine semigroup ring by the “⊇” containment already proved. The surjection must be an isomorphism because, as we noted in Example 2.13, k[NA]α has dimension 1 whenever Fα is nonempty. Corollary 2.15. The toric ideal IA is prime. Proof. k[NA] is an integral domain, being contained in k[Zd ].

Example 2.16. In the 0123 case, with variables {a, b, c, d} instead of {x1 , x2 , x3 , x4 }, ⎡ ⎤ 1 0

⎢ −2 1 ⎥ 1111 2 2 ⎥ A= and B = ⎢ ⎣ 1 −2 ⎦ =⇒ IA = ac − b , bd − c , ad − bc , 0123 0 1 where ker A is the image of B in Z4 . The presence of binomials ac − b2 and bd − c2 in IA translates the statement “the columns of the matrix B lie in the kernel of A”. However, note that IA is not generated by these two binomials; it is a complicated problem, in general, to determine a minimal generating set for IA . Example 2.17. In the 1100 0111 case, using variables {a, b, c, d} instead of {x1 , x2 , x3 , x4 }, ⎡ ⎤ 1 1

⎢ −1 −1 ⎥ 1100 ⎥ A= and B = ⎢ ⎣ 1 0 ⎦ =⇒ IA = ac − b, c − d , 0111 0 1

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where again ker A is the image of B in Z4 . In this case, IA is indeed generated by two binomials corresponding to a basis for the kernel of A, but not the given basis appearing as the columns of B. In Sect. 5, we shall be interested in the ideal generated by the two binomials corresponding to the columns of B. Example 2.18. In the square cone case, using {a, b, c, d} instead of {x1 , x2 , x3 , x4 }, ⎡ ⎤ ⎡ ⎤ 1 0101 ⎢ −1 ⎥ ⎥ A = ⎣ 0 0 1 1 ⎦ and B = ⎢ ⎣ −1 ⎦ =⇒ IA = ad − bc , 1111 1 where ker A is the image of B. In the codimension 1 case, when ker A has rank 1, the toric ideal is always principal, just as any codimension 1 prime ideal in k[x] is.

2.3 Prime Binomial Ideals At last it is time to define binomial ideals precisely. Definition 2.19. An ideal I ⊆ k[x] is a binomial ideal if it is generated by binomials xu − λ xv

with u, v ∈ Nn and λ ∈ k.

Note that λ = 0 is allowed: monomials are viable generators of binomial ideals. This may seem counterintuitive, but it is forced by allowing arbitrary nonzero constants λ , and in any case, even ideals generated by differences of monomials (“pure-difference binomials”) have associated primes containing monomials. Example 2.20. The ideal x3 − y2 , x3 − 2y2 ⊆ k[x, y] is generated by binomials but equals the monomial ideal x3 , y2 , no matter the characteristic of k. Worse, the ideal I = x2 − xy, xy − 2y2 ⊆ k[x, y] is generated by “honest” binomials that are not linear combinations of monomials in I, and yet I contains monomials, because I contains both of x2 y − xy2 and x2 y − 2xy2 , so the monomials x2 y and xy2 lie in I. Example 2.21. The pure-difference binomial ideal I = x2 − xy, xy − y2 ⊆ k[x, y] has a monomial associated prime ideal x, y , since I = x − y ∩ x2 , y . In general, which binomial ideals are prime? We have seen that toric ideals IA are prime, but for binomial primary decomposition in general it is important to know all of the other binomial primes, as well. The answer was given by Eisenbud and Sturmfels [ES96, Corollary 2.6]. Theorem 2.22. When k is algebraically closed, a binomial ideal I ⊆ k[x1 , . . . , xm ] is prime if and only if it is the kernel of a surjective A-graded homomorphism π : k[x] k[NA] in which the variables are homogeneous. The surjection π in Theorem 2.22 need not equal πA . For example, π (x j ) = 0 is allowed, so I could contain monomials. Furthermore, even when π (x j ) = 0, the


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image of x j could be λ tα for some λ = 1. Thus, if xu → λ tα and xv → μ tα , then xu − λμ xv ∈ I: the binomial generators of I need not be pure differences. However, assuming I is prime, we are not free to assign the coefficients λ and μ at will: xu − λμ xv ∈ I =⇒ xu+w − λμ xv+w ∈ I for w ∈ Nn and xru − λμ r xrv ∈ I for r ∈ N. r

The first line means that the coefficient on xv in xu − xv depends only on u − v, and the second means essentially that the assignment u − v → μ /λ constitutes a homomorphism ker A → k∗ . The precise statement requires a definition. Definition 2.23. A character on a sublattice L ⊆ Zn is a homomorphism ρ : L → k∗ . If L ⊆ ZJ for some subset J ⊆ {1, . . . , n}, then

where and

Iρ ,J = Iρ + mJ , Iρ = xu − ρ (u − v)xv | u − v ∈ L mJ = xi | i ∈ J .

Corollary 2.24. A binomial ideal I ⊆ k[x] with k algebraically closed is prime if and only if I = Iρ ,J for a character ρ : L → k∗ defined on a saturated lattice L ⊆ ZJ . In other words, every prime binomial ideal in the polynomial ring k[x] over an algebraically closed field k is toric after forgetting some of the variables (those outside of J) and rescaling the rest (by the character ρ ). Remark 2.25. Given any sublattice L ⊆ Zn , a character ρ is defined as a homomorphism L → k∗ . On the other hand, rescaling the variables x j for j ∈ J amounts to a homomorphism ZJ → k∗ . When L ZJ , there is usually no unique way to extend ρ to a character ZJ → k∗ (there can be a unique way if k has positive characteristic). However, there is always at least one way when L is saturated—so the inclusion L → Zn is split—because the natural map Hom(ZJ , k∗ ) → Hom(L, k∗ ) is surjective. √

Example 2.26. Let ω = 1+ 2 −3 ∈ C be a primitive cube root of 1. If L ⊆ Z4 ⊆ Z5 is spanned by the columns of the matrix B, below, and the character ρ takes the indicated values on these generators of L, then Iρ ,J for J = {1, 2, 3, 4} is as indicated. ⎡ ⎤ −1 −1 0 ⎢ 1 2 −1 ⎥ ⎢ ⎥ ⎥ B=⎢ 2 2 2 ⎢ 1 −1 2 ⎥ =⇒ I ρ ,{1,2,3,4} = bc − ω ad, b − ω ac, c − ω bd, e . ⎣ −1 0 −1 ⎦ 0 0 0

ρ : ω2 ω ω For instance, when u = (0, 1, 1, 0) and v = (1, 0, 0, 1), we get ρ (u − v) = ω 2 . Compare this example to the 0123 case in Example 2.16.

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3 Monomial Ideals and Primary Binomial Ideals The lattice-point geometry of binomial primary decomposition generalizes the geometry of monomial ideals. For primary binomial ideals, the connection is particularly clear. To highlight it, this section discusses what it looks like for a primary binomial ideal to have a monomial associated prime. The material in this section is developed in the context of an arbitrary affine semigroup ring, because that generality will be crucial in the applications to binomial ideals in polynomial rings k[x]. In return for the generality, there are no restrictive hypotheses on the characteristic or algebraic closure of k to contend with; except in Example 3.19 and Theorem 3.20, k can be arbitrary. Definition 3.1. An ideal I ⊆ k[Q] in the monoid algebra of an affine semigroup Q over an arbitrary field is a monomial ideal if it is generated by monomials tα , and I is a binomial ideal if it is generated by binomials tα − λ tβ with α , β ∈ Q and λ ∈ k.

3.1 Monomial Primary Ideals Given a monomial ideal, it is convenient to have terminology and notation for certain sets of monomials and lattice points. Definition 3.2. If I ⊆ k[Q] is a monomial ideal, then write std(I) for the set of exponent vectors on its standard monomials, meaning those outside of I. Example 3.3. Here is a monomial ideal in k[Q] for Q = N2 :

The reason for using z instead of y will become clear in Example 3.4. The bottom of the cross-hatched region is the staircase of I; its lower corners are the (lattice points corresponding to) the generators of I. The lattice points below the staircase correspond to the standard monomials of I. Example 3.4. The same monomial generators can result in a higher-dimensional picture if the ambient monoid is different. Consider the generators from Example 3.3 but in k[Q] for Q = N3 :


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The “front face” of the picture—corresponding to the xz-plane—coincides with Example 3.3. The surface cross-hatched in thick lines is the staircase of I; its minimal elements again correspond to the generators of I, drawn as solid dots. Below the staircase sit the lattice points corresponding to the standard monomials of I. There are infinitely many standard monomials, but they occur along finitely many rays parallel to the y-axis, each emanating from a point drawn as a bold hollow dot; these are the points in std(I) in the xz-plane. Example 3.5. In the square cone case, Example 2.4, let a, b, c, d be the generators of the affine semigroup ring, as indicated in the figure there. Thus k[Q] = k[a, b, c, d]/ad − bc . The monomial ideal c, d ⊆ k[Q] is prime; in fact, the composite map k[a, b] → k[Q] k[Q]/c, d is an isomorphism. The phenomenon in Example 3.5 is general; the statement requires a definition. Definition 3.6. A face F ⊆ Q of an affine semigroup Q ⊆ Zd is a subset F = Q ∩ H obtained by intersecting Q with a halfspace H ⊆ Rd such that Q ⊆ H + is contained in one of the two closed halfspaces H + , H − defined by H in Rd . Lemma 3.7. A monomial ideal I in an affine semigroup ring is prime ⇔ std(I) is a face. The prime pF for F ⊆ Q induces an isomorphism k[F] → k[Q] k[Q]/pF . For a proof of the lemma, and lots of additional background on the connections between faces of cones and the algebra of affine semigroup rings, see [MS05, §7.2]. Before jumping to the question of when an arbitrary binomial ideal is primary, let us first consider the monomial case. In a polynomial ring k[x], there is an elementary algebraic description as well as a satisfying geometric one. Both will be important in later sections, but it is the geometric description that generalizes most easily to arbitrary affine semigroup rings. Proposition 3.8. A monomial ideal I ⊆ k[x] is primary if and only if mr 1 I = xm i1 , . . . , xir , some other monomials in xi1 , . . . , xir .

A monomial ideal I ⊆ k[Q] for an arbitrary affine semigroup Q ⊆ Zd is pF -primary for a face F ⊆ Q if and only if there are elements α1 , . . . , α ∈ Q with

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std(I) =

(αk + ZF) ∩ Q,

k=1

where ZF is the subgroup of Zd generated by F. Proof. The statement about k[x] is a standard exercise in commutative algebra. The statement about k[Q] is the special case of Theorem 3.23, below, in which the binomial ideal I is generated by monomials. What does a set of the form (α + ZF) ∩ Q look like? Geometrically, it is roughly the (lattice points in the) intersection of an affine subspace with a cone. In the polynomial ring case, where Q = Nn , a set (α + ZF) ∩ Q is always β + F for some lattice point β ∈ Nn : the intersection of a translate of a coordinate subspace with the nonnegative orthant is a translated orthant. In fact, F = NJ = Nn ∩ ZJ for some subset J ⊆ {1, . . . , n}, and then β is obtained from α by setting all coordinates from J to 0 (if α has negative coordinates outside of J, then α + ZF fails to meet Nn ). For general Q, on the other hand, (α + ZF) ∩ Q need not be a translate of F. Example 3.9. The prime ideal in Example 3.5 corresponds to the face F consisting of the nonnegative integer combinations of e1 + e3 and e3 , where e1 , e2 , e3 are the standard basis of Z3 . In terms of the depiction in Example 2.4, these are the lattice points in Q that lie in the xz-plane. The subgroup ZF ⊆ Z3 comprises all lattice points in the xz-plane. Now suppose that α = e2 , the first lattice point along the y-axis. Then

is a union of two translates of F. For reference, the square over which Q is the cone is drawn lightly, while dotted lines fill out part of the vertical plane α + ZF.


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Proposition 3.10. Every monomial ideal I ⊆ k[Q] in an affine semigroup ring has a unique minimal primary decomposition I = P1 ∩ · · · ∩ Pr as an intersection of monomial primary ideals Pi with distinct associated primes. Proof sketch. I has a unique irredundant decomposition I = W1 ∩ · · · ∩Ws as an intersection of irreducible monomial ideals W j . The existence of an irredundant irreducible decomposition can be proved the same way irreducible decompositions are produced for arbitrary submodules of noetherian modules. The uniqueness of such a decomposition, on the other hand, is special to monomial ideals in affine semigroup rings [MS05, Corollary 11.5]; it follows from the uniqueness of irreducible resolutions [Mil02, Theorem 2.4]. See [MS05, Chap. 11] for details. Given the uniqueness properties of minimal monomial irreducible decompositions, the (unique) monomial primary components are obtained by intersecting all irreducible components sharing a given associated prime. Remark 3.11. In polynomial rings, uniqueness of monomial irreducible decomposition occurs for approximately the same reason that monomial ideals have unique minimal monomial generating sets: the partial order on irreducible ideals is particularly simple [Mil09, Proposition 1.4]. See [MS05, §5.2] for an elementary derivation of existence and uniqueness of monomial irreducible decomposition by Alexander duality.

3.2 Congruences on Monoids The uniqueness of irreducible and primary decomposition of monomial ideals rests, in large part, on the fine grading on k[Q], in which the nonzero components k[Q]α have dimension 1 as vector spaces over k. Similar gradings are available for quotients modulo binomial ideals, except that the gradings are by general noetherian commutative monoids, rather than by free abelian groups or by affine semigroups. Our source for commutative monoids is Gilmer’s excellent book [Gil84]. For the special case of affine semigroups, by which we mean finitely generated submonoids of free abelian groups, see [MS05, Chap. 7]. For motivation, recall from Lemma 2.7 that the fibers of a monoid morphism from Nn to Zd have nice structure, and that the polynomial ring k[x] becomes graded by Zd via such a morphism. The fibers are the equivalence classes in an equivalence relation, as is the case for any map π : Q → Q of sets; but when π is a morphism of monoids, the equivalence relation satisfies an extra condition. Definition 3.12. A congruence on a commutative monoid Q is an equivalence relation ∼ that is additively closed, in the sense that u ∼ v ⇒ u+w ∼ v+w

for all w ∈ Q.

The quotient Q/∼ is the set of equivalence classes under addition.

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Lemma 3.13. The quotient Q = Q/∼ of a monoid by a congruence is a monoid. Any congruence ∼ on Q induces a Q-grading on the monoid algebra k[Q] = u∈Q k · tu in which the monomial tu has degree u ∈ Q whenever u → u. Proof. This is an easy exercise. It uses that the multiplication on k[Q] is given by tu tv = tu+v for u, v ∈ Q. Definition 3.14. In any monoid algebra k[Q], a binomial ideal I ⊆ k[Q] generated by binomials tu − λ tv with λ ∈ k induces a congruence ∼ (often denoted by ∼I ) in which u ∼ v if tu − λ tv ∈ I for some λ = 0. Lemma 3.15. Fix a binomial ideal I ⊆ k[Q] in a monoid algebra. Then I and k[Q]/I are both graded by Q = Q/∼. The Hilbert function Q → N, which for any Q-graded vector space M takes q → dimk Mq , satisfies 0 if q = {u ∈ Q | tu ∈ I} dimk (k[Q]/I)q = 1 otherwise. The proof of the lemma is another simple exercise. To rephrase, it says that every pair of monomials in a given congruence class under ∼I are equivalent up to a nonzero scalar modulo I, and the only monomials sent to 0 lie in I. A slightly less set-theoretic and more combinatorial way to think about congruences uses graphs. Definition 3.16. Any binomial ideal I ⊆ k[Q] defines a graph GI whose vertices are the elements of the monoid Q and whose (undirected) edges are the pairs (u, v) ∈ Q × Q such that tu − λ tv ∈ I for some nonzero λ ∈ k. Write π0 GI for the set of connected components of GI . Thus C ∈ π0 GI is the same thing as a congruence class under ∼I . The moral of the story is that combinatorics of the graph GI controls the (binomial) primary decomposition of I. Example 3.17. Each of the two binomial generators of the ideal

determines a collection of edges of the graph GI , indicated in the figure, by additivity of the congruence ∼I . In reality, GI has many more edges than those depicted: since ∼I is an equivalence relation, every connected component is a complete graph


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on its vertex set. However, in examples, it is convenient to draw—and more helpful to see—only edges determined by monomial multiples of generating binomials. The connected components of GI are the fibers of the monoid morphism of N2 to

the monoid N with the element 1 “doubled”. The ideal I has primary decomposition I = x2 , xy, y2 ∩ x − y . The first primary component reflects the three singleton components of GI near the origin. The other primary component reflects the diagonal connected components of GI marching off to infinity. Although the N-graded Hilbert function of k[x, y]/I takes values 1, 2, 1, 1, 1, . . ., the N2 -graded Hilbert function takes only the value 1. Example 3.18. When I = IA is the toric ideal for a matrix A, the connected components of GI are the fibers Fα for α ∈ NA. Example 3.19. If I = Iρ ,J is a binomial prime in a polynomial ring k[x] over an algebraically closed field k, and C ∈ π0 GI is a connected component, then either C = Nn NJ or else C = (u + L) ∩ NJ for some u ∈ NJ . When ρ is the trivial (only) character on the lattice L = {0} ⊆ Z{3} and J = {3} ⊆ {1, 2, 3}, for instance, then

In this case Nn NJ consists of the monomials off of the vertical axis (i.e., those in the region outlined by bold straight lines), whereas every component (u + L) ∩ NJ of GI is simply a single lattice point on the vertical axis.

3.3 Binomial Primary Ideals with Monomial Associated Primes The algebraic characterization of monomial primary ideals in the first half of Proposition 3.8 has an approximate analogue for primary binomial ideals, although it requires hypotheses on the base field k.

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Theorem 3.20. Fix k algebraically closed of characteristic 0. If I ⊆ k[x] is an Iρ ,J primary binomial ideal, then I = Iρ + B for some binomial ideal B ⊇ mJ with > 0. Proof. This is the characterization of primary decomposition [ES96, Theorem 7.1] applied to a binomial ideal that is already primary. The content of the theorem is that I contains both Iρ and a power of each vari/ J. (In positive characteristic, I contains a Frobenius power of Iρ , but able xi for i ∈ not necessarily Iρ itself.) A more precise analogue of the algebraic part of Proposition 3.8 would characterize which binomial ideals B result in primary ideals. However, there is no simple way to describe such binomial ideals B in terms of generators. The best that can be hoped for is an answer to a pair of questions: (a) What analogue of standard monomials allows us to ascertain when I is primary? (b) What property of the standard monomials characterizes the primary condition? Preferably the answers should be geometric, and suitable for binomial ideals in arbitrary affine semigroup rings, as in the second half of Proposition 3.8. Lemma 3.15 answers the first question. Indeed, if I ⊆ k[Q] is a monomial ideal, then std(I) is exactly the subset of Q such that k[Q]/I has Q-graded Hilbert function dimk (k[Q]/I)u = 1 for u ∈ std(I) and 0 for tu ∈ I. In the monomial case, we could still define the monoid quotient Q = Q/∼I , whose classes are all singleton monomials except for the class of monomials in I. Therefore, for general binomial ideals I, the set of non-monomial classes of the congruence ∼I plays the role of std(I). Note that Proposition 3.8 answers the second question for monomial ideals in affine semigroup rings, whose associated primes are automatically monomial. The next step relaxes the condition on I but not on the associated prime: consider a binomial ideal I in an affine semigroup ring k[Q], and ask when it is pF -primary for a face F ⊆ Q. The answer in this case relies, as promised, on the combinatorics of the graph GI from Definition 3.16 and its set π0 GI of connected components; however, as Proposition 3.8 hints, the group generated by F enters in an essential way. Definition 3.21. For a face F of an affine semigroup Q, and any k[Q]-module M, M[ZF] = M ⊗k[Q] k[Q + ZF] is the localization by inverting all monomials not in pF . If I ⊆ k[Q] is a binomial ideal, then a connected component C ∈ π0 GI is F-finite if C = C ∩ Q for some finite connected component C of the graph GI[ZF] for the localization I[ZF] ⊆ k[Q][ZF]. Thus, for example, k[Q][ZF] = k[Q + ZF] is the monoid algebra for the affine semigroup Q + ZF obtained by inverting the elements of F in Q. In Proposition 3.8, where I ⊆ k[Q] is a monomial ideal, all of the connected components of GI inside of std(I) are singletons, and the same is true of GI[ZF] . See Sect. 4.1 for additional information and examples concerning geometry and combinatorics of localization. Example 3.22. Connected components of GI can be finite but not F-finite for a given face F of Q. For instance, if I = x − y ⊆ k[x, y] = k[N2 ], then each connected component of GI is finite—comprising the set of monomials in k[x, y] of some fixed total


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degree—but not F-finite if F is the horizontal axis of Q = N2 : once x is inverted, π0 GI[x−1 ] consists of infinite northwest-pointing rays in the upper half-plane. Theorem 3.23. Fix a monomial prime ideal pF = tu | u ∈ / F in an affine semigroup ring k[Q] for a face F ⊆ Q. A binomial ideal I ⊆ k[Q] is pF -primary if and only if (a) Every connected component of GI other than {u ∈ Q | tu ∈ I} is F-finite. (b) F acts on the set of F-finite components semifreely with finitely many orbits. Thus the primary condition is fundamentally a finiteness condition—or really a pair of finiteness conditions. A proof is sketched after Example 3.26; but first, the terminology requires precise explanations. Semifreeness, for example, guarantees that the set of F-finite components is a subset of a set acted on freely by ZF; this is part of the characterization of semifree actions in [KM10]. Definition 3.24. An action of a monoid F on a set T is a map F × T → T , written ( f ,t) → f + t, that satisfies 0 + t = t for all t ∈ T and respects addition: ( f + g) + t = f + (g + t). The monoid action is semifree if t → f + t is an injection T → T for each f ∈ F, and f → f + t is an injection F → T for each t ∈ T . In contrast to group actions, monoid actions do not a priori define equivalence relations, because the relation t ∼ f +t can fail to be symmetric. The relation is already reflexive and transitive, however, precisely by the two axioms for monoid actions. Definition 3.25. An orbit of a monoid action of F on T is an equivalence class under the symmetrization of the relation {(s,t) | f + s = t for some f ∈ F} ⊆ T × T . Combinatorially, if F acts on T , one can construct a directed graph with vertex set T and an edge from s to t if t = f +s for some f ∈ F. Then an orbit is a connected component of the underlying undirected graph. Example 3.26. The ideal

is p-primary for p = pF = x, y , where F is the z-axis of N3 . The monoid F acts on the F-finite connected components of GI with two orbits: one on the z-axis, where each connected component is a singleton; and one adjacent orbit, where every connected component is a pair. The monomial class in this example (outlined by bold straight lines) is the set of monomials in x2 , xy, y2 .

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Example 3.27. Fix notation as in Examples 3.5 and 3.9. The ideal I = c2 , cd, d 2 ⊆ k[Q] is pF -primary. In contrast to Example 3.26, this time the F-finite connected components are all singletons, but the two orbits are not isomorphic as sets acted on by the face F: one orbit is F itself, while the other is the set depicted in Example 3.9. Proof sketch for Theorem 3.23. This theorem is the core conclusion of [DMM10, Theorem 2.15 and Proposition 2.13]. The argument is summarized as follows. For any set T , let k{T } denote the vector space over k with basis T . If π0 GI satisfies the two conditions, then k[Q]/I has finite a filtration, as a k[Q]-module, whose associated graded pieces are the vector spaces k{T } for the finitely many F-orbits T of F-finite components of GI . In fact, semifreeness guarantees that for each orbit T , the vector space k{T } is naturally a torsion-free module over k[F] = k[Q]/pF . Finiteness of the number of orbits guarantees that the associated graded module of k[Q]/I is a finite direct sum of modules k{T }, so it has only one associated prime, namely pF . Consequently k[Q]/I itself has just one associated prime. For the other direction, when I is pF -primary, one proves that inverting the monomials and binomials outside of pF annihilates the Q-graded pieces of k[Q]/I for which the connected component in GI[ZF] is infinite [DMM10, Lemmas 2.9 and 2.10]. Since the elements outside of pF act injectively on k[Q]/I by definition of pF -primary, every class of ∼I that is not F-finite must therefore already consist of monomials in I. The semifree action of F on the F-finite components derives simply from the fact that k[Q]/I is torsion-free as a k[F]-module, where the k[F]-action is induced by the inclusion k[F] ⊆ k[Q]. Generalities about Q-gradings of this sort imply that k[Q]/I possesses a filtration whose associated graded pieces are as in the previous paragraph. The minimality of pF over I implies that the length of the filtration is finite.

4 Binomial Primary Decomposition General binomial ideals induce more complicated congruences than primary binomial ideals. This section completes the combinatorial analysis of binomial primary decomposition by describing how to pass from an arbitrary binomial ideal to its primary components. There are crucial points where characteristic zero or algebraically closed hypotheses are required of the field k, but those will be mentioned explicitly; if no mention is made, then k is assumed to be arbitrary.

4.1 Monomial Primes Minimal over Binomial Ideals The first step is to consider again the setting from the previous section, particularly Theorem 3.23, where a monomial prime ideal pF is associated to I in an arbitrary affine semigroup ring k[Q], except that now the binomial ideal I is not assumed to be primary. The point is to construct its pF -primary component. The nature of Theorem 3.23 splits the construction into two parts:


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(a) ensuring that the (non-monomial) connected components are all F-finite, and (b) forcing the face F to act in the correct manner on the components. These operations will be carried out in reverse order, with part 2 being accomplished by localization, and then part 1 being accomplished by simply lumping all of the connected components that are not F-finite together. Definition 4.1. For a face F of an affine semigroup Q and a binomial ideal I ⊆ k[Q], (I : tF ) = I[ZF] ∩ k[Q] is the kernel of the composite map k[Q] → k[Q]/I → (k[Q]/I)[ZF]. Remark 4.2. The notation (I : tF ) is explained by an equivalent construction of this ideal. Indeed, the usual meaning of the colon operation for an element y ∈ k[Q] is that (I : y) = {z ∈ k[Q] | yz ∈ I}. Here, (I : tF ) = (I : t f ) for any lattice point f lying sufficiently far in the relative interior of F. Equivalently, (I : tF ) = (I : (t f )∞ ) = rf r∈N (I : t ) for any lattice point f ∈ F that does not lie on a proper subface of F. Combinatorially, the passage from I to (I : tF ) has a concrete effect. Lemma 4.3. The connected components of the graph G(I:tF ) defined by (I : tF ) are obtained from π0 GI by joining together all pairs of components Cu and Cv such that Cu+ f = Cv+ f for some f ∈ F, where Cu is the component containing u ∈ Q. Proof. Two lattice points u, v ∈ Q lie in the same component of G(I:tF ) exactly when there is a binomial tu − λ tv ∈ k[Q] such that t f (tu − λ tv ) ∈ I for some f ∈ F. Roughly speaking: join the components if they become joined after moving them up by an element f ∈ F. Illustrations of lattice point phenomena related to binomial primary decomposition become increasingly difficult to draw in two dimensions as the full nature of the theory develops, but a small example is possible at this stage. Example 4.4. The ideal I = xz − yz, x2 − x3 ⊆ k[x, y, z] yields the same graph as Example 3.4 except for two important differences: • here there are many fewer edges in the xy-plane (z = 0); and • every horizontal (z = constant ≥ 1) slice of the big region is a separate connected component, in contrast to Example 3.4, where the entire big region was a single connected component corresponding to the monomials in the ideal.

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Only the second generator, , is capable of joining pairs of points in the xyalso plane, and it does so parallel to the x-axis, starting at x = 2. Of course, joins pairs of points in the same manner at positive heights z ≥ 1, but again only starting at x = 2. The first generator, , has the same effect as did in only joins pairs of lattice points at height z = 1 or Example 3.4, except that more. In summary, every connected component of GI in this example is contained in a single horizontal slice, and the horizontal slices of GI are

The outline of the big region is drawn as a dotted line in the z ≥ 1 slice illustration, which depicts only enough of the edges to elucidate its three connected components. Let F be the part of N3 in the xy-plane, so pF = z . The ideal (I : tF ) = (I : ∞ z ) = (I : z) = x−y, x2 −x3 again has the property that every connected component of G(I:z) is contained in a single horizontal slice, but now all of these slices look like the z ≥ 1 slices of GI . Compare this to the statement of Lemma 4.3. / F in an affine semigroup ring Theorem 4.5. Fix a monomial prime pF = tu | u ∈ k[Q] for a face F ⊆ Q. If pF is minimal over a binomial ideal I ⊆ k[Q] and π 0 GI[ZF] is the set of finite components of the graph GI[ZF] , then I has pF -primary component / π 0 GI[ZF] . (I : tF ) + tu | Cu ∈ The exponents on the monomials in this primary component are precisely the elements of Q that lie in infinite connected components of the graph GI[ZF] . Proof. This is [DMM10, Theorem 2.15]. The pF -primary component of I is equal to the pF -primary component of (I : tF ) because primary decomposition is preserved by localization (see [AM69, Proposition 4.9], for example), so we may as well assume that I = (I : tF ). It is elementary to check that F acts on the connected components. The action is semifree on the F-finite components, for if f + u ∼ g + u for some f , g ∈ F, then u ∼ n( f − g) + u for all n ∈ N, whence Cu ∈ π0 GI[ZF] is infinite; and if f + u ∼ f + v then u ∼ v, because x f is a unit on I[ZF]. It is also elementary, though nontrivial, to check that the kernel of the usual localization homomorphism k[Q]/I → k[Q]pF /IpF —inverting all polynomials outside of pF , not just monomials—contains every monomial tu for which Cu ∈ / π 0 GI[ZF] [DMM10, F u / π 0 GI[ZF] is already priLemmas 2.9 and 2.10]. Now note that (I : t ) + t | Cu ∈ mary by Theorem 3.23.


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Remark 4.6. The graph of I[ZF] has vertex set Q[ZF], which naturally contains Q. That is why, in Definition 3.21 and Theorem 4.5, it makes sense to say that an element of Q lies in a connected component of GI[ZF] . Example 4.7. The pF -primary component of the ideal I in Example 4.4 is the ideal I in Example 3.4: the differences that remain, after the localization operation in Example 4.4 is complete, are erased by lumping together the infinite connected components into a single monomial component. Example 4.8. Starting with (I : tF ) in Theorem 4.5, it is not enough to throw in the monomials whose exponents lie in infinite components of G(I:tF ) ; i.e., a connected component of G(I:tF ) could be finite but nonetheless equal to the intersection with Q of an infinite component of GI[ZF] . This occurs for I = xz − yz ⊆ k[x, y, z], with F being the xy-coordinate plane of Q = N3 , so pF = z . Every connected component of GI is finite, even though I = (I : tF ) is not primary. When x and y are inverted to form z ≥ 1 become cosets of the line spanned by 1 I[ZF], the components at height 3 , whose intersections with N are bounded. Hence the z -primary component −1 of I is I + z = z , as is clear from the primary decomposition I = z ∩ x − y .

4.2 Primary Components for Arbitrary Given Associated Primes For this subsection, fix a binomial ideal I ⊆ k[x] in a polynomial ring with a binomial associated prime Iρ ,J for some character ρ : L → k∗ defined on a saturated sublattice L ⊆ ZJ . Now it is important to assume that the field k is algebraically closed of characteristic 0, for these hypotheses are crucial to the truth of Theorem 3.20, and that theorem is the tool that reduces the current general situation to the special case in Sect. 4.1. The logic is as follows. Every binomial Iρ ,J -primary ideal contains Iρ by Theorem 3.20. Since we are trying to construct a binomial Iρ ,J -primary component of I starting from I itself, the first step should therefore be to enlarge I by throwing in Iρ . The following is a formal statement. Proposition 4.9. Fix a binomial ideal I ⊆ k[x] with k algebraically closed of characteristic 0. If P is any binomial Iρ ,J -primary component of I, then P is the preimage in k[x] of a binomial (Iρ ,J /Iρ )-primary component of (I + Iρ )/Iρ ⊆ k[x]/Iρ . An alternate phrasing makes the point of considering the quotient k[x]/Iρ in Proposition 4.9 clearer. Proposition 4.10. If P is an Iρ ,J -primary binomial ideal in k[x], with k algebraically closed of characteristic 0, then the image of P in the affine semigroup ring k[Q] = k[x]/Iρ is a binomial ideal pF -primary to the monomial prime pF = Iρ ,J /Iρ in k[Q]. Proof. This is an immediate consequence of Theorem 3.20 and Theorem 2.22 along with Corollary 2.24: the affine semigroup Q is (NJ/L) × NJ , and the face F is the copy of NJ/L = (NJ/L) × {0} in Q.

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Thus the algebra of general binomial associated primes for polynomial rings is lifted from the algebra of monomial associated primes in affine semigroup rings. The final step is isolating how the combinatorics, namely Theorem 3.23, lifts. Since the algebra of quotienting k[x] modulo Iρ corresponds to the quotient of Nn modulo L, we expect the lifted finiteness conditions to involve cosets of L. Definition 4.11. A subset of Nn is L-bounded for a sublattice L ⊆ Zn if the subset is contained in a finite union of cosets of L. Corollary 4.12. Fix a binomial ideal I ⊆ k[x] with k algebraically closed of characteristic 0. If Iρ ,J is minimal over I, then the Iρ ,J -primary component of I is P = I + xu | Cu ∈ π0 GI [ZJ ] is not L-bounded , where I [ZJ ] is the localization along NJ , and I is defined, using xJ = ∏ j∈J x j , to be J I = (I + Iρ ) : x∞ J = (I + Iρ )[Z ] ∩ k[x]. If Iρ ,J is associated to I but not minimal over I, then for any monomial ideal K containing a sufficiently high power of mJ = xi | i ∈ / J , an Iρ ,J -primary component of I is defined as P is, above, but using IK in place of I , where IK = (I + Iρ + K) : x∞ J . Proof sketch. This is [DMM10, Theorem 3.2]. The key is to lift the monomial minimal prime case for affine semigroup rings in Theorem 4.5 to the current binomial associated prime case in polynomial rings using Propositions 4.9 and 4.10. For an embedded prime Iρ ,J , one notes that any given Iρ ,J -primary component of I must contain a sufficiently high power of mJ , so it is logical to begin the search for an Iρ ,J -primary component by simply throwing such monomials along with Iρ into I. But then Iρ ,J is minimal over the resulting ideal I + Iρ + K, so the minimal prime case applies. The definition of P in the theorem says that GP has two types of connected components: the ones that are L-bounded upon localization along ZJ , and the connected component consisting of exponents on monomials in P. The theorem says that GP shares all but its monomial component with the graph GI , and that the other connected components of GI fail to remain L-bounded upon localization along ZJ . In the case where Iρ ,J is an embedded prime, the graph-theoretic explanation is that GI has too many connected components that remain L-bounded upon localization; in fact, there are infinitely many NJ -orbits. The hack of adding K throws all but finitely many NJ -orbits into the L-infinite “big monomial” connected component. Example 4.13. A primary decomposition of the ideal was already given in Example 3.17. Analyzing it from the perspective of Corollary 4.12 completes the heuristic insight. First let Iρ ,J = x − y , so J = {1, 2} and ρ : L → k∗ is trivial on the lattice L generated by −11 . Then I = I +x−y is already prime. Passing from I to I has the sole effect of joining the two isolated points (the basis vectors) on the axes together.


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Now let Iρ ,J = x, y , so J = ∅ and ρ is the trivial (only) character defined on L = ZJ = {0}. Every connected component of GI in Example 3.17 remains Lbounded upon localization along ZJ , but there are infinitely many such components. Choosing K = x, y e , so that IK = I + x, y e , kills off all but finitely many, to get

In particular, taking e = 2 recovers the primary decomposition from Example 3.17.

4.3 Finding Associated Primes Combinatorially The constructions of binomial primary components in previous sections assume that a monomial or binomial associated prime of a binomial ideal has been given. To conclude the discussion of primary decomposition of binomial ideals, it remains to examine the set of associated primes. The existence of binomial primary decompositions hinges on a fundamental result, due to Eisenbud and Sturmfels [ES96, Theorem 6.1], that was a starting point for all investigations involving primary decomposition of binomial ideals. Theorem 4.14. Every associated prime of a binomial ideal in k[x] is a binomial prime if the field k is algebraically closed. Although the statement is for polynomial rings, a simple reduction implies the existence of binomial primary decomposition in the generality of monoid algebras as defined in Sect. 3.2, given the construction of binomial primary components. Corollary 4.15. Fix a finitely generated commutative monoid Q and an algebraically closed field k of characteristic 0. Every binomial ideal I in k[Q] admits a binomial primary decomposition: I = P1 ∩ · · · ∩ Pr for binomial ideals P1 , . . . , Pr . Proof. Choose a presentation Nn Q. The kernel of the induced presentation k[x] k[Q] is a binomial ideal in k[x]. Therefore the preimage of I in k[x] is a binomial ideal I . The image in k[Q] of any binomial primary decomposition of I is a binomial primary decomposition of I. Therefore it suffices to prove the case where Q = Nn and I = I. Since every associated prime of I is binomial by Theorem 4.14, the result follows from Corollary 4.12. Corollary 4.15 is stated only for characteristic 0 to demonstrate the connection between prior results in this survey. However, the restriction is unnecessary.

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Theorem 4.16. Corollary 4.15 holds for fields of positive characteristic, as well. Proof. For polynomial rings this is [ES96, Theorem 7.1], and the case of general monoids Q follows by the argument in the proof of Corollary 4.15. What’s missing in positive characteristic cases is combinatorics of primary ideals. Open Problem 4.17. Characterize primary binomial ideals and primary components of binomial ideals combinatorially in positive characteristic. Note, however, that a solution to this problem would still not say how to discover—from the combinatorics—which primes are associated. The same is true in characteristic 0. Thus Corollary 4.12 is unsatisfactory for two reasons: • it requires strong hypotheses on the field k; and • it assumes we know which primes Iρ ,J are associated to I. Fortunately, there is a combinatorial, lattice-point method to recognize associated primes—or at least, to reduce the recognition to a finite problem. The main point is Theorem 4.26: the combinatorics of the graph GI can be used to construct a decomposition of I as an intersection of “primary-like” binomial ideals in a manner requiring no hypotheses on the characteristic or algebraic closure of the base field. The statement employs some additional concepts. Definition 4.18. A subset of Zn is J-bounded if it intersects only finitely many cosets of ZJ in Zn . Lemma 4.19. If I ⊆ k[x] is a binomial ideal, then ZJ acts on the set of J-bounded components of the graph GI[ZJ ] on ZJ × NJ induced by localizing I along NJ . Proof. In fact, ZJ acts on all of the connected components, because the Laurent monomials xu for u ∈ ZJ are units modulo I[ZJ ] = I[ZF] for the face F = NJ . Definition 4.20. A witness for a sublattice L ⊆ ZJ potentially associated to I is any element in a J-bounded connected component of GI[ZJ ] whose stabilizer is L. Example 4.21. The binomial ideal

induces the depicted congruence. Its potentially associated lattices are all contained in Z = Z{1} , parallel to the x-axis. The lattices are generated by 0, by 20 , and by 10 .


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The subset J is part of the definition of potentially associated sublattice; it is not enough to specify L alone. The notion of associated lattice, without the adverb “potentially”, would require further discussion of primary decomposition of congruences on monoids; see the definition of associated lattice in [KM10]. That said, the set of potentially associated lattices, which contains the set of associated ones, suffices for the purposes here, although sharper results could be stated with the more precise notion. Proposition 4.22. Every binomial ideal I ⊆ k[x] has finitely many potentially associated lattices L ⊆ ZJ . If K = (I : xu ) ⊆ k[x] is the annihilator of xu in k[x]/I for a witness u ∈ Nn , then K[ZJ ] + mJ = Iσ ,J [ZJ ] for a uniquely determined witness character σ : L → k∗ . Given I, each L ⊆ ZJ determines finitely many witness characters. Proof. This is proved in [KM10] on the way to the existence theorem for combinatorial mesoprimary decomposition. The finiteness of the set of potentially associated lattices traces back to the noetherian property for congruences on finitely generated commutative monoids. The conclusion concerning K is little more than the characterization of binomial ideals in Laurent polynomial rings [ES96, Theorem 2.1]. The finiteness of the number of witness characters occurs because witnesses for L ⊆ ZJ with distinct witness characters are forced to be incomparable in Nn . In Proposition 4.22, the domain L of the character σ appearing in Iσ ,J need not be saturated (see Definition 2.23), and no hypotheses are required on the field k. Deducing combinatorial statements about associated primes or primary decompositions of binomial ideals is often most easily accomplished by reducing to the case of ideals with the simplest possible structure in this regard. Definition 4.23. A binomial ideal with a unique potentially associated lattice is called mesoprimary. A mesoprimary decomposition of a binomial ideal I ⊆ k[x] is an expression of I as an intersection of finitely many mesoprimary binomial ideals. Example 4.24. Primary binomial ideals in polynomial rings over algebraically closed fields of characteristic 0 are mesoprimary. That is the content of Theorem 3.23 for such fields, given Proposition 4.10. More precisely, combinatorics of mesoprimary ideals is just like that of primary ideals, except that instead of an affine semigroup acting semifreely, an arbitrary finitely generated cancellative monoid acts semifreely; see the characterizations of mesoprimary congruences in [KM10]. Definition 4.23 stipulates constancy of the combinatorics, not the arithmetic— meaning the witness characters—but the arithmetic constancy is automatic. Lemma 4.25. If I is a mesoprimary ideal, then the witnesses for the unique potentially associated lattice all share the same witness character. Proof. This follows from the same witness incomparability that appeared in the proof of Proposition 4.22. The statement is equivalent to one direction of the characterization of mesoprimary binomial ideals as those with precisely one associated mesoprime; see [KM10], where a complete proof can be found.

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Theorem 4.26. Every binomial ideal I ⊆ k[x] admits a mesoprimary decomposition in which the unique associated lattice and witness character of each mesoprimary component is potentially associated to I. Proof. This is a weakened form of the existence theorem for combinatorial mesoprimary decomposition in [KM10]. The power of Theorem 4.26 lies in the crucial conceit that the combinatorics of the graph GI controls everything, so the lattices associated to the mesoprimary components are severely restricted. Over an algebraically closed field of characteristic 0, for instance, every primary decomposition is a mesoprimary decomposition, but usually the lattices are not associated to I. This is the case for a lattice ideal IL , as long as the lattice L is not saturated: the ideal IL is already mesoprimary, but the associated lattice of every associated prime is the saturation Lsat = (L ⊗Z Q) ∩ Zn , the smallest saturated sublattice of Zn containing L. In general, the combinatorial control is what allows Theorem 4.26 to be devoid of hypotheses on the field. As in any expression of an ideal I as an intersection of larger ideals, information about associated primes of I can just as well be read off of the intersectands. For mesoprimary decompositions this is especially effective because primary decomposition of mesoprimary ideals [KM10] is essentially as simple as that of lattice ideals [ES96, Corollary 2.5]. In particular, when the field is algebraically closed, potentially associated lattices yield associated primes by way of saturation. The point is that only finitely many characters Lsat → k∗ restrict to a given fixed character L → k∗ . In fact, when k is algebraically closed, these characters are in bijection with the finite group Hom(Lsat /L, k∗ ). Thus Theorem 4.26 reduces the search for associated primes of I to the combinatorics of the graph GI , along with a minimal amount of arithmetic. Corollary 4.27. If k is algebraically closed, then every associated prime of I ⊆ k[x] is Iρ ,J for some character ρ : Lsat → k∗ whose restriction to L is one of the finitely many witness characters defined on a potentially associated lattice L ⊆ ZJ of I. Proof. Every associated prime of I is associated to a mesoprime in the decomposition from Theorem 4.26. Now apply either the primary decomposition of mesoprimary ideals [KM10] or the witness theorem for cellular ideals [ES96, Theorem 8.1], using the fact that mesoprimary ideals are cellular [KM10]. Remark 4.28. In the special case where I is cellular, meaning that every variable is either nilpotent or a nonzero divisor modulo I, Corollary 4.27 coincides with [ES96, Theorem 8.1]. Every binomial ideal in any polynomial ring over any field is an intersection of cellular ideals [ES96, Theorem 6.2], with at most one cellular component for each subset J ⊆ {1, . . . , n}, so it suffices for many purposes to understand the combinatorics of cellular ideals. (Theorem 4.26 strengthens this approach, since mesoprimary ideals are cellular and their combinatorics is substantially simpler.) The way witnesses and witness characters are defined above, however, it is not quite obvious that the information extracted from witnesses for the original ideal I and those for its cellular components coincides. That this is indeed the case constitutes


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a key ingredient proved in preparation for the existence theorem for combinatorial mesoprimary decomposition in [KM10]. Exercise 4.29. The ideal I = xz − yz, x2 − x3 ⊆ k[x, y, z] from Example 4.4 has primary decomposition I = x − y, x2 , xy, y2 ∩ x − 1, y − 1 ∩ x2 , z ∩ x − 1, z . The reader is invited to find all associated lattices L ⊆ ZJ of I and match them to the associated primes of I. Then the reader can verify, using Corollary 4.12, that the given primary decomposition of I really is one. Hint: take J ∈ {3}, {1, 2, 3}, {2}, {1, 2} . The upshot of Sects. 2–4 is that the lattice-point combinatorics of congruences on monoids lifts to combinatorics of monomial and binomial primary and mesoprimary decompositions of binomial ideals in monoid algebras. From there, binomial primary decomposition is a small arithmetic step, having to do with group characters for finitely generated abelian groups.

Applications 5 Hypergeometric Series The idea for lattice-point methods in binomial primary decomposition originated in the study of hypergeometric systems of differential equations, particularly their series solutions. The literature on these systems and series is so vast—owing to its connections with physics, numerical analysis, combinatorics, probability, number theory, complex analysis, and algebraic geometry—that one section in a survey lacks the ability to lend proper perspective. Therefore, the goal of this section is to make a beeline for the connections to binomial primary decomposition, with just enough background along the way to allow the motivations and conclusions to shine through. Much of the exposition is borrowed from [DMM07, DMM10 ], sometimes nearly verbatim. The extended abstract [DMM07] presents a broader, more complete historical overview.

5.1 Binomial Horn Systems Horn systems are certain sets of linear partial differential equations with polynomial coefficients. Their development grew out of the ordinary univariate hypergeometric theory going back to Gauss (see [SK85], for example) and Kummer [Kum1836], through the bivariate versions of Appell, Horn, and Mellin [App1880, Hor1889, Hor31, Mel21]. These formulations had no apparent connection to binomials, but through a relatively simple change of variables, Gelfand, Graev, Kapranov, and Zelevinsky brought binomials naturally into the picture [GGZ87, GKZ89]. The data required to write down a binomial Horn system consist of a basis for a sublattice L ⊆ Zn and a homomorphism β : Zn /L → C. Focus first on the basis,

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which is traditionally arranged in an integer n × m matrix B, where m = rank(L). If b ∈ Zn is a column of B, then b determines a binomial: b ∈ Zn ∂ b+ − ∂ b− ∈ C[∂ ] = C[∂1 , . . . , ∂n ], where b = b+ − b− expresses the vector b as a difference of nonnegative vectors with disjoint support. Elements of the polynomial ring C[∂ ] are to be viewed as differential operators on functions Cn → C. Therefore the matrix B determines a system of m binomial differential operators, one for each column. The interest is a priori in solutions to differential systems, not really the systems themselves, so it is just as well I(B) = ∂ b+ − ∂ b− | b = b+ − b− is a column of B ⊆ C[∂ ] generated by these binomials, because any function annihilated by the m binomials is annihilated by all of I(B). Example 5.1. In the 0123 situation from Examples 2.2 and 2.16, with variables ∂ = ∂1 , ∂2 , ∂3 , ∂4 instead of a, b, c, d or x1 , x2 , x3 , x4 yields I(B) = ∂1 ∂3 − ∂22 , ∂2 ∂4 − ∂32 . Example 5.2. In the 1100 0111 situation from Examples 2.3 and 2.17 with ∂ variables, I(B) = ∂1 ∂3 − ∂2 , ∂1 ∂4 − ∂2 . The set of homomorphisms Zn /L → C is a complex vector space Hom(Zn /L, C) of dimension d := n − m. Choosing a basis for this vector space is the same as choosing a basis for (Zn /L) ⊗Z C, which is the same as choosing a d × n matrix A with AB = 0. Let us now, once and for all, fix such a matrix A with entries ai j for i = 1, . . . , d and j = 1, . . . , n. The situation is therefore just as it was in Examples 2.16 and 2.17, and our homomorphism Zn /L → C becomes identified with a complex vector β ∈ Cd . Together, A and β determine d differential operators E1 − β1 , . . . , Ed − βd , where Ei = ai1 x1 ∂1 + · · · + ain xn ∂n . Note that ai j x j ∂ j is the operator on functions f (x1 , . . . , xn ) : Cn → C that takes the partial derivative with respect to x j and multiplies the resulting function by ai j x j . Definition 5.3. The binomial Horn system H(B, β ) is the system I(B) f = 0 E1 f = β1 f .. . Ed f = βd f of differential equations on functions f (x) : Cn → C determined by the lattice basis ideal I(B) and the Euler operators E1 − β1 , . . . , Ed − βd . The goal is to find, characterize, or otherwise understand the solutions to H(B, β ).


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Example 5.4. In the 0123 case from Example 2.16, H(B, β ) has lattice basis part I(B) f = 0 ⇔ (∂1 ∂3 − ∂22 ) f = 0 and (∂2 ∂4 − ∂32 ) f = 0 and the Euler operators yield the following equations: (x1 ∂1 + x2 ∂2 + x3 ∂3 + x4 ∂4 ) f = β1 f ( x2 ∂2 + 2x3 ∂3 + 3x4 ∂4 ) f = β2 f . Example 5.5. In the 1100 0111 case from Example 5.2, H(B, β ) has lattice basis part I(B) f = 0 ⇔ (∂1 ∂3 − ∂2 ) f = 0 and (∂1 ∂4 − ∂2 ) f = 0 and the Euler operators yield the following equations: ) f = β1 f (x1 ∂1 + x2 ∂2 ( x2 ∂2 + x3 ∂3 + x4 ∂4 ) f = β2 f . Since Horn systems are linear, their solution spaces are complex vector spaces. More precisely, the term solution space in what follows means the vector space of local holomorphic solutions defined in a neighborhood of a (fixed, but arbitrary) point in Cn that is nonsingular for the Horn system. Example 5.6. In the 0123 case from Example 5.4, for any parameter vector β , the β /3 β /3 Puiseux monomial f = x1 1 x4 2 is a solution of H(B, β ). Indeed,

∂1 ∂3 ( f ) = ∂2 ( f ) = ∂2 ∂4 ( f ) = ∂32 ( f ) = 0, so I(B) f = 0, and (E1 − β1 ) f = (E2 − β2 ) f = 0 because x1 ∂1 ( f ) = (β1 − 13 β2 ) f x2 ∂2 ( f ) = 0 x3 ∂3 ( f ) = 0 x4 ∂4 ( f ) = 13 β2 f . Erdélyi produced this solution and similar ones in other examples [Erd50], but he furnished no explanation for why it should exist or how he found it. In this particular example, the Horn system has, in addition to the Puiseux monomial f , three linearly independent fully supported solutions, in the following sense. Definition 5.7. A Puiseux series solution f to a Horn system H(B, β ) is fully supported if there is a normal affine semigroup Q of dimension m and a vector γ ∈ Cn such that the translate γ + Q consists of vectors that are exponents on monomials with nonzero coefficient in f .

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The integer m = rank(L) in the definition is the maximum possible: the Euler operator equations impose homogeneity on Puiseux series solutions, meaning that every solution must be supported on a translate of L ⊗Z C. In fact, the translate is by any vector γ ∈ Cn satisfying Aγ = β . Questions 5.8. Consider the family of Horn systems B determines with varying β . (a) For which parameters β does H(B, β ) have finite-dimensional solution space? (b) What is a combinatorial formula for the minimum solution space dimension, over all possible choices of the parameter β ? (c) Which β are generic in the sense that the minimum dimension is attained? (d) Which monomials occur in solutions expanded as series centered at the origin? These questions arise from classical work done in the 1950s, such as Erdélyi’s, and earlier. Implicit in Question 3 is that the dimension of the solution space rises above the minimum for only a “small” subset of parameters β . Example 5.9. In the 1100 0111 case from Example 5.5, if β1 = 0, then any (local holomorphic) bivariate function f (x3 , x4 ) satisfying x3 ∂3 f + x4 ∂4 f = β2 f is a solution of the Horn system H(B, β ). The space of such functions is infinite-dimensional; in fact, it has uncountable dimension, as it contains all Puiseux monomials xw3 3 x4w4 with w3 , w4 ∈ C and w3 + w4 = β2 . When β1 = 0, the solution space has finite dimension. The 1100 0111 example has vast numbers of linearly independent solutions expressible as Puiseux series with small support, but only for special values of β . In contrast, in the 0123 case there are many fewer series solutions of small support, but they appear for arbitrary values of β . This dichotomy is central to the interactions of Horn systems with binomial primary decomposition.

5.2 True Degrees and Quasidegrees of Graded Modules The commutative algebraic version of the dichotomy just mentioned arises from elementary (un)boundedness of Hilbert functions of A-graded modules (recall Definition 2.12, Example 2.13, and Lemma 3.15); see Definition 5.16. For the remainder of this section, fix a matrix A ∈ Zd×n of rank d whose affine semigroup NA ⊆ Zd is pointed (Definition 6.10). Lemma 5.10. A binomial ideal I ⊆ C[∂ ] is A-graded if and only if it is generated by binomials xu − λ xv for which Au = Av. It is of course not necessary—and it almost never happens—that every binomial of the form xu − λ xv with Au = Av lies in I. Example 5.11. I = I(B) is always A-graded when B is a matrix for ker(A). The set of degrees where a graded module is nonzero should, for the purposes of the applications to Horn systems, be considered geometrically.


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Definition 5.12. For any A-graded module M, tdeg(M) = {α ∈ Zd | Mα = 0} is the set of true degrees of M. The set of quasidegrees of M is qdeg(M) = tdeg(M), the Zariski closure in Cd of the true degree set of M. The Zariski closure here warrants some discussion. By definition, the Zariski closure of a subset T ⊆ Cd is the largest set T of points in Cd such that every polynomial vanishing on T also vanishes on T . All of the sets T that we shall be interested in are sets of lattice points in Zd ⊆ Cd . When T consists of lattice points on a line, for example, its Zariski closure T is the whole line precisely when T is infinite. When T is contained in a plane, its Zariski closure is the whole plane only if T is not contained in any algebraic curve in the plane. In the cases that interest us, T will always be a finite union of translates of linear subspaces of Cd . Lemma 5.13. If M is a finitely generated A-graded module over C[∂ ], then qdeg(M) is a finite arrangement of affine subspaces of Cd , each one parallel to ZAJ for some J ⊆ {1, . . . , n}, where AJ is the submatrix of A comprising the columns indexed by J. Proof. This is [DMM10 , Lemma 2.5]. Since M is noetherian, it has a finite filtration whose successive quotients are A-graded translates of C[∂ ]/p for various Agraded primes p. Finiteness of the filtration implies that qdeg(M) is the union of the quasidegrees of these A-graded translates. But tdeg(C[∂ ]/p) is the affine semigroup generated by {deg(∂i ) | ∂i ∈ / p}, because tdeg(C[∂ ]/p) is an integral domain. Example 5.14. If p = IL + mJ is a binomial prime ideal, then qdeg(C[∂ ]/p) = CAJ is the complex vector subspace of Cd spanned by the columns of A indexed by J. Example 5.15. In the 1100 0111 case from Examples 2.3 and 2.17, consider the A-graded module M = C[∂1 , ∂2 , ∂3 , ∂4 ]/I for varying ideals I. Then the true degree sets and quasidegree sets can be depicted as follows. (a) When I = IA = ∂1 ∂3 − ∂2 , ∂3 − ∂4 , so that M ∼ = C[∂1 , ∂3 ] by the homomorphism sending ∂2 → ∂1 ∂3 and ∂4 → ∂3 ,

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(b) When I = ∂1 , ∂2 , so that M ∼ = C[∂3 , ∂4 ] by the homomorphism sending ∂1 → 0 and ∂2 → 0,

(c) When I = ∂12 , ∂2 , so that M ∼ = C[∂3 , ∂4 ] ⊕ ∂1 C[∂3 , ∂4 ],

Definition 5.16. An A-graded prime ideal p ⊆ C[∂ ] is • toral if the Hilbert function u → dimC (C[∂ ]/p)u is bounded for u ∈ Zd , and • Andean if the Hilbert function is unbounded. The adjective “Andean” indicates that Andean A-graded components sit like a high, thin mountain range on Zd of unbounded elevation over cosets of lattices ZAJ . Example 5.17. Consider the situation from Example 5.15. (a) The prime ideal IA in Example 5.15.1 is toral because the Hilbert function of C[∂ ]/IA only takes the value 1 on the true degrees. By definition, the Hilbert function vanishes outside of the true degree set. 5.15.2 is Andean because the Hilbert (b) The prime ideal I = ∂1 , ∂2 in Example function of C[∂ ]/I is unbounded: 0k → k. Example 5.18. In the 0123 situation from Examples 2.2 and 2.16, the ideal IA = ∂1 ∂3 − ∂22 , ∂2 ∂4 − ∂32 , ∂1 ∂4 − ∂2 ∂3 is toral under the A-grading. Indeed, for any


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matrix A the toric ideal IA is toral under the A-grading by Lemma 3.15: the Q-graded Hilbert function of the algebra k[Q] takes the constant value 1 and is thus bounded. Theorem 5.19. Fix an A-graded ideal I ⊆ C[∂ ]. Given that NA is pointed, every associated prime of I is A-graded, and I admits a decomposition as an intersection of A-graded primary ideals. The intersection IAndean of the primary components of I with Andean associated primes is well-defined. Proof. The A-graded conclusion on the associated primes is [MS05, Prop. 8.11]. The Andean part is well-defined because if p ⊇ q for some Andean p then q is also Andean. (“The set of Andean primes is closed under going down.”) Corollary 5.20. If I ⊆ C[∂ ] is an A-graded ideal, then I admits a decomposition I = Itoral ∩ IAndean into toral and Andean parts, where Itoral is the intersection of the primary components of I with toral associated primes in any fixed primary decomposition of I. Definition 5.21. The Andean arrangement of an ideal I is qdeg(C[∂ ]/IAndean ). The Andean arrangement is a union of affine subspaces of Cd by Lemma 5.13. It is well-defined by Theorem 5.19. Example 5.22. The 1100 0111 lattice ideal I(B) = ∂1 ∂3 − ∂2 , ∂1 ∂4 − ∂2 from Example 5.2 has primary decomposition I(B) = IA ∩ ∂1 , ∂2 = Itoral ∩ IAndean , with toral part Itoral = IA and Andean part IAndean = ∂1 , ∂2 by Example 5.17. Therefore the Andean arrangement of I(B) is the thick vertical line in Example 5.15.2.

5.3 Counting Series Solutions The distinction between toral and Andean primes provides the framework for the answers to Questions 5.8. Throughout the remainder of this section, fix a matrix B ∈ Zn×m of rank m = n − d such that AB = 0. Assume that B is mixed, meaning that every nonzero integer vector in the span of the columns of B has two nonzero entries of opposite sign. The mixed condition is a technical hypothesis arising while constructing series solutions to H(B, β ); its main algebraic consequence is that it forces NA to be pointed. The multiplicity of a prime ideal p in an ideal I is, by definition, the length of the largest submodule of finite length in the localization C[∂ ]p /Ip . This number is nonzero precisely when p is associated to I. Combinatorially, when p and I are binomial ideals, the multiplicity of p in I counts connected components of graphs

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related to GI , such as those in Corollary 4.12. For the purposes of Horn systems, the most relevant number is derived from multiplicities of prime ideals as follows. Definition 5.23. The multiplicity μ (L, J) of a saturated sublattice L ⊆ ZJ is the product ι μ , where ι is the index |L/(ZB ∩ ZJ )| of the sublattice ZB ∩ ZJ in L, and μ is the multiplicity of the binomial prime ideal IL + mJ in the lattice ideal I(B). The factor ι = |L/(ZB ∩ ZJ )| counts the number of partial characters ρ : L → C∗ for which Iρ ,J is associated to I(B). It is the penultimate combinatorial input required to count solutions to Horn systems. The final one is polyhedral. Definition 5.24. For any subset J ⊆ {1, . . . , n}, write vol(AJ ) for the volume of the convex hull of AJ and the origin, normalized so a lattice simplex in ZAJ has volume 1. Example 5.25. In the 0123 case, vol(A) = 3, since the columns of A span Z2 and the convex hull of A with the origin is a triangle that is a union of three lattice triangles.

When J = {1, 4}, in contrast, vol(AJ ) = 1, since the first and last columns of A form a basis for the lattice (of index 3 in Z2 ) that they span. Answers 5.26. The answers to Questions 5.8 for the systems H(B, β ) are as follows. (a) The dimension is finite exactly when β lies in the Andean arrangement of I(B). (b) The generic (minimum) dimension is ∑ μ (L, J) · vol(AJ ), the sum being over all saturated L ⊆ ZJ such that IL + mJ is a toral binomial prime with CAJ = Cd . (c) The minimum rank is attained precisely when β lies outside of an affine subspace arrangement determined by certain local cohomology modules, with the same flavor as (and containing) the Andean arrangement. (d) If the configuration A lies in an affine hyperplane not containing the origin, and β is general, then the solution space of H(B, β ) has a basis containing precisely ∑J μ (L, J) · vol(AJ ) Puiseux series supported on finitely many cosets of L. Example 5.27. To illustrate Answer 5.26.1 in the 1100 0111 case, compare Example 5.9 to Example 5.22: the solution space has finite dimension precisely when the parameter lies off the vertical axis, which is the Andean arrangement in this case. Example 5.28. In contrast, both associated primes are toral in the 0123 case, where I(B) = IA ∩ ∂2 , ∂3 . Indeed, the quotient of C[∂ ] modulo each of these components is an A-graded affine semigroup ring, with the second component yielding C[∂ ]/∂2 , ∂3 ∼ = C[NA{1,4} ]. It follows that the solution space has finite dimension for all parameters β .


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On the other hand, Answer 5.26.2 is interesting in this 0123 case: Example 5.25 implies that H(B, β ) has generic solution space of dimension 3 + 1 = μ ZB, {1, 2, 3, 4} + μ {0}, {1, 4} , with the first summand giving rise to solution series of full support, and the second summand giving rise to one solution series with finite support—that is, supported on finitely many cosets of {0}—by Answer 5.26.4. Compare Example 5.6. Proof of Answers 5.26. These are some of the main results of [DMM10 ], namely: (a) (b) (c) (d)

Theorem 6.3. Theorem 6.10. Definition 6.9 and Theorem 6.10. Theorem 6.10, Theorem 7.14, and Corollary 7.25.

The basic idea is to filter C[∂ ]/I(B) with successive quotients that are A-graded translates of C[∂ ]/p for various binomial primes p. There is a functorial (“Euler– Koszul”) way to lift this to a filtration of a corresponding D-module canonically constructed from H(B, β ). The successive quotients in this lifted filtration are Ahypergeometric systems of Gelfand, Graev, Kapranov, and Zelevinsky [GGZ87, GKZ89]. The solution space dimension equals the volume for such hypergeometric systems, and the factor μ (L, J) simply counts how many times a given such hypergeometric system appears as a successive quotient in the D-module filtration. That, together with series solutions constructed by GGKZ, proves Answers 2 and 4. When a successive quotient is C[∂ ]/p for an Andean prime p, the Euler operators span a vector space of too small dimension; they consequently fail to cut down the solution space to finite dimension: at least one “extra” Euler operator is needed. Without this extra Euler operator, its (missing) continuous parameter allows an uncountable family of solutions as in Example 5.9; this proves Answer 1. Answer 3 is really a corollary of the main results of [MMW05], and is beyond the scope of this survey. To explain Erdélyi’s observation (Example 5.6), note that only | ker A/ZB| · vol(A) many of the series solutions from Answer 5.26.4 have full support, where ker(A) = (ZB)sat is the saturation of the image of B. The remaining solutions have smaller support. Most lattice basis ideals have associated primes other than IZB [HS00], so most Horn systems H(B, β ) have spurious solutions, whether they be of the toral kind (finite-dimensional, but small support, perhaps for special parameters β ) or Andean kind (uncountable dimensional). The “hyperplane not containing the origin” condition in Answer 5.26.4 amounts to a homogeneity condition on I(B): the generators should be homogeneous under the standard N-grading of C[∂ ], as in Example 5.28. More deeply, this condition is equivalent to regular holonomicity of the corresponding D-module [SW08]. As soon as the support of a Puiseux series solution is specified, hypergeometric recursions determine the coefficients up to a global scalar.

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The recursive rules governing the coefficients of series solutions to Horn hypergeometric systems force the combinatorics of lattice-point graphs upon binomial primary decomposition of lattice basis ideals, via the arguments in the proof of Answers 5.26. Granted the generality of Sects. 2–4, it subsequently follows that the questions as well as the answers work essentially as well for arbitrary A-graded binomial ideals, with little adjustment [DMM10 ].

6 Combinatorial Games Combinatorial games are two-player affairs in which the sides alternate moves, both with complete information and no element of chance. The germinal goal of Combinatorial Game Theory (CGT) is to find strategies for such games. After briefly reviewing the foundations and history of CGT using some key examples (Sect. 6.1), this section gives an overview of how to phrase the theory in terms of lattice points in polyhedra (Sect. 6.2). Exploring data structures for strategies via generating functions (Sect. 6.3) or misère quotients (Sect. 6.4) leads to conjectures and computational open problems involving binomial ideals and related combinatorics.

6.1 Introduction to Combinatorial Game Theory There are many different ways to represent games and winning strategies by combinatorial structures. To understand their formal definitions, it is best to have in mind some concrete examples. Example 6.1. The quintessential combinatorial game is N IM. The players—you and I, say—are presented with a finite number of heaps of beans, such as

when there are three heaps, of sizes , , and . Any finite set of heaps is a position in the game of N IM. The game is played by alternating turns, where each turn consists of picking one of the heaps and removing at least one bean from it. For instance, if you play first, then you could remove one bean from the -heap, or three beans from the -heap, or all of the beans from the -heap, to get one of the following positions:

The goal of the game is to play last. As it turns out, if I play first in the game, then you can always force a win by ensuring that you play last. How? Take the nim sum of the heap sizes: express each heap size in binary and add these binary


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numbers digit by digit, as elements of the field F2 of cardinality 2: 11 111 100 000 Whatever move I make will alter only one of the summands and hence will leave a nonzero nim sum, at which point you can always remove beans from a heap to reset the nim sum to zero; I can’t win because removing the last bean leaves a zero nim sum. This general solution to N IM is one of the oldest formal contributions to combinatorial game theory [Bou1902]. More general “heap games”, in which players take beans from heaps according to specified rules, constitute a core class of examples for the theory. Example 6.2. C HESS and its variants give rise to a rich bounty of combinatorial games. One of the most famous, other than C HESS itself, is DAWSON ’ S C HESS, played on a 3 × d board with an initial position of opposing pawns facing one another with a blank rank (row) between [Daw34]. When d = 7, the initial position is:

Moves in DAWSON ’ S C HESS are as usual for C HESS pawns. The only additional rule is that a capture must be made if one is possible. For example, if white moves first (as usual) and chooses to push the third pawn, then a game might start as follows:

And now it is black’s turn; with no captures available, black is allowed to push any pawn in file (column) 1, 5, 6, or 7 down to the middle rank. Another bloodbath ensues, and then it is white’s turn to push a pawn freely. The goal of DAWSON ’ S C HESS is to force your opponent to play last. Thus, in contrast to N IM, a player has won when their turn arrives and no moves are available. Definition 6.3. An impartial combinatorial game is a rooted directed graph on which two players alternate moves along edges. One wins by moving to a node (position) with no outgoing edges. A game is finite if its graph is finite with no directed cycles.

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The root of the directed graph corresponds to the initial position. Any finite impartial game can equivalently be represented as a rooted tree in which the children of each position correspond to its options: the endpoints of its outgoing edges in any directed graph representation. A given position might be repeated in the tree representation. Given a tree representation, an optimally efficient directed graph representation can be constructed by identifying all vertices whose descendant subtrees are isomorphic. Example 6.4. In the language of combinatorial game theory, N IM is not a finite game but a family of finite games that specifies a consistent “rule set” for how to play starting from any particular initial position among an infinite number of possibilities. In the situation of Example 6.1, using i jk for 0 ≤ i, j, k ≤ 9 to represent three heaps of sizes i, j, and k, the top of the game tree is

Note that positions such as , and are repeated in the tree (even in the small bit of the tree depicted here), at the same level or at multiple levels; their descendant subtrees can be identified to form another directed graph representation of this game of N IM. Every leaf of the tree corresponds to the position . Remark 6.5. It is natural to wonder why the games in Definition 6.3 are called “impartial”. The term is meant to indicate that the players have the same options available from each position, as opposed to partizan games, where the players can have distinct sets of options. “But only one player is allowed to play from each position,” you may argue, “so how can you tell the difference between impartial and partizan games?” The answer is to put two games side by side; this is called the disjunctive sum of the two games: the player whose turn it is chooses one game and plays any of their legal moves in that game. The result is that a player can end up making consecutive moves in a single game, if the intervening move took place in the other game. In a partizan game, such as ordinary C HESS, white’s options from any given position are different from black’s options. When white makes two consecutive moves on the same board, they are two consecutive moves of white pieces only. In contrast, in an impartial game such as N IM, a move by either player could have been made by the other player, if the other player had the chance. Beyond the ability to distinguish between impartial and partizan games, what are disjunctive sums good for? The answer is that these sums arise naturally when positions decompose, in the course of play, into smaller independent subgames.


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Example 6.6. The final position in Example 6.2 can be represented as a disjoint union of two DAWSON ’ S C HESS boards, one with one file and one with three,

except that now it is black’s turn to move. Similarly, an initial move at either end of the board obliterates the two columns at that end, leaving the other player to move. In addition, any move on a 3 × 1 or 3 × 2 board obliterates the entire board, as does a move on the middle file of a 3 × 3 board. This description implies that DAWSON ’ S C HESS is a heap game, by restricting to the ordinary (non-capturing) pawn moves. Indeed, any connected board in its initial position is a heap whose size is its number of files (that is, its width), and any position between bloodbaths is a disjunctive sum of such boards. The rules allow any player to • eliminate any heap of size 1, 2, or 3; • take two or three from any heap of size at least 3; or • split any heap of size d ≥ 3 into two heaps of sizes k and d − 3 − k. Dawson’s choice for the ending of his fairy chess game was both unfortunate and fortuitous. It was unfortunate because it made the game hard: over three quarters of a century after Dawson published his little game, its solution remains elusive, both computationally and in a closed form akin to Bouton’s N IM solution. Open Problem 6.7. Determine a winning strategy for DAWSON ’ S C HESS and find a polynomial-time algorithm to calculate it. Dawson’s choice was fortuitous because the simple change of ending uncovered a remarkable phenomenon: the vast difference between trying to lose and trying to win. Definition 6.8. Given a finite combinatorial game, the misère play version declares the winner to be the player who does not move last. Thus misère play is what happens when both players try to lose under the normal play rules. It fosters amusing titles such as “Advances in losing” [Pla09]. DAWSON ’ S C HESS motivated substantial portions of the development of CGT over the past few decades. Misère games are generally much more complex than their normal-play counterparts. Heuristically, the reason is that, in contrast to the unique “zero position” in normal play, the multiple “penultimate positions” that become winning positions in misère play cause ramifications in positions expanding farther from the zero position, and these ramifications interfere with one another in relatively unpredictable ways. Regardless of the reason, aspects of the fact of misère difficulty were formalized by Conway in the 1970s (see [Con01]). There are, for example, many more nonisomorphic impartial misère games than impartial normal play games of any given

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birthday (the height of the game tree). For comparison, note that a complete structure theory for normal play games was formulated in the late 1930s [Spr36, Gru39]. It is based on the Sprague–Grundy theorem, building on Bouton’s solution of N IM by reducing all finite impartial games to it: every impartial game under normal play is, in a precise sense, equivalent to a single N IM heap of some size. (The details of this theory would be more appropriate for a focused exposition on the foundations of CGT, such as Siegel’s highly recommended lecture notes [Sie06], which proceed quickly to the substantive aspects from an algebraic perspective. Additional background and details can be found in [ANW07, BCG82].) Because the “zero position” is declared off-limits in misère structure theory, the elegant additivity of normal play under disjunctive sum fails for misère play, and what results is algebraically complicated in that case; see Sect. 6.4.

6.2 Lattice Games Impartial combinatorial games admit a reformulation in terms of lattice points in polyhedra. For the purpose of Open Problem 6.7, the idea is to bring to bear the substantial algorithmic theory of rational polyhedra [BW03]. The transformation begins by a simple change of perspective on N IM, DAWSON ’ S C HESS, and other heap games. Example 6.9. For certain families of games, the game tree is an inefficient encoding. For heap games, it is better to arrange the numbers of heaps of each size into a nonnegative integer vector whose ith entry is the number of heaps of size i. Thus the and positions from Example 6.4 become

The moves “make a heap of size j into a heap of size i < j” and “remove a heap of size j” correspond to other (not necessarily positive) integer vectors, namely (. . . , 1, . . . , −1, . . .) = ei − e j for i < j, and (. . . , −1, . . .) = −e j for all j ≥ 1, where e1 , . . . , ed is the standard basis of Zd . All entries in the moves are zero except for the 1 and −1 entries indicated. N IM looks curiously like it could be connected to root systems of type A, but nothing has been made of this connection. Example 6.9 says that N IM positions with heaps of size at most d are points in Nd , and moves between them are vectors in Zd . The idea behind lattice games is to polyhedrally formalize the relationship between the positions and moves. To that end, for the rest of this section, fix a pointed rational cone C ⊆ Zd of dimension d, and write Q = C ∩ Zd for the normal affine semigroup of integer points in C. As in earlier sections, basic knowledge of polyhedra is assumed; see [Zie95] for additional


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background. For simplicity, this survey restricts attention to lattice games that are played on (the lattice points in) cones instead of arbitrary polyhedra, and to special rule sets for which every position has a path to the origin (cf. [GM10, Lemma 3.5]); see [GM10, §2] for full generality. Briefly, a lattice game is played by moving a token on a game board comprising all but finitely many of the lattice points in a polyhedral cone. The allowed moves come from a finite rule set consisting of vectors that generate a pointed cone containing the game board cone. The game ends when no legal moves are available; the winner is the last player to move. The misère condition is encoded by the finitely many disallowed lattice point positions. To define these properly, it is necessary to define rule sets first. Definition 6.10. A rule set is a finite subset Γ ⊂ Zd {0} such that (a) the affine semigroup NΓ is pointed, meaning that its unit group is trivial, and (b) every lattice point p ∈ Q has a Γ -path to 0 in Q, meaning a sequence 0 = p0 , . . . , pr = p in Q, with pi+1 − pi ∈ Γ , as illustrated in the following figure.

With these conventions, moves correspond to elements of −Γ rather than of Γ itself. The sign is a choice that must be made, and neither option is fully convenient. The choice in Definition 6.10 prevents unpleasant signs in the next lemma. Lemma 6.11. NΓ contains Q and induces a partial order on Zd in which p q whenever q − p ∈ NΓ . Proof. The containment is immediate from Definition 6.10. The partial order occurs because NΓ is a pointed affine semigroup. Definition 6.12. A lattice game played on a normal affine semigroup Q = C ∩Zd has • a rule set Γ , • defeated positions D ⊆ Q that constitute a finite Γ -order ideal, and • game board B = Q D.

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Lattice points in Q are referred to as positions; note that these might lie off the game board. A position p ∈ Q has a move to q if p−q ∈ Γ ; the move is legal if q ∈ B. The order ideal condition, which means by definition that q ∈ D ⇒ p ∈ D if p q, guarantees that a legal move must originate from a position on the game board B. Example 6.13. A heap game in which the heaps have size at most d is played on Q = Nd . Under normal play, the game board is all of Nd , so D = ∅. To get misère play, let the set of defeated positions be D = {0}, so B = Nd {0}. Larger sets of defeated positions allow generalizations of misère play not previously considered. Example 6.14. DAWSON ’ S C HESS is a lattice game on Nd when the heap sizes are bounded by d. The game board in this case is Nd {0}, corresponding to misère play. The rule set is composed of the following vectors, by Example 6.6: • e1 ; • e2 and e j − e j−2 for j ≥ 3; • e3 and e j − e j−3 for j ≥ 4 and e j − ei − e j−3−i for j ≥ 5. Historically, the abstract theory of combinatorial games was developed more with set theory than combinatorics. Formally, a finite impartial combinatorial game is often defined as a set consisting of its options, each being, recursively, a finite impartial combinatorial game. Using this language, the disjunctive sum of games G and H is the game G+H whose options comprise the union of {G +H | G is an option of G} and {G+H | H is an option of H}. A set of games is closed if it is closed under taking options and under disjunctive sum. In particular, the closure of a single game G is the free commutative monoid on G and its followers, meaning the games obtained recursively as an option, or an option of an option, etc. See [PS07] and its references for more details on closure and on the historical development of CGT. Theorem 6.15. Any position in a lattice game determines a finite impartial combinatorial game. Conversely, the closure of an arbitrary finite impartial combinatorial game, in normal or misère play, can be encoded as a lattice game played on Nd . Proof. The first sentence is a consequence of Lemma 6.11. The second is [GM10, Theorem 5.1]: if the game graph has d nodes, then lattice points in Q = Nd correspond to disjunctive sums of node-positions. The proof of Theorem 6.15 clarifies an important point about the connection between lattice games and games given by graphs: lattice game encodings are efficient only when the nodes of the graph represent “truly different” positions. N IM game in Example 6.4 by using all d Example 6.16. The encoding of the of the followers of as coordinate directions in Nd is woefully inefficient. On position is encoded efficiently in N7 because it lies in the the other hand, the closure of the single N IM heap of size 7, whose followers are “truly different” from one another. This explains part of the reason for allowing arbitrary normal affine semigroups as game boards: more classes of combinatorial games beyond heap games can be


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encoded efficiently. That said, heap games are now—and have been for decades— key sources of motivation and examples. As such, the encoding of heap games is particularly efficient for the following class of games [GM10, §6–§7]. Definition 6.17. A lattice game is squarefree if it is played on Nd and the maximum entry of any vector in the rule set is 1. Equivalently, a squarefree game represents a heap game in which each move destroys at most one heap of each size. Multiple heaps of different sizes can be destroyed, and a destroyed heap can be replaced with multiple heaps of other sizes, as long as the moves still form a rule set. Example 6.18. Squarefree games are the natural limiting generalizations of octal games, invented by Guy and Smith [GS56 ] with DAWSON ’ S C HESS as a motivating example. For each k ∈ N, an octal game specifies whether or not any heap of size • k may be destroyed; • j ≥ k + 1 may be turned into a heap of j − k; and • j ≥ k + 2 may be turned into two heaps of sizes summing to j − k. These constitute three binary choices, and hence are conveniently represented by an octal digit 0 ≤ dk ≤ 7. DAWSON ’ S C HESS is “.137” as an octal game, where the digits correspond, in order, to the types of moves in Example 6.14. For example, 7 = 111 in binary indicates that all options are allowed for k = 3, while 3 = 011 indicates that only the top two are available for k = 2. The dot in “.137” is a place holder.

6.3 Rational Strategies What does a strategy for a combinatorial game look like? Abstractly, the finiteness condition ensures that one of the players can force a win. The argument explaining why is recursive and elementary. But how does one describe such a strategy? Lattice games provide malleable data structures for this purpose. Definition 6.19. Two subsets W, L ⊆ B are winning and losing positions for a lattice game with game board B if • B = W ∪· L is the disjoint union of W and L; and • (W + Γ ) ∩ B = L. Winning positions are the desired spots to move to. The first condition says that every position in B is either a winning position (the player who moved to that spot can force a win) or a losing position (the player who moves from that spot can force a win by moving to a winning position). The second condition says that losing positions are precisely those with (legal) moves to winning positions. Example 6.20. Consider IM2 of N IM with heaps of size at most 2. The the game N rule set in this case is [ 10 ], [ 01 ], [ −11 ] . The negatives of these vectors—representing the legal moves from a generic position—are depicted in the following figure, along with the winning positions in N IM2 for both for normal and misère play. An easy

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way to verify that the depicted sets W are forced is to figure out what happens on the bottom row first, from left to right, and then proceed upward, row by row.

The defeated position, labeled by D in the misère play diagram, causes the bottom row of winning positions to be shifted over one unit to the right. Remark 6.21. The disarray caused by the defeated position in Example 6.20 becomes substantially worse with more complicated rule sets in higher dimensions. Much of the study of misère combinatorial games amounts to analyzing, quantifying, computing, and controlling the disarray. Theorem 6.22. Given a lattice game with rule set Γ ⊂ Zd and game board B, there exist unique sets W and L of winning and losing positions for B. Proof. This is [GM10, Theorem 4.6]. The main point is that the cones generated by Q and NΓ point in the same direction, so recursion is possible after declaring the Γ -minimal positions in B to be winning. Thus everything there is to know about a lattice game is encoded by its set of winning positions: given a pointed normal affine semigroup, Theorem 6.22 implies that specifying a rule set and defeated positions is the same as specifying a valid set of winning positions, at least abstractly. But the rule set encodes winning strategies only implicitly, while the set of winning positions—or better, a generating function fW (t) = ∑w∈W tw for the winning positions—encodes the strategy explicitly. The following is [GM10, Conjecture 8.5]. Conjecture 6.23. Every lattice game has a rational strategy: a generating function for its winning positions expressed as a ratio of polynomials with integer coefficients. Example 6.24. Resume Example 6.20. In normal play N IM2 , a rational strategy is fW (a, b) =

1 , (1 − a2 )(1 − b2 )

the rational generating function for the affine semigroup 2N2 . In misère play, a rational strategy is


fW (a, b) =

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a b2 + , 1 − a2 (1 − a2 )(1 − b2 )

where the first term enumerates the odd lattice points on the horizontal axis, and the second enumerates normal play winning positions that lie off the horizontal axis. Example 6.25. For a squarefree game, if W0 = W ∩ {0, 1}d ⊆ Nd then W = W0 + 2Nd . This is [GM10, Theorem 6.11]. It implies that the game has a rational strategy fW (t) =

tw . 2 2 w∈W0 (1 − t1 ) · · · (1 − td )

∑

The reader is encouraged to reconcile this statement with Example 6.1. A rational strategy has a reasonable claim to the title of “solution to a lattice game” since it can be manipulated algorithmically and has potential to be compact. Theorem 6.26. A rational strategy for a lattice game produces algorithms to • determine whether a position is winning or losing, and • compute a legal move to a winning position, given any losing position. These algorithms are efficient when the rational strategy is a short rational function, in the sense of Barvinok and Woods [BW03]. Proof. This is a straightforward application of the theory developed by Barvinok and Woods; see [GM10 ] for details. The efficiency in the theorem is in the sense of complexity theory. Short rational generating functions have not too many terms in their numerator and denominator polynomials. They are algorithmically efficient to manipulate and—when they enumerate lattice points in polyhedra—to compute. Since computations of lattice points in rational polyhedra are efficient, it would be better to get a polyhedral decomposition of the set W of winning positions. In fact, examples of lattice games exhibit a finer structural phenomenon than is indicated [GM10, Conjecture 8.9] & [GMW09]. Conjecture 6.27. Every lattice game has an affine stratification: an expression of its winning positions as a finite union of translates of affine semigroups. Roughly speaking, winning positions should be finite unions of sets of the form (lattice ∩ cone). This definition of affine stratification differs from [GM10, Definition 8.6] but is equivalent [Mil10, Theorem 2.6]; it would also be equivalent to require the union to be disjoint, or (independently of disjointness) the affine semigroups to be normal. Example 6.28. Consider again the situation from Examples 6.20 and 6.24. An affine positions stratification for this game is W = 2N2 ; that is, the entire set of winning forms an affine semigroup. In misère play, W = (1, 0) + N(2, 0) ∪· (0, 2) + 2N2

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is the disjoint union of W1 = 1 + 2N (along the first axis) and W2 , which equals the translate by twice the second basis vector of the affine semigroup 2N2 . Remark 6.29. Conjecture 6.27 bears a resemblance to statements about local cohomology of finitely generated ZQ-graded modules M over an affine semigroup ring k[Q] with support in a monomial ideal: the local cohomology HIi (M) is supported on a finite union of translates of affine semigroups [HM05]. If Conjecture 6.27 is true, then perhaps it would be possible to develop a homological theory for winning positions in combinatorial games that explains why. Theorem 6.30. A rational strategy can be efficiently computed from any given affine stratification. Proof. See [GM10 , §5]. As with Theorem 6.26, this is reasonably straightforward, applying the methods from [BW03] with care. Algorithms for dealing with affine stratifications and rational strategies are stepping stones toward a higher aim, which would be to prove the existence of affine stratifications (Conjecture 6.27), and hence rational strategies (Conjecture 6.23), in an efficient algorithmic manner and in enough generality for DAWSON ’ S C HESS (Open Problem 6.7). Part of the problem to overcome for DAWSON ’ S C HESS is the need to deal with increasing heap size d. That problem is key, since the algorithm for DAWSON ’ S C HESS is supposed to be polynomial in d as d → ∞, but affine stratifications and rational strategies at present are designed for fixed d.

6.4 Misère Quotients The development of lattice games and rational strategies outlined in the previous subsections were motivated by—and continue to take cues from—exciting recent advances in misère theory by Plambeck and Siegel [Pla05, PS07] pertaining to misère quotients (Definition 6.31). The lattice point methods are also beginning to return the favor, spawning new effective methods for misère quotients. This final subsection on combinatorial games ties together the lattice point perspectives on games and binomial ideals with misère quotients, particularly in Theorem 6.36. Definition 6.31. Fix a lattice game with winning positions W ⊆ Q in a pointed normal affine semigroup. Two positions p, q ∈ Q are indistinguishable, written p ∼ q, if (p + Q) ∩W − p = (q + Q) ∩W − q. In other words, p + r ∈ W ⇔ q + r ∈ W for all r ∈ Q. The misère quotient of G is the quotient Q = Q/∼ of the affine semigroup Q modulo indistinguishability. Geometrically, indistinguishability means that the winning positions in the cone above p are the same as those above q, up to translation by p − q.


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Lemma 6.32. Indistinguishability is a congruence in the sense of Definition 3.12, so the misère quotient Q is a monoid. Example 6.33. In the situation of Example 6.25, the misère quotient is a quotient of (Z/2Z)d = Nd /2Nd [GM10, Proposition 6.8]. As with Example 6.25, the reader is encouraged to reconcile this statement with Example 6.1. Example 6.34. For N IM2 (Examples 6.20, 6.24, and 6.28), the misère quotient is the commutative monoid with presentation Q = a, b | a2 = 1, b3 = b , in multiplicative notation. This monoid has six elements because it is a | a2 = 1 × b | b3 = b , and the second factor has order 3. The presentation of Q can be seen geometrically in the right-hand figure from Example 6.20: translating the grid two units to the right moves W bijectively to the part of W outside of the leftmost two columns (this is a2 = 1), and translating the grid up by two units takes the part of W above the first row bijectively to the part of W above the third row (this is b3 = b). Misère quotients were introduced by Plambeck [Pla05] as a less stringent way to collapse the set of games than had been proposed earlier by Grundy and Smith [GS56], in view of Conway’s proof that very little simplification results when the collapsing is attempted in too large a universe of games [Con01, Theorem 77]. Misère quotients have subsequently been studied and applied to computations by Plambeck and Siegel [PS07, Sie07], the point being that taking quotients often leaves a much smaller—and sometimes finite—set of positions to consider, when it comes to strategies. The Introduction of [PS07] contains an excellent account of the history, including personal accounts from some of the main players. The first contribution of lattice games to misère theory is the following. Proposition 6.35. A short rational strategy for a game played on Q results in an efficient algorithm for determining the indistinguishability of any pair of positions in Q. In particular, an affine stratification results in such an algorithm. Proof. This is the main result in [GM10 , §6]. If f = ∑q∈Q ϕq tq and g = ∑q∈Q ψq tq are two short rational functions, then their Hadamard product f star g = ∑q∈Q ϕq ψq tq can be efficiently computed as a short rational function [BW03].

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The final result in this section combines three main themes thus far in the survey: lattice games, misère quotients, and—for the proof—binomial combinatorics. Theorem 6.36. Lattice games with finite misère quotients have affine stratifications. Proof. This is [Mil10, Corollary 4.5], given that we work with games played on normal affine semigroups. The proof proceeds via a general result [Mil10, Theorem 3.1] of interest here: the fibers of any projection Q → Q from an affine semigroup Q to a monoid Q all possess affine stratifications. This is proved using combinatorial mesoprimary decompositions of congruences—the (simpler) monoid analogues of binomial mesoprimary decomposition—whose combinatorics, as in Example 4.24, give rise to affine stratifications of fibers. When the misère quotient of a lattice game is finite, the winning positions automatically comprise a finite union of fibers. Open Problem 6.37. Find an algorithm to compute the misère quotient of any lattice game starting from an affine stratification. Algorithms for computing finite misère quotients are known and useful [PS07, Appendix A]. In addition, Weimerskirch has algorithmic methods that apply in the presence of certain known periodicities [Wei08], although for infinite quotients the methods fail to terminate. Binomial primary decomposition, or at least the combinatorial aspects present in mesoprimary decomposition of congruences on monoids, is likely to play a role in further open questions on misère quotients, including when finite quotients occur, and more complex “algebraic periodicity” questions, which have yet to be formulated precisely [PS07, Appendix A.5].

7 Mass-Action Kinetics in Chemistry Toss some chemicals into a vat. Stir. What products are produced? How fast? If the process is repeated, can the result differ? These questions belong to the study of chemical reaction dynamics. One of the earliest theories of such dynamics, the law of mass action, was formulated by Guldberg and Waage in 1864 [GW1864]. It is widely observed to hold in real-life chemical systems (as distinguished from, say, biochemical systems; see Remark 7.9). Over the years, mass-action kinetics has matured, especially certain mathematical aspects following seminal work by Horn, Jackson, and Feinberg [HJ72, Fei87] from the 1970s and onward. In the past decade, the resulting mathematical formalizations have seen increasing amounts of algebra, particularly of the binomial sort. This section provides a brief overview of mass-action kinetics (Sect. 7.1), covering just enough basics to understand the relevance of binomial algebra. From there, the main goal is to explain the Global Attractor Conjecture (Sect. 7.2), which posits that a system of reversible chemical reactions always reaches the same steady state if one exists, with a view to how binomial primary decomposition could be relevant to its solution.


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Length constraints prevent many substantial details, as well as examples demonstrating key phenomena, from being included. For an elementary introduction to chemical reaction network theory, in mathematical language, the reader is referred to the well-written notes by Gunawardena [Gun03]. For details on an abstract formalization of the law of mass-action in terms of binomials, see [AGHMR09].

7.1 Binomials from Chemical Reactions Before presenting mass-action kinetics in general, it is worthwhile to study a small sample reaction. Example 7.1. Consider the breakdown of hydrogen peroxide into water and oxygen: λ

2H2 O + O2 2H2 O2 μ The λ and μ here are rate constants: λ indicates that two molecules of peroxide decompose into two molecules of water and one molecule of oxygen at some rate, and μ indicates that the reverse reaction also occurs, though at another (in this case, slower) rate: two molecules of water and one molecule of oxygen react to from two molecules of peroxide. To be precise, let x = [H2 O2 ], y = [H2 O], and z = [O2 ] be the concentrations of peroxide, water, and oxygen in some medium. These concentrations are viewed as functions of time, and as such, they satisfy a system of ordinary differential equations: x˙ = 2μ y2 z − 2λ x2 y˙ = 2λ x2 − 2μ y2 z z˙ = λ x2 − μ y2 z. The right-hand side of x˙ says that after an infinitesimal unit of time, • for every two molecules of H2 O2 that came together (this is the x2 term), two molecules of H2 O2 disappear (this is the −2 in the coefficient of x2 ) some fraction of the time (this is the meaning of λ in the coefficient of x2 ); and • for every two H2 O molecules and one O2 that came together (the y2 z term), two molecules of H2 O2 are formed (the 2 on y2 z) some fraction of the time (μ ). The fact that x and y are squared in all of the right-hand sides is for the same reason that y2 is multiplied by z: the products represent concentrations of chemical complexes. Thus y2 z can be thought of as an “effective concentration” of 2H2 O + O2 . Example 7.2. For comparison, it is instructive to see what happens when a new species is introduced to the chemical equation. Suppose that the reaction 2H2 O2 2H2 O + O2 in Example 7.1 had a fictional additional term on the right-hand side:

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The right-hand sides of the equations governing evolution of the species H2 O2 , H2 O, and O2 would remain binomial:

as would the new equation

governing evolution of the species A.

In general, a chemical reaction involves species s1 , . . . , sn with corresponding concentrations [si ] = xi , each viewed as a function xi = xi (t) of time. In Example 7.1, the species are peroxide, water, and oxygen. A reaction A B occurs between chemical complexes A = a1 s1 + · · · + an sn and B = b1 s1 + · · · + bn sn . Thus the complex A is composed of ai molecules of si for i = 1, . . . , n, and similarly for B. In Example 7.1, the complexes are H2 O2 and 2H2 O + O2 . Definition 7.3. A reaction A B between complexes A = a1 s1 + · · · + an sn and B = b1 s1 + · · · + bn sn evolves under mass action kinetics [GW1864] if species si is lost at ai times a rate proportional to the concentration xa11 · · · xnan of A, and gained at bi times a rate proportional to the concentration xb11 · · · xbnn of B. The reaction is reversible if the reaction rates in both directions are strictly positive. As in Example 7.1, the concentration of a complex A is the product of species concentrations with exponents corresponding to the multiplicities of the species in A. Proposition 7.4. The differential equation governing the evolution of the reversible reaction A B from Definition 7.3 under mass-action kinetics is x˙i = (bi − ai )(λ xa − μ xb ), with rate constants λ , μ > 0. In vector form, with λ = λab and μ = λba , this becomes x˙ = (b − a)(λab xa − λba xb ) Proof. This is merely a translation of Definition 7.3 into symbols.

The factor of λ xa − μ xb in Proposition 7.4 is a scalar quantity; the only vector quantity on the right-hand side is b − a. A single reaction under mass-action kinetics reaches a steady state when the binomial on the right-hand side of its evolution equation vanishes. Thus the set of steady states for a single reaction is the zero set of a binomial. General reaction systems involve more than one reaction at a time: in a given vat of chemicals, simultaneous transformations take place involving different pairs of chemical complexes using the given set of species in the vat. Definition 7.5. For multiple reactions on a set of species, in which each reaction A B involves complexes with species vectors a = (a1 , . . . , an ) and b = (b1 , . . . , bn ), the law of mass-action is obeyed if the species evolve according to the binomial sum:


x˙ =

149

∑ (b − a)(λabxa − λba xb ).

AB

Thus, when more than one reaction is involved, the ith entry of the vector field on the right-hand side is not a binomial but a sum of binomials, one for each reaction in which species si occurs. Consequently, the set of steady states need not be binomial [DM10]. Definition 7.6. A point ξ = (ξ1 , . . . , ξn ) ∈ Rn is a detailed balanced equilibrium for a reversible reaction system if it is strictly positive, meaning ξi > 0 for all i, and every binomial summand on the right-hand side in Definition 7.5 vanishes at ξ . A system is detailed balanced if it is reversible and has a detailed balanced equilibrium. Definition 7.6 creates the bridge from chemistry to binomial algebra: the chemical interest lies in equilibria, and these are varieties of binomial ideals. Detailed balanced equilibria lie interior to the positive orthant in Rn . For such concentrations of the species, each reaction A B in the system rests at equilibrium with both A and B present at some nonzero concentration. In fact, more is true. Theorem 7.7. A detailed balanced equilibrium is a locally attracting steady state. Proof. The main point is that a detailed balanced equilibrium possesses an explicit strict Lyapunov function (given by Helmholtz free energy) [HJ72, Fei87]. Remark 7.8. Chemical reaction network theory (CRNT) works in more general settings than systems of reactions each of which is reversible; see [Gun03] for an introduction. The theory is most successful when the system of reactions is weakly reversible: each individual reaction A → B need not be reversible, but it must be possible to reach the reactant complex A from the product complex B through a sequence of reactions in the system. See also Remark 7.11. Remark 7.9. Mass-action kinetics fails for more complicated chemical systems, such as biochemical ones. Indeed, it must fail, for life abhors chemical equilibrium: an organism whose chemical reactions are at steady-state is otherwise known as dead. The failure of mass-action kinetics in biochemical systems occurs for a number of reasons. For one, the reaction medium is not homogeneous—that is, the reactants are not well-mixed). In addition, the molecules are often too big, and the number of them too small, for the natural discreteness to be smoothed; see [Gun03, §2].

7.2 Global Attractor Conjecture Polynomial dynamical systems—linear ordinary differential equations with polynomial right-hand sides—behave quite poorly and unpredictably, in general. The famous chaotic Lorenz attractor, for example, is defined by a vector field whose entries are simple 3-term cubics. However, the binomial nature of mass-action chemistry lends a striking tameness to the dynamics.

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The reversibility hypothesis for detailed balanced systems is natural from the perspective of chemistry: every reaction can, in principle, be reversed (although the activation energy required might be prohibitive under standard conditions). Overwhelming experience says that typical chemical reactions—well-mixed, at constant temperature, as in chemical manufacturing—approach balanced steady states, and the same products really do emerge every time. But this is surprisingly unknown theoretically for detailed balanced systems under mass-action kinetics, even though their equilibria are local attractors by Theorem 7.7. Conjecture 7.10 (Global Attractor Conjecture [HJ72, Hor74]). If a reversible reaction system as in Definition 7.5 has a detailed balanced equilibrium, then every trajectory starting from strictly positive initial concentrations reaches it in the limit. The Global Attractor Conjecture is “the fundamental open question in the field” [AGHMR09, §1], since it would close the book on fundamentally justifying massaction kinetics. It is known that the conjecture holds when the binomial ideal is prime [Gop09]. It is also known, for detailed balanced reaction systems with fixed positive initial species concentrations, that (a) the detailed balanced equilibrium is unique [Fei87], and (b) each trajectory tends toward some equilibrium [Son01, Cha03]. Thus it suffices to bound every strictly positive trajectory away from all boundary equilibria, for then the trajectory limits are forced toward the detailed balanced one. Remark 7.11. Detailed balancing is a stronger hypothesis than required for the known results listed above, including Theorem 7.7. Weak reversibility as in Remark 7.8, or even a less stringent condition, often suffices. Detailed balancing is also weaker than the hypothesis in the strongest (and still widely believed) form of Conjecture 7.10, which stipulates a condition called complex-balancing that implies weak reversibility; see, for instance, [AS09, §4.2] for a precise statement. What do the boundary equilibria look like? The restriction of a detailed balanced reaction system to a coordinate subspace of Rn amounts to forcing the concentrations of some reactant species to be zero. Such restrictions still constitute detailed balanced reaction systems, and the binomials whose vanishing describes the equilibria come from the binomials in the original system. This discussion can be rephrased as follows. Proposition 7.12. Boundary equilibria of detailed balanced reaction systems are zeros of associated primes of the ideal generated by the binomials in Definition 7.5. Conjecture 7.10 holds if and only if every trajectory with positive initial concentrations remains bounded away from the zero set of every associated prime. Proposition 7.12 brings binomial primary decomposition to bear on the chemistry of mass-action kinetics. The details of how this occurs are illustrated by certain special cases of Conjecture 7.10 whose proofs are known. The characterizations of these cases rely on a polyhedral concept hiding in the dynamics.


151

Definition 7.13. The stoichiometric compatibility subspace of the reaction system in Definition 7.5 is the real span S of the vectors a − b over all reactions A B in the system. The stoichiometric compatibility class (or invariant polyhedron) of a species concentration vector ξ ∈ Rn is the intersection of the nonnegative orthant with ξ + S. Lemma 7.14. Trajectories for any reaction system are constrained to lie in the invariant polyhedron of the vector of initial species concentrations. Proof. This is immediate from the equation for x˙ in Definition 7.5.

The known cases of Conjecture 7.10 include all systems whose initial species concentration vectors have invariant polyhedra of dimension 2 or less [AS09, Corollary 4.7]. In the language of Proposition 7.12, the method of proof is to bound all trajectories away from the zero set of every associated prime whose intersection with the relevant invariant polyhedron is • a vertex [And08, CDSS09] or • interior to a facet [AS09]. For a comprehensive review of known cases of the Global Attractor Conjecture, see [AS09, §1 and §4]. The combinatorics of binomial primary decomposition might contribute further than merely the statement of Proposition 7.12. For example, mesoprimary decomposition (Definition 4.23; see [KM10]) provides decompositions of binomial ideals over the rational or real numbers, and therefore takes steps toward primary decomposition over the reals. Mesoprimary decomposition also characterizes the associated lattices combinatorially, without a priori knowing the primary decomposition. Both could be important for applications to the Global Attractor Conjecture: perhaps finiteness conditions surrounding associated lattices (Example 4.24) indicates how to produce the desired trajectory bounds, with the reality (i.e., defined over R) of the components forcing progress away from the boundary, as opposed to (say) periodicity of some kind. In an amazing convergence, graphs associated to event systems [AGHMR09, Definition 2.9] provide chemical interpretations of the graphs GI from Sects. 3 and 4 (particularly Definition 3.16) in this survey. Reversibility of the reaction system means that it is correct for GI to be undirected. The characterization of naturality in [AGHMR09, Theorem 5.1] is a condition on the primary decomposition of the event ideal. This convergence is cause for optimism that lattice-point point combinatorics will be instrumental in proving Conjecture 7.10 via Proposition 7.12.

References [AGHMR09] Leonard Adleman, Manoj Gopalkrishnan, Ming-Deh Huang, Pablo Moisset, and Dustin Reishus, On the mathematics of the law of mass action, SIAM Review, to appear, 2009.

152 [And08] [AS09] [App1880]

[ANW07] [AM69] [BW03] [BCG82]

[Bou1902] [Cha03]

[Con01] [CDSS09] [Daw34] [DMM07]

[DMM10] [DMM10 ] [DM10] [DSS09] [ES96] [Erd50] [Fei87]

[GGZ87] [GKZ89]

[Gil84]

E. Miller David F. Anderson, Global asymptotic stability for a class of nonlinear chemical equations, SIAM J. Appl. Math. 68, no. 5 (2008), 1464–1476. David F. Anderson and Anne Shiu, The dynamics of weakly reversible population processes near facets, preprint. arXiv:math.DS/0903.0901 Paul Appell, Sur les séries hypergéometriques de deux variables et sur des e´ quations différentielles linéaires aux dérivées partielles, Comptes Rendus 90 (1880), 296– 298. Michael H. Albert, Richard J. Nowakowski, and David Wolfe, Lessons in play: An introduction to the combinatorial theory of games, A K Peters, Wellesley, 2007. M.F. Atiyah and I.G. Macdonald, Introduction to commutative algebra, Addison– Wesley, Reading, MA–London–Don Mills, ON, 1969. Alexander Barvinok and Kevin Woods, Short rational generating functions for lattice point problems, J. Amer. Math. Soc. 16, no. 4 (2003), 957–979. (electronic) Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy, Winning ways for your mathematical plays. Vol. 1: Games in general, Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers), London–New York, 1982. Charles L. Bouton, Nim, a game with a complete mathematical theory, Ann. of Math. (2) 3, no. 1–4 (1901/02), 35–39. Madalena Chavez, Observer design for a class of nonlinear systems, with applications to chemical and biological networks, Ph.D. Thesis, Rutgers University, New Brunswick, NJ, 2003. John H. Conway, On numbers and games, Second edition, A K Peters, Ltd., Natick, MA, 2001. Gheorghe Craciun, Alicia Dickenstein, Anne Shiu, and Bernd Sturmfels, Toric dynamical systems, J. Symbolic Computation 44 (2009), 1551–1565. Thomas Dawson, Fairy Chess Supplement, The Problemist: British Chess Problem Society 2, no. 9 (1934), p. 94, Problem No. 1603. Alicia Dickenstein, Laura Felicia Matusevich, and Ezra Miller, Extended abstract: Binomial D-modules, Proceedings MEGA (Effective Methods in Algebraic Geometry), Strobl, Austria, 2007, 13 pages. http://www.ricam.oeaw.ac.at/mega2007/ electronic/electronic.html Alicia Dickenstein, Laura Felicia Matusevich, and Ezra Miller, Combinatorics of binomial primary decomposition, Mathematische Zeitschrift 264, no. 4 (2010), 745– 763. Alicia Dickenstein, Laura Felicia Matusevich, and Ezra Miller, Binomial Dmodules, Duke Mathematical Journal 151, no. 3 (2010), 385–429. Alicia Dickenstein and Mercedes Pérez Millán, How far is complex balancing from detailed balancing?, preprint. arXiv:math.DS/1001.0947 Mathias Drton, Bernd Sturmfels, and Seth Sullivant, Lectures on algebraic statistics, Oberwolfach seminars, vol. 39, Springer, (Birkhäuser, Basel), 2009. David Eisenbud and Bernd Sturmfels, Binomial ideals, Duke Math. J. 84, no. 1 (1996), 1–45. Arthur Erdélyi, Hypergeometric functions of two variables, Acta Math. 83 (1950), 131–164. Martin Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors, I. The deficiency zero and deficiency one theorems, Chem. Eng. Sci., 42, no. 10 (1987), 2229–2268. I.M. Gelfand, M.I. Graev, and A.V. Zelevinski˘ı, Holonomic systems of equations and series of hypergeometric type, Dokl. Akad. Nauk SSSR 295, no. 1 (1987), 14–19. I.M. Gelfand, A.V. Zelevinski˘ı, and M.M. Kapranov, Hypergeometric functions and toric varieties, Funktsional. Anal. i Prilozhen. 23, no. 2 (1989), 12–26. Correction in ibid, 27, no. 4 (1993), 91. Robert Gilmer, Commutative semigroup rings, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1984.

Theory and Applications of Lattice Point Methods for Binomial Ideals [Gop09] [Gru39] [GS56] [GW1864] [Gun03]

[GM10] [GM10 ] [GMW09]

[GS56 ] [HM05] [HJ72] [Hor74]

[Hor1889] [Hor31] [HS00]

[KM10] [Kum1836] [Mel21] [MMW05] [Mil02] [Mil09] [Mil10] [MS05] [Pla05]

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Manoj Gopalkrishnan, An algebraic generalization of the atomic hypothesis, preprint, 2009. Patrick M. Grundy, Mathematics and games, Eureka 2 (1939), 6–8; reprinted 27 (1964), 9–11. Patrick M. Grundy and C.A.B. Smith, Disjunctive games with the last player losing, Proc. Cambridge Philos. Soc. 52 (1956), 527–533. Cato M. Guldberg and Peter Waage (translated by H.I. Abrash), Studies concerning affinity, Journal of chemical education 63 (1986), 1044–1047. Jeremy Gunawardena, Chemical reaction network theory for in-silico biologists, preprint, 2003. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.121. 126&rep=rep1&type=pdf Alan Guo and Ezra Miller, Lattice point methods for combinatorial games, Adv. in Appl. Math., 19 pages, to appear. arXiv:math.AC/0908.3473 Alan Guo and Ezra Miller, Algorithms for lattice games, in preparation, 2010. Alan Guo, Ezra Miller, and Mike Weimerskirch, Potential applications of commutative algebra to combinatorial game theory, in Kommutative Algebra, abstracts from the April 19–25, 2009 workshop, organized by W. Bruns, H. Flenner, and C. Huneke, Oberwolfach rep. 22 (2009), 23–26. R.K. Guy and C.A.B. Smith, The G-values of various games, Proc. Cambridge Philos. Soc. 52 (1956), 514–526. David Helm and Ezra Miller, Algorithms for graded injective resolutions and local cohomology over semigroup rings, J. Symbolic Computation 39 (2005), 373–395. Friedrich Horn and Roy Jackson, General mass action kinetics, Arch. Ration. Mech. Anal., 47, no. 2 (1972), 81–116. Friedrich Horn, The dynamics of open reaction systems, in Mathematical aspects of chemical and biochemical problems and quantum chemistry (Proc. SIAM–AMS Sympos. Appl. Math., New York, 1974), pp. 125–137. SIAM–AMS Proceedings, vol. VIII, Amer. Math. Soc., Providence, R.I., 1974. ¨ J. Horn, Uber die konvergenz der hypergeometrischen Reihen zweier und dreier Veränderlichen, Math. Ann. 34 (1889), 544–600. J. Horn, Hypergeometrische Funktionen zweier Veränderlichen, Math. Ann. 105, no. 1 (1931), 381–407. Serkan Hos¸ten and Jay Shapiro, Primary decomposition of lattice basis ideals, J. Symbolic Comput. 29, no. 4–5 (2000), 625–639, Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998). Thomas Kahle and Ezra Miller, Decompositions of commutative monoid congruences and binomial ideals, in preparation, 2010. ¨ Ernst Eduard Kummer, Uber die hypergeometrische Reihe F(α , β , x), J. Reine Angew. Math. 15 (1836). Hjalmar Mellin, Résolution de l’équation algébrique générale a` l’aide de la fonction Γ , C.R. Acad. Sc. 172 (1921), 658–661. Laura Felicia Matusevich, Ezra Miller, and Uli Walther, Homological methods for hypergeometric families, J. Amer. Math. Soc. 18, no. 4 (2005), 919–941. Ezra Miller, Cohen–Macaulay quotients of normal semigroup rings via irreducible resolutions, Math. Res. Lett. 9, no. 1 (2002), 117–128. Ezra Miller, Alexander duality for monomial ideals and their resolutions, Rejecta Mathematica 1, no. 1 (2009), 18–57. arXiv:math.AC/9812095 Ezra Miller, Affine stratifications from finite misère quotients, preprint, 2010. arXiv:math.CO/1009.2199 Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. Thane E. Plambeck, Taming the wild in impartial combinatorial games, Integers 5, no. 1 (2005), G5, 36 pp. (electronic)

154 [Pla09]

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[Spr36] [SK85]

[Wei08] [Zie95]

E. Miller Thane E. Plambeck, Advances in losing, In Games of no chance 3, papers from the Combinatorial Game Theory Workshop held in Banff, AB, June 2005, edited by Michael H. Albert and Richard J. Nowakowski, MSRI Publications, Cambridge University Press, Cambridge, forthcoming. arXiv:math.CO/0603027 Thane E. Plambeck and Aaron N. Siegel, Misère quotients for impartial games, J. Combin. Theory Ser. A 115, no. 4 (2008), 593–622. arXiv:math.CO/0609825v5 Mathias Schulze and Uli Walther, Irregularity of hypergeometric systems via slopes along coordinate subspaces, Duke Math. J. 142, no. 3 (2008), 465–509. Aaron N. Siegel, Misère games and misère quotients, unpublished lecture notes. arXiv:math.CO/0612616 Aaron N. Siegel, The structure and classification of misère quotients, preprint. arXiv:math.CO/0703070 Eduardo Sontag, Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction, IEEE Trans. Automat. Control, 46 (2001), 1028–1047. ¨ Roland P. Sprague, Uber mathematische Kampfspiele [On mathematical war games], Tôhoku Math. Journal 41 (1935–1936), 438–444. H.M. Srivastava and Per W. Karlsson, Multiple Gaussian hypergeometric series, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, 1985. Michael Weimerskirch, An algorithm for computing indistinguishability quotients in misère impartial combinatorial games, preprint, 31 August 2008. Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995.

Equations Defining Secant Varieties: Geometry and Computation Jessica Sidman and Peter Vermeire

Abstract In the 1980’s, work of Green and Lazarsfeld (Invent. Math., 83, 1 (1985), 73–90; Compositio Math., 67, 3 (1988), 301–314), helped to uncover the beautiful interplay between the geometry of the embedding of a curve and the syzygies of its defining equations. Similar results hold for the first secant variety of a curve, and there is a natural conjectural picture extending to higher secant varieties as well. We present an introduction to the algebra and geometry used in (Sidman and Vermeire, Algebra Number Theory, 3, 4 (2009), 445–465) to study syzygies of secant varieties of curves with an emphasis on examples of explicit computations and elementary cases that illustrate the geometric principles at work.

1 Introduction Intuition about the behavior of equations defining a secant variety embedded in projective space arises from consideration of both algebra and geometry, and our main goal is to bring together some of these ideas. In this paper we will be concerned with syzygies of secant varieties of smooth curves. We will begin with the algebraic point of view with the aim of giving the reader tools for computing secant varieties of curves using Macaulay 2 [9]. We then turn to the geometric point of view, based on work of Aaron Bertram [1], which led to the second author’s original conjectures on cubic generation of secant ideals and linear syzygies [26]. These conjectures were refined and strengthened in [20] using Macaulay 2 [9]. Bertram’s setup is also used by Ginensky [7] to study determinantal equations for curves and their secant Jessica Sidman Department of Mathematics and Statistics, Mount Holyoke College, South Hadley, MA 01075, USA e-mail: [email protected] Peter Vermeire Department of Mathematics, 214 Pearce, Central Michigan University, Mount Pleasant, MI 48859, USA e-mail: [email protected] G. Fløystad et al. (eds.), Combinatorial Aspects of Commutative Algebra and Algebraic Geometry, Abel Symposia 6, DOI 10.1007/978-3-642-19492-4 9, © Springer-Verlag Berlin Heidelberg 2011

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varieties. Our hope is to make some of the geometric intuition accessible to readers familiar with [3] and [12], and that the examples we discuss will be of help in reading the existing literature. We want to study the minimal free resolution of the homogeneous coordinate ring of a secant variety. If X ⊂ Pn is a variety, by which we mean a reduced, but not necessarily irreducible, scheme, then we define its kth secant variety, denoted Σ k , to be the Zariski closure of the union of the k-planes in Pn meeting X in at least k + 1 points. We often write Σ for Σ 1 . We are primarily interested in the situation in which the maps Γ (X, OX (k)) → Γ (Pn , OPn (k)) are surjective, or equivalently that the homogeneous coordinate ring SX is normally generated, as then SX ∼ = ⊕Γ (X, OX (k)) and geometric techniques can be used to study SX . We will assume throughout that if X ⊂ Pn is a curve, then it is embedded via a complete linear system. Two examples of notions that have geometric and algebraic counterparts are Castelnuovo–Mumford regularity, or regularity, and the Cohen–Macaulay property, both of which can be defined algebraically in terms of minimal free resolutions. We begin algebraically, and let M be a finitely generated graded module over the standard graded ring S = k[x0 , . . . , xn ]. The module M has a minimal free resolution 0 → ⊕ j S(− j)βn, j → · · · → ⊕ j S(− j)β1, j → ⊕ j S(− j)β0, j → M → 0. The computer algebra package Macaulay 2 [9] computes minimal free resolutions and displays the graded Betti numbers βi, j in a Betti table arranged as below 0

1

2

0 β0,0 β1,1 1 β0,1 β1,2 .. .

···

j

β2,2 · · · β j, j β2,3 · · · β j,1+ j

i β0,i β1,i+1 β2,i+2 · · · β j,i+ j

Definition 1.1. The regularity of a finitely generated graded module M is the maximum d such that some β j,d+ j is nonzero. Example 1.2 (The graded Betti diagram of a curve, Example 1.4 in [20]). For example, we can compute the graded Betti diagram of a curve of genus 2 embedded in P7 using Macaulay 2. 0 total: 1 0: 1 1: . 2: .

1 19 . 19 .

2 58 . 58 .

3 75 . 75 .

4 44 . 44 .

5 6 11 2 . . 5 . 6 2

As the diagram shows that β5,7 and β6,8 are nonzero, we see that the regularity of the homogeneous coordinate ring is 2 and the homogeneous ideal of the curve has regularity 3. We will discuss this computation in greater depth in Example 2.3.

Equations Defining Secant Varieties: Geometry and Computation

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The regularity of the geometric counterpart of a finitely generated graded module, a coherent sheaf on projective space, has a geometric definition which originally appeared on pg. 99 of [18]. Definition 1.3. The regularity of a coherent sheaf F on Pn is defined to be the infimum of all d such that H i (Pn , F (d − i)) = 0 for all i > 0. If M = ⊕ j≥0Γ (Pn , F ( j)), then the regularity of M is the maximum of the regularity of F and zero. The reader will find a good discussion in Chap. 4 of [3]. It is well-known that if X ⊆ Pn is a smooth curve of genus g and degree d ≥ 2g + 1 then the regularity of IX is 2 if X is a rational normal curve and 3 otherwise, and the reader may find a nice exposition in [3]. A similar result is true for the first secant variety of a smooth curve. Remark 1.4. As this article was going to press, we learned from Adam Ginensky and Mohan Kumar that the image of a normal variety over an algebraically closed field under a proper morphism with reduced, connected fibers may fail to be normal. This invalidates the proof of the normality of Σ in Lemma 3.2 in [24]. The arguments in [24] and the subsequent papers [20, 26] go through under the additional hypothesis that Σ is normal, which we add below in Theorems 1.5 and 1.8. Theorem 1.5 ([20, 26]). Let X ⊆ Pn be a smooth curve of genus g and degree d ≥ 2g + 3. If Σ is normal, the regularity of IΣ is 3 if X is a rational normal curve and 5 otherwise. Moreover, it is natural to conjecture: Conjecture 1.6 ([20, 26]). Let X ⊆ Pn be a smooth curve of genus g and degree d ≥ 2g + 2k + 1. The regularity of IΣk is 2k + 1 if X is a rational normal curve and 2k + 3 otherwise. This conjecture holds for genus 0 curves, as the kth secant variety of a rational normal curve has ideal generated by the maximal minors of a matrix of linear forms, and thus the ideal is resolved by an Eagon–Northcott complex. The result for genus 1 is proved in [5, 8]. Definition 1.7. We say that a variety X ⊂ Pn is arithmetically Cohen–Macaulay if the depth of the irrelevant maximal ideal of S = k[x0 , . . . , xn ] on SX is equal to the Krull dimension of SX . Via the Auslander–Buchsbaum theorem, this is equivalent to saying that the length of a minimal free resolution of SX is equal to codim X. Using the correspondence between local and global cohomology, one can see that this is the same as requiring H i (Pn , IX (k)) = 0 for all 0 < i ≤ dim X. If X ⊂ Pn is a normally generated smooth curve of degree d and genus g, then it is arithmetically Cohen–Macaulay as normal generation implies H 1 (Pn , IX ( j)) = 0 for j ≥ 1. (The cohomology groups vanish automatically for j ≤ 0.) The main result of [20] is Theorem 1.8 ([20]). If X ⊂ Pn is a smooth curve of genus g and degree d ≥ 2g + 3, and Σ is normal, then Σ is arithmetically Cohen–Macaulay.

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Remark 1.9. As we know that the singular locus of Σ is the curve C, its normality is equivalent to the arithmetically Cohen–Macaulay condition via Serre’s condition. Indeed, Theorem 1.8 holds if we assume normality of Σ , and if we could prove that Σ is arithmetically Cohen–Macaulay without assuming normality, we would get the depth condition in Serre’s criterion. In fact, we know that the ideal of a secant variety of a rational normal curve has a resolution given by an Eagon–Northcott complex, and we also know the graded Betti diagram of the secant varieties of elliptic normal curves via [8], so in these two cases, we do know normality. We conjecture that Σ k is arithmetically Cohen–Macaulay if d ≥ 2g + 2k + 1 and hope that we can use cohomology to limit both the number of rows and the number of columns in the graded Betti diagram of IΣk in general. The main difficulty in the cohomological program is that our hypotheses are solely in terms of the positivity of a line bundle on a smooth curve X, and we need to prove vanishings in the cohomology of sheaves on its secant varieties which will necessarily have singularities. We begin Sect. 2 by giving the definition of the ideal of a secant variety and discussing how it may be computed via elimination and prolongation. We then give several examples of ideals of smooth curves and their secant varieties. In Sect. 3 we discuss the geometry of the desingularization of the secant varieties of a curve and how Terracini recursion may be used to study them. We have not made an attempt to survey the vast literature on secant varieties of higher dimensional varieties here, choosing instead to limit our attention to secant varieties of smooth curves. Acknowledgements. Code for the computation of prolongations used in our examples was written in conjunction with the paper [19] with the help of Mike Stillman. We are grateful to Seth Sullivant and Mike Stillman for allowing us to include this code here. The first author is partially supported by NSF grant DMS-0600471 and the Clare Booth Luce program. She also thanks the organizers of the conference on Hilbert functions and syzygies in commutative algebra held at Cortona in 2007 as well as organizers of the Abel Symposium. We thank Mohan Kumar and Adam Ginensky for their communications.

2 Computing Secant Varieties In this section we will describe how secant ideals may be computed via elimination and via prolongation. In Sect. 2.1 we will see that the ideal of Σk (X) can be defined as the intersection of an ideal in k + 1 sets of variables with a subring corresponding to the original ambient space. Thus, it is theoretically possible to compute the secant ideal of any variety whose homogeneous ideal can be written down. However, elimination orders are computationally expensive, so this method will be unwieldy for large examples. If X is defined by homogeneous forms of the same degree, then the method of prolongation can be used to compute the graded piece of I(Σk (X)) of minimum possible degree. This computation is fast, and in many cases yields a


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set of generators of I(Σk (X)). We will discuss prolongation in Sect. 2.2 and give a Macaulay 2 implementation in Appendix A.

2.1 Secant Varieties via Elimination Let X ⊂ Pn be a variety with homogeneous ideal I ⊂ k[x]. We can define the kth secant ideal of I so that it can be computed via elimination. We work in a ring with k + 1 sets of indeterminates yi = (yi,0 , . . . , yi,n ) and let I(yi ) denote the image of the ideal I under the ring isomorphism x j → yi, j . We define the ideal of the ruled join of X with itself k times as in Remark 1.3.3 in [6]: J = I(y1 ) + · · · + I(yk+1 ). Geometrically, we embed X into k + 1 disjoint copies of Pn in a projective space of dimension (k + 1)(n + 1) − 1. If a point is in the variety defined by J, we will see a point of X in each set of y-variables. If we project to the linear space [y1,0 +· · ·+y1,n : · · · : yk+1,0 + · · · + yk+1,n ] then the points in the image are points whose coordinates are sums of k + 1 points of X. In practice, we make the change of coordinates which is the identity on the first k sets of variables and is defined by yk+1, j → yk+1, j − y1, j − · · · − yk, j on the last set of variables. This has the effect of sending y1, j + · · · + yk+1, j to yk+1, j , so that the ideal of the kth secant variety is the intersection of I(y1 ) + · · · + I(yk ) + I(yk+1 − y1 − · · · − yk ) with k[yk+1,0 , . . . , yk+1,n ]. Example 2.1 (The secant variety of two points in P2 ). Consider X = {[1 : 0 : 0], [0 : 1 : 0]} with defining ideal I = x0 x1 , x2 . Using the definition above, the ideal of the join is J = y1,0 y1,1 , y1,2 + y2,0 y2,1 , y2,2 Under the change of coordinates y2, j → y2, j − y1, j we have J˜ = y1,0 y1,1 , y1,2 + (y2,0 − y1,0 )(y2,1 − y1,1 ), y2,2 − y1,2 ˜ consists of points of the form The variety V (J) [a : 0 : 0 : a : b : 0], [c : 0 : 0 : d : 0 : 0], [0 : e : 0 : 0 : f : 0], [0 : g : 0 : h : g : 0], where [a : b], [c : d], [e : f ], [g : h] ∈ P1 . Eliminating the first three variables projects ˜ under this projection is the line ˜ into P2 , and we see that the image of V (J) V (J) joining the two points of X. Sturmfels and Sullivant use a modification of this definition of the secant ideal in which they first define an ideal in the ring k[x, y1 , . . . , yk+1 ], which has k + 2 sets of variables, and then eliminate. Using the notation from before they work with secant ideals by first defining J = I(y1 ) + · · · + I(yk+1 ) + y1 + · · · + yk+1 − x and then

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computing I(Σ k (X)) = J ∩ k[x]. Eliminating the y-variables produces an ideal in k[x] that vanishes on all points in Pn that can be written as the sum of k + 1 points of X, and hence defines the secant ideal of Σ k (X). Example 2.2 (The secant variety of two points in P2 revisited). Consider X = {[1 : 0 : 0], [0 : 1 : 0]} with defining ideal I = x0 x1 , x2 . Using the definition of Sturmfels and Sullivant, the secant variety of X is J = y1,0 y1,1 , y1,2 + y2,0 y2,1 , y2,2 + y1,0 + y2,0 − x0 , y1,1 + y2,1 − x1 , y1,2 + y2,2 − x2 . The variety V (J ) consists of points of the form [a + b : 0 : 0 : a : 0 : 0 : b : 0 : 0], [c : d : 0 : c : 0 : 0 : 0 : d : 0], [0 : e + f : 0 : 0 : e : 0 : 0 : f : 0], [h : g : 0 : 0 : g : 0 : h : 0 : 0], where [a : b], [c : d], [e : f ], [g : h] ∈ P1 . Eliminating the y-variables projects V (J ) into P2 , and we see that the image of V (J ) under this projection is the line joining the two points of X. From the point of view of computation, using the first definition of a secant ideal is probably better as it involves computing an elimination ideal in an ambient ring with fewer variables. The advantage of the second definition reveals itself in proofs, especially involving monomial ideals. Indeed, as the linear form y1, j + · · · + yk+1, j − x j is in J , we see that (y1, j + · · · + yk+1, j )m is equivalent to xmj modulo J . Therefore, a monomial f (x) is in J ∩ k[x] if and only if f (y1 + · · · + yk+1 ) ∈ J . This is a key observation in Lemma 2.3 of [21].

2.2 Secant Varieties via Prolongation A sufficiently positive embedding of a variety X has an ideal generated by quadrics. We expect the ideal of Σk (X) to be generated by forms of degree k + 2. There are many ways of seeing that I(Σ k (X)) cannot contain any forms of degree less than k + 2. This fact was made explicit algebraically by Catalano-Johnson [2] who showed that I(Σ k (X)) is contained in the (k + 1)st symbolic power of I(X) and cites an independent proof due to Catalisano. The stronger statement, that I(Σk (X))k+2 = I(X)(k+1) appears in [16, 17] and a proof of a generalization for ideals generated by forms of degree d is in [19]. The connection between symbolic powers and the ideals of secant varieties, at least in the case of smooth curves, is implicit in work of Thaddeus [22]. Specifically, in Sect. 5.3 he constructs a sequence of flips whose exceptional loci are the transforms of secant varieties and then identifies the ample cone at each stage. The identification of sections of line bundles on the transformed spaces with those on the original in Sect. 5.2 then provides the connection. Though not made algebraically


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explicit, this connection is used in Sect. 2.12 of [27], is present throughout [24] and is discussed on pg. 80 of [25]. The observation that I(Σk (X)) and the (k + 1)st symbolic power agree in degree k + 2 leads to an algorithm for quickly computing all forms of degree k + 2 in I(Σk (X)). First, we define the prolongation of a vector space V of homogeneous forms of degree d to be the space of a forms of degree d + 1 whose first partial derivatives are all in V. An easy way to compute the prolongation of V is to compute the vector space Vi of forms of degree d + 1 formed by integrating the elements of V with respect to xi . Then V1 ∩ · · ·∩Vn is the prolongation of V. We provide Macaulay2 code for the computation of prolongations in Appendix A. If V = I(X)2 , then the prolongation of V is I(Σ 1 (X))3 . The prolongation of I(Σk (X))k+2 is I(Σk+1 (X))k+3 . As each of these spaces is just the intersection of a set of finite dimensional linear spaces, the vector spaces I(Σ k (X)k+2 ) can be computed effectively in many variables.

2.3 Compute the Ideal of a Smooth Curve In this section we discuss the examples that we have been able to compute so far, most of which also appear in [20]. Essentially we need a mechanism for writing down the generators of I(X). A curve of degree ≥ 4g+3 has a determinantal presentation by [4], where matrices whose 2-minors generate I(X) for elliptic and hyperelliptic curves are given. We can also re-embed plane curves into higher dimensional spaces. We give explicit examples of each class of example below. We computed the ideals of the secant varieties using the idea of prolongation described in the previous section. In each case, the projective dimension of the ideal generated by prolongation is equal to the codimension of the secant variety. Hence, we can deduce that the ideal of the secant variety is generated by the prolongation. Example 2.3 (Re-embedding a plane curve with nodes: g = 2, d = 5, Example 1.5 in [20]). Suppose we have a plane quintic with 4 nodes. If we blow up the nodes we have a smooth curve of genus 2. We provide Macaulay 2 [9] code below that shows how to compute the ideal of the embedding of such a curve in P7 . This method of finding the equations of a smooth curve is due to F. Schreyer and was suggested to the first author by D. Eisenbud. --The homogeneous coordinate ring of Pˆ2. S= ZZ/32003[x_0..x_2] --These are the ideals of the 4 nodes I1 = ideal(x_0, x_1) I2 = ideal(x_0, x_2) I3 = ideal(x_1, x_2) I4 = ideal(x_0-x_1, x_1-x_2)

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--Forms in I vanishing twice at each --of our chosen nodes I = intersect(I1ˆ2, I2ˆ2, I3ˆ2, I4ˆ2); --The degree 5 piece of I. M = flatten entries gens truncate(5, I) --The target of the rational map given by M. R = ZZ/32003[y_0..y_8] --The rational map given by M and the ideal --of its image. f = map(S, R, M) K = ker f --A random linear change of coordinates on Pˆ8. g = map(R, R, {random(1, R),random(1, R),random(1, R), random(1, R),random(1, R),random(1, R),random(1, R), random(1, R),random(1, R)}) --Add in the element y_8 and then eliminate it. --J = ideal of the cone over our plane quintic in Pˆ7. J =eliminate(y_8, g(K)+ideal(y_8)); The graded Betti diagram of the ideal of the curve is Example 1.2 and also Example 1.5 in [20]. Its secant ideal has graded Betti diagram given below. total: 0: 1: 2: 3: 4:

0 1 2 3 4 1 12 16 8 3 1 . . . . . . . . . . 12 16 . . . . . 4 . . . . 4 3

Example 2.4 (A determinantal curve: g = 2, d = 12, Example 4.8 in [20]). Following [4], we can write down matrices whose 2 × 2 minors generate the ideal of a hyperelliptic curve. We give such a matrix below for a curve with g = 2 and d = 12. ⎞ ⎛ y0 x0 x1 x2 x3 ⎟ ⎜x1 x2 x3 x4 y1 ⎟ ⎜ ⎟ ⎜x2 x3 x4 x5 y2 ⎟ ⎜ ⎠ ⎝x3 x4 x5 x6 y3 y0 y1 y2 y3 x1 + x2 + x3 + x4 + x5 The varieties Σk , k = 0, 1, 2, have the graded Betti diagrams below. Notice that the index of the final row of each diagram is 2k, indicating that the homogeneous


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coordinate ring of Σk has regularity 2k and that the ideal of Σ k has regularity 2k + 1 as predicted by Conjecture 1.6. Moreover, our curve sits in P10 , and dim Σ k = 2k +1, so the index of the last column is the codimension of Σ k , indicating that each variety is arithmetically Cohen–Macaulay. We can also see that I(Σk ) is generated in degree k + 2 and has syzygies up to stage p where 12 = 2g + 2k + 1 + p and that linear β9−2k,11 = g+k k+1 as in Conjecture 1.4 in [20]. total: 0: 1: 2:

total: 0: 1: 2: 3: 4:

total: 0: 1: 2: 3: 4: 5: 6:

0 1 2 3 4 5 6 7 8 9 1 43 222 558 840 798 468 147 17 2 1 . . . . . . . . . . 43 222 558 840 798 468 147 8 . . . . . . . . . 9 2 0 1 2 3 4 5 6 7 1 70 283 483 413 155 14 3 1 . . . . . . . . . . . . . . . . 70 283 483 413 155 . . . . . . . . 7 . . . . . . . 7 3

0 1 2 3 4 5 1 41 94 61 11 4 1 . . . . . . . . . . . . . . . . . . 41 94 61 . . . . . . . . . . . . 6 . . . . . 5 4

Example 2.5 (A Veronese re-embedding of a plane curve: g = 3, d = 12). Let X be a smooth plane curve of degree 4 and genus 3. Re-embedding this curve via the degree 3 Veronese map we have a curve of degree 12 in P9 . Below we give the graded Betti diagram of the curve and its first two secant varieties. 0 1 2 3 4 5 6 7 8 total: 1 33 144 294 336 210 69 16 3 0: 1 . . . . . . . . 1: . 33 144 294 336 210 48 . . 2: . . . . . . 21 16 3 0 1 2 3 4 5 6 total: 1 38 108 102 43 18 6 0: 1 . . . . . . 1: . . . . . . .

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2: . 38 108 102 10 . . 3: . . . . 30 . . 4: . . . . 3 18 6

total: 0: 1: 2: 3: 4: 5: 6:

0 1 1 . . . . . .

1 2 3 4 8 23 26 10 . . . . . . . . . . . . 8 . . . . 6 . . . 16 10 . . 1 16 10

Examples 2.3, 2.4 and 2.5 together suggest an additional conjecture, that row 2k of the Betti diagram of Σ k has precisely g nonzero elements.

3 Secant Varieties as Vector Bundles: Terracini Recursion The definition of a secant variety as the Zariski closure of the union of secant lines of X does not lend itself to thinking about a secant variety geometrically. A more elegant point of view is to realize that a secant line to a projective variety X is just the span of a length two subscheme of X. Thus we should think of Σ (X) as the image of a P1 -bundle over the space of length two subschemes of X, Hilb2 X. One nice consequence of this point of view is that Hilb2 X = BlΔ (X × X)/S2 is smooth as long as X is, and we obtain a geometric model for the secant variety on which we can apply standard cohomological techniques. Perhaps more importantly, this P1 -bundle can be constructed explicitly via blowing up. Under mild hypotheses on the positivity of the embedding of X, the blowup of Pn at X produces a desingularization of Σ (X) as a P1 bundle over Hilb2 X. Thinking of this bundle embedded inside the blowup of Pn at X we can also examine how it meets the exceptional divisor. What we see above a point p ∈ X is a Pn−2 that meets the proper transform of Σ (X) in the projection of X into Pn−2 from the tangent space at p. In this section we provide explicit examples illustrating this point of view. In Sect. 3.1 we illustrate the geometry of the blowups desingularizing secant varieties when X is a set of 5 points in P3 . Although the secant varieties of rational normal curves are well-understood algebraically, we discuss them here as we may make explicit computations and give proofs which highlight the main ideas used in the more general case in [26] but are much simpler. We make computations with rational normal curves of degrees 3 and 4 in Sects. 3.2 and 3.3. In Sect. 3.4 we discuss how to think about the cohomology along the fibers of these blowups and how cohomology may be used to show that the secant variety is projectively normal.


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3.1 Secant Varieties of Points Let X ⊂ Pn be a finite set of points in linearly general position. We will analyze the successive blowups of X and the proper transforms of its secant varieties, moving up one dimension at each stage. It is instructive to consider the geometry of the blowups of a finite set of points and its secant varieties as we can easily restrict our attention to the picture above a single point. We think the general picture of the geometry of the blowups will become transparent if we illustrate the construction in a concrete example. Although this computation may seem quite special, if X consists of n + 2 points in linearly general position, then a result of Kapranov [13] tells us that the sequence of blowups actually gives a realization of M 0,n . 3.1.1 Points in P3 Let B0 = P3 and let Σ0 = X be a set of 5 linearly general points denoted p1 , p2 , p3 , p4 , p5 . Let B1 be the blowup of B0 at Σ 0 . We let Ei denote the exceptional divisor above pi and Σ j be the proper transform of Σ j for j = 1, 2.

3.1.2 The First Recursion The exceptional divisor Ei is a P2 in which Σ 1 ∩ Ei is a set of 4 points in P2 and Σ 2 ∩ Ei is the union of lines joining these points in Σ 1 . Below we give a diagram depicting the exceptional divisor above a point p1 together with the strict transforms of the span of p1 with two other points p2 and p3 .

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The picture in E1 can be found by projecting Σ1 and Σ2 into P2 away from p1 . Taking a more global picture, we see that Σ 1 is a smooth variety consisting of the disjoint unions of proper transforms of lines. However, Σ 2 is not smooth as the components of Σ 2 intersect the Ei in lines which meet at points.

3.1.3 The Second Recursion Let us now define B2 to be the blowup of B1 at Σ 1 . We will abuse notation and let Ei denote its own proper transform in B2 . To analyze B2 , it may be helpful to restrict our attention locally to a single point p = [0 : 0 : 0 : 1] and examine the fiber over p after blowing up p and then blowing up the proper transform of a line containing p. Using bihomogeneous coordinates x and y on P3 × P2 , the blowup of P3 is defined by I(B p ) = xi y j − x j yi | i = j ∈ {0, 1, 2}. Blowing up the proper transform of the line L defined by x0 , x1 yields a subvariety of P3 × P2 × P1 which can be given in tri-homogeneous coordinates x, y, z by I(B p,L ) = xi y j − x j yi | i = j ∈ {0, 1, 2} + x0 z1 − x1 z0 , y1 z0 − y0 z1 . To understand the fiber above p, we add the ideal of the point to I(B p,L ) to get

x0 , x1 , x2 , y1 z0 −y0 z1 . The x coordinates alone cut out p×P2 ×P1 , and the equation in the y and z-variables cuts out the blowup of the point p×[0 : 0 : 1] in p×P2 . Thus, we can see that if blow up a point p and a line containing it, above p we get a P2 in which we have blown up one point. Turning back to the global picture, we see that when we have blown up Σ 1 , the proper transform of Σ 2 in B2 is smooth. Moreover, above each pi we have a copy of our global picture projected into P2 away from a point of X. Indeed, each Ei is a P2 in which we have blown up 4 points. These 4 points correspond to the intersection of Σ 1 with Ei . The intersection of the proper transform of Σ 2 with Ei consists of the union of the exceptional divisors of the 4 points that are blown up in Ei .

3.2 Blowing up a Rational Normal Curve of Degree 3 We begin with the twisted cubic X ⊂ P3 . In this case the equations of the blowup are easy to write down and we can realize the blowup of P3 along X as a P1 bundle over Hilb2 P1 explicitly. The secant variety Σ (X) = P3 is smooth, but the embedding is positive enough for I(X) to have linear first syzygies, which is the condition required for the setup in [23]. The three quadrics x0 x2 − x12 , x0 x3 − x1 x2 , x1 x3 − x22 , which generate I(X) give a rational map P3 P2 . The blowup of P3 at X is the graph of this map in P3 × P2 with bihomogeneous coordinates x and y. This graph is defined set-theoretically by equations coming from Koszul relations, for example y0 (x0 x3 − x1 x2 ) − y1 (x0 x2 − x12 ). But these relations are generated by the two relations coming from linear syzy-


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gies: x0 y2 − x1 y1 + x2 y0 , x1 y2 − x2 y1 + x3 y0 , which generate the ideal of the blowup.

3 denote the blowup of P3 along X and E denote the exceptional divisor. Let P Further, let q1 and q2 denote the restrictions of the projections from P3 × P2 to the

3 . Then q1 : P

3 → P3 is the blowup map and q2 : P

3 → P2 first and second factors to P is the morphism induced by |2H − E|. We analyze the fibers of both maps explicitly below. Above any point p ∈ X, the fiber of q1 is a P1 . For example, if p = [0 : 0 : 0 : 1], then q−1 1 (p) is defined by adding x0 , x1 , x2 to the ideal of the blowup to get

x0 , x1 , x2 , x3 y0 = x0 , x1 , x2 , x3 ∩ x0 , x1 , x2 , y0 . The first primary component is irrelevant, and the second defines the P1 with points ([0 : 0 : 0 : 1], [0 : y1 : y2 ]). Moreover, we can see from these equations that the blowup of P3 along the twisted cubic is a P1 -bundle over P2 . To see q−1 2 ([1 : 0 : 0]), add the ideal y1 , y2 to the ideal of the blowup to get the ideal y0 x2 , x3 . This shows that the fiber above [1 : 0 : 0] consists of points of the form ([a : b : 0 : 0], [1 : 0 : 0]). As a length n subscheme of P1 has an ideal generated by a single form of degree n, Hilbn P1 = Pn , and so this matches exactly what we expect from the description above. Choosing a less trivial example, in the next section we will begin to see a recursive geometric picture analogous to what we saw when we blew up a finite set of points and its secant varieties. Again, we will see the projection of our global picture in the fibers above points of our original variety. Note that we are projecting to the projectivized normal bundle and hence away from the tangent space to a point on our variety. (In our earlier example, we projected from a single point because a zero-dimensional variety has a zero-dimensional tangent space.)

3.3 Blowing up a Rational Normal Curve of Degree 4 Let X ⊂ P4 be a rational normal curve with defining ideal minimally generated by the six 2 × 2 minors of the matrix x0 x1 x2 x3 x1 x2 x3 x4 These six quadrics are a linear system on P4 with base locus X, so they give a rational map P4 P5 . The closure of the graph of this map in P4 × P5 is B = BX (P4 ), P4 blown up at X.

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3.3.1 The Ideal of the Blowup Let R = k[x, y]. Since I(B) defines a subscheme of P4 × P5 it is a bihomogeneous ideal. It must contain homogeneous forms in the y-variables that define the image of P4 in P5 . Since each yi is the image of a quadric vanishing on X, the ideal I(B) will also contain bihomogeneous forms that are linear in the yi corresponding to syzygies. In our example, we get an ideal with 9 generators by running the Macaulay 2 [9] code --The coordinate ring of Pˆ4 x Pˆ5. S = ZZ/32003[x_0..x_4, y_0..y_5] --The coordinate ring of Pˆ4 with an extra parameter. --The parameter t ensures that the kernel --is bi-homogeneous. B = ZZ/32003[x_0..x_4,t] --The map S -> B. f = map(B,S,{x_0, x_1, x_2, x_3, x_4, t*(-x_1ˆ2+x_0*x_2), t*(-x_1*x_2+x_0*x_3), t*(-x_2ˆ2+x_1*x_3), t*(-x_1*x_3+x_0*x_4), t*(-x_2*x_3+x_1*x_4), t*(-x_3ˆ2+x_2*x_4)}) --The ideal defining the blowup in S K = ideal mingens ker f One generator, y22 − y2 y3 + y1 y4 − y0 y5 , in the y-variables alone cuts out the image of P4 in P5 . The other 8 generators are constructed from linear syzygies on generators of I(X) as in the previous example. We can use the ideal to analyze what happens when we take the pre-image of a point p ∈ P4 under the blowup map. There are two cases depending on whether p is contained in X. Case 1: Suppose that p = [0 : 1 : 0 : 0 : 0] ∈ / X. When we compute I(p) + I(B) and then use local coordinates where x1 = 1, we get the ideal x0 , x2 , x3 , x4 , y0 , y1 , y2 , y3 , y4 . Thus, the pre-image of p is a single point as expected. Case 2: Suppose that p = [1 : 0 : 0 : 0 : 0] ∈ X. Again, we compute I(p) + I(B) and then use local coordinates where x0 = 1. We get the ideal x1 , x2 , x3 , x4 , y2 , y4 , y5 which defines a P2 . 3.3.2 The Intersection of Σ 1 with the Exceptional Divisor To examine what happens to the secant variety of X algebraically, we add the equation of Σ1 (X) to I(B). Above the point [1 : 0 : 0 : 0 : 0] we have the intersection of the P2 with coordinates [y0 : y1 : y3 ] with the hypersurface defined by y21 − y0 y3 . The conic y21 − y0 y3 is the projection of our original curve away from the line with points


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of the form [a : b : 0 : 0 : 0]. We have [y0 : y1 : y3 ] = [t 2 : t 3 : t 4 ], which is defined by the given equation. Schematically, we have

3.4 Cohomology Along the Fibers It was observed by the second author in [24] that one could use Bertram’s Terracini Recursiveness to obtain cohomological relationships between different embeddings of the same curve. For example, let X be a smooth curve embedded in Pn by a line bundle L of degree at least 2g + 3. As discussed above, blowing up Pn along X desingularizes Σ1 , and thus blowing up again along the proper transform of Σ1 yields a smooth variety B2 π

π

B2 →2 B1 →1 Pn = PΓ (X, L). Let π = π1 ◦ π2 . If x ∈ X, then π1−1 (x) ∼ = Pn−2 = PΓ (X, L(−2x)). By Terracini Recursiveness, −1 π (x) is the blow up of PΓ (X, L(−2x)) along a copy of X embedded by L(−2x); equivalently π −1 (x) is precisely what is obtained by the projection Pn Pn−2 from the line tangent to X at x. In fact, it can be shown that the exceptional divisor of the desingularization π : Σ → Σ is precisely X × X where the restriction π : X × X → X is projection [23, Lemma 3.7]. Because B2 is obtained from Pn by blowing up twice along smooth subvarieties, we know that Pic(B2 ) = ZH + ZE1 + ZE2 where H is the proper transform of a hyperplane section, E1 is the proper transform of the exceptional divisor of the first blow-up π1 , and E2 is the exceptional divisor of the second blow-up. Note that we

+ ZE 1 . We examine the relationships among line similarly have Pic(π −1 (x)) = ZH n bundles on P , B1 , and B2 . Because the generic hyperplane in Pn misses x ∈ X, the restriction of H to π −1 (x) is trivial. Because E1 is the proper transform of the exceptional divisor of the first

Finally, by Terracini Recursiveness, blow-up, the restriction of E1 to π −1 (x) is −H.

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the restriction of E2 to π −1 (x) is E 1 . Thus a typical effective line bundle on B2 of

− cE 1 ). the form OB2 (aH − bE1 − cE2 ) restricts to Oπ −1 (x) (bH 3.4.1 Regularity and Projective Normality of the Secant Variety to a Rational Normal Curve In this section will illustrate how to apply the first stage of Bertram’s Terracini Recursiveness in the special case where X is a rational normal curve. Suppose that X ⊂ Pn is a rational normal curve with L = OX (1) = OP1 (n) of degree at least 4 so that IΣ is 3-regular, and Σ is projectively normal. It follows [26, Proposition 9] from the description of the exceptional divisor of the desingularization π : Σ → Σ above that Σ has rational singularities; i.e. Ri π∗ OΣ = 0 for i > 0. Thus by Leray–Serre we immediately have H i (Σ , OΣ (k)) = H i (Σ , OΣ (kH)) for all i, k. Proposition 3.1. Let X ⊂ Pn be a rational normal curve. Then IΣ is 3-regular. Proof. We show directly that H i (Pn , IΣ (3−i)) = 0 for i ≥ 1. As Σ is 3-dimensional, we have only to show the four vanishings 1 ≤ i ≤ 4. For each i ≥ 2 and all k we have H i (Pn , IΣ (k)) = H i−1 (Σ , OΣ (k)) = H i−1 (Σ , OΣ (kH)). The restriction of OΣ (kH)) to a fiber of the map from p : Σ → P2 is OP1 (kH). When i = 4, all cohomology along the fiber vanishes and so in particular we have H 3 (Σ , OΣ (−H)) = 0. In general, if k ≥ −1, all of the higher cohomology along the fibers vanishes. This implies that the higher direct image sheaves vanish and that H i (Σ , OΣ (kH)) = H i (P2 , p∗ OΣ (kH))). When i = 3, k = 0, and H 2 (Σ , OΣ ) = H 2 (P2 , p∗ OΣ ) = H 2 (P2 , OP2 ) = 0. For i = 2, we blow up B1 along Σ and use Terracini recursiveness. Consider the sequence 0 → OB2 (H − E1 − E2 ) → OB2 (H − E2 ) → OE1 (H − E2 ) → 0. As discussed earlier, the restriction of OE1 (H − E2 ) to a fiber of the flat morphism E1 → X is Oπ −1 (x) (−E 1 ), but H i (π1−1 (x), Oπ −1 (x) (−E 1 )) = H i (Pn−2 , IX ) = 0 for i ≥ 0. Thus it follows that H i (B2 , OB2 (H − E1 − E2 )) = H i (B2 , OB2 (H − E2 )) = H i (B1 , IΣ (H)) = H i (Pn , IΣ (1)), where the last equality is a consequence of Σ having rational singularities. From the sequence on B2


171

0 → OB2 (H − E1 − E2 ) → OB2 (H − E1 ) → OE2 (H − E1 ) → 0 Once again considering Σ as a P1 -bundle over P2 , we see that OΣ (H − E1 ) is OP1 (−1) along the fibers, thus H i (Σ , OΣ (H − E1 )) = 0 for i ≥ 0. Putting this together, we have H i (Pn , IΣ (1)) = H i (B2 , OB2 (H − E1 − E2 )) = H i (B1 , OB1 (H − E1 )) = H i (Pn , IX (1)) = 0. For i = 1, in a similar fashion it is enough to show H 1 (B2 , OB2 (2H − E1 − E2 )) = 0. From the sequence 0 → OB2 (2H − E1 − E2 ) → OB2 (2H − E1 ) → OE2 (2H − E1 ) → 0 and the fact that H 1 (B2 , OB2 (2H − E1 )) = H 1 (Pn , IX (2)) = 0, it suffices to show H 0 (B2 , OB2 (2H − E1 )) → H 0 (E2 , OE2 (2H − E1 )) is surjective. However, we know that H 0 (B2 , OB2 (2H − E1 − E2 )) = H 0 (B1 , IΣ (2)) = 0, which show that the map is injective. Thus, if we shows the two spaces have the same dimension we are done. We have the well-known identification H 0 (B2 , OB2 (2H − E1 )) = H 0 (Pn , IX (2)). Further, letting ϕ : B1 → Ps be the map given by quadrics vanishing on X (i.e. the morphism induced by the linear system |2H − E1 |), then we have the restriction ϕ : Σ → P2 . Note that Σ ⊂ Pn × Ps is a P1 -bundle over P2 . It is, further, a nice exercise to show that the double cover P1 × P1 = Σ ∩ E1 → P2 is, in situ, the natural double cover re-embedded by the Veronese vn−2 . Therefore, OPs (1)|P2 = OP2 (n − 2), and this implies that ϕ ∗ OP2 (n − 2) = OΣ (2H − E1 ). Hence, H 0 (Σ , OΣ (2H − E1 )) = H 0 (P2, O P2 (n − 2)). A quick computation gives h0 (Pn , IX (2)) = h0 (P2 , OP2 (n − 2)) = n2 . In fact, we see from the proof that H 1 (Pn , IΣ (1)) = 0. As 3-regularity implies that H 1 (Pn , IΣ (k)) = 0 for all k ≥ 2 we also have: Corollary 3.2. Σ is projectively normal.

Appendix A Code for Computing Prolongations The code in this section was written to produce examples for joint work of the first author and Seth Sullivant. Mike Stillman wrote the intersection.m2 package and helped rewrite the code for prolongations. The code in intersection.m2 was written to allow us to specify a degree d and intersect the degree d piece of a list of ideals.

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-- file name: intersection.m2 -- intersect needs a degree limit intersection = method(Dispatch => Thing, TypicalValue=>Ideal, Options => {DegreeLimit=>{}}) intersection Sequence := intersection List := o -> L -> ( if not all(L, x -> instance(x,Ideal)) then error "expected a list of ideals"; B := directSum apply(L, generators); A := map(target B, 1, (i,j) -> 1); ideal syz(A|B, SyzygyRows => 1, DegreeLimit=>o.DegreeLimit) ) --INPUT: L = list of ideals -d = integer --OUTPUT: generators for the intersection --of the ideals in L in degrees ( ans := L_0; for i from 1 to #L-1 do time ans = intersection(ans,L_i,DegreeLimit=>d); ans) --INPUT: L = list of ideals, each generated --in degree d -d = integer --OUTPUT: the ideal generated by the intersection --of the degree d parts of each ideal intersectSpaces = (d,L) -> ( S := ring L_0; k := coefficientRing S; monoms := basis(d,S); time L = apply(L, I -> ( (mn,cf) := coefficients(gens I, Monomials=>monoms); image substitute(cf,k))); time M := intersect L; ideal(monoms * (gens M)) ) The code for computing prolongations follows below. If X ⊂ Pn has ideal I generated by quadrics, then the prolongation P of a basis for the quadrics in I contains


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all forms of degree 3 in I(Σ1 (X)). Of course, P is not guaranteed to generate the ideal of the secant variety. In practice, it is often easy to verify that the ideal of the secant variety is generated by P by computing the projective dimension, degree, and dimension of S/ P. If we see that the degree and dimension are as expected and S/ P is Cohen–Macaulay, then we conclude that I(Σ1 (X)) = P. load "intersection.m2" --INPUT: f = polynomial in S -i = index of i-th variable in S --OUTPUT: an antiderivative of f integrated --with respect to the i-th variable in S. integration = (i,f) ->( S:=ring f; sum apply (terms f, g -> (1/((flatten exponents g)_i+1)) * g*S_i) ) --INPUT: A = a list of forms of degree d. --OUTPUT: a basis for the prolongation of A. prolong = (A) ->( S:= ring A_0; n:= numgens S; d:= (degree(A_0))_0; L :={}; for i from 0 to (n-1) do( M := ideal drop (flatten entries vars S, {i,i}); intA := ideal apply(A, f -> integration(i, f))+Mˆ(d+1); L = append(L, intA); );

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