Birational Geometry, Rational Curves, and Arithmetic

Simons Symposia Fedor Bogomolov Brendan Hassett Yuri Tschinkel Editors Birational Geometry, Rational Curves, and Arith...

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Simons Symposia

Fedor Bogomolov Brendan Hassett Yuri Tschinkel Editors

Birational Geometry, Rational Curves, and Arithmetic

Simons Symposia

Fedor Bogomolov • Brendan Hassett Yuri Tschinkel Editors

Birational Geometry, Rational Curves, and Arithmetic

123

Editors Fedor Bogomolov Courant Institute of Mathematical Sciences New York University New York, NY, USA

Brendan Hassett Department of Mathematics Rice University Houston, TX, USA

Yuri Tschinkel Courant Institute of Mathematical Sciences New York University New York, NY, USA Simons Foundation New York, NY, USA

ISBN 978-1-4614-6481-5 ISBN 978-1-4614-6482-2 (eBook) DOI 10.1007/978-1-4614-6482-2 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2013935219 Mathematics Subject Classification (2010): 14E05, 14E08, 14E30, 14G05, 14G27, 14G20, 14M20 © Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This volume is an outgrowth of the Simons Symposium “Geometry over NonClosed Fields,” which took place in a magnificent setting on the island of St. John in February 2012. It gathered in a tropical paradise a small group of experts working at the interface of algebraic geometry and number theory and experts from complementary branches of these fields. In addition to a series of very interesting and high-level lectures, the meeting was characterized by intense discussions and collaborative research. These lively discussions were greatly enhanced by the multitude of points of view and approaches advocated at the meeting. This volume reflects the main topics of the symposium: interactions between birational geometry, the theory of rational curves, and arithmetic. Among the actively studied problems in higher-dimensional birational geometry are the questions of rationality, unirationality, and birational geometry of varieties naturally appearing in geometric and arithmetic problems such as moduli spaces of various origins. An important role is played by the theory of birational automorphisms and birational models; these themes are taken up in several papers of the volume. Specifically, the survey paper by I. Arzhantsev, H. Flenner, S. Kaliman, K. Kutzschebauch, and M. Zaidenberg compares the notions of flexible and infinitely transitive algebraic varieties. Their main result is that smooth varieties X of dimension ≥2 with a transitive action of the subgroup SAut(X) of algebraic automorphisms generated by additive one-parameter subgroups are flexible, and in fact the action of SAut(X) is infinitely transitive. This theme is also treated in the article of F. Bogomolov, I. Karzhemanov, and K. Kuyumzhiyan. They formulate a provocative conjecture linking unirationality of the function field K = k(X) of an algebraic variety X over an algebraically closed field k with the existence of an infinitely transitive model after stabilization, i.e., addition of sufficiently many algebraically independent elements to K. Furthermore, they prove new results on the existence of infinitely transitive models for unirational function fields with many cancellations and provide evidence for their conjecture for some interesting classes of unirational functions fields, e.g., function fields of cubic hypersurfaces and some complete intersections. v

vi

Preface

Birational models and fibration structures of a variety X are encoded, to a large extent, in the convex, linear geometry of the cones of ample and effective divisors of X. The article of Kovács supplies a characteristic independent proof of the basic result concerning the shape of the ample cone of a K3 surface: either it consists of all vectors with square more than zero, in the upper half-space of the second cohomology, or it is a cone whose faces are dual to smooth rational (−2)-curves. The interplay between cones of curves and divisors and finite-generation questions is at the forefront of Lazić’s contribution. He discusses implications of his method of proof of finite generation of canonical rings to other problems in higher-dimensional algebraic geometry, specifically concerning varieties of zero or negative Kodaira dimension. In particular, he discusses applications to the Mori program which conjecturally constitute the core in the class of Fano varieties and relations between abundance conjecture and the cone conjecture for Calabi– Yau manifolds. He also develops a new viewpoint on the birational properties of Mori Dream spaces, i.e., varieties for which the total coordinate ring (over all line bundles) is finitely generated. The article of Bertram and Coskun discusses in detail cones of curves and ample divisors for Hilbert schemes of points on surfaces, with a special emphasis on del Pezzo surfaces. They show Hilbert schemes of del Pezzo and Hirzebruch surfaces are Mori Dream spaces. Moreover, in many cases, birational models can be interpreted as moduli spaces for certain choices of Bridgeland stability data, and this allows one to identify classes of extremal curves. The paper by Cheltsov, Katzarkov, and Przhyjalkowski develops a new categorical approach to birational properties of algebraic varieties, combining ideas from homological mirror symmetry and techniques which evolved in the study of derived categories of coherent sheaves on projective varieties. Their expectation is that the presence of phantom subcategories, as almost direct summands of the derived category of a variety, is indicative of nonrationality. They verify this for many representative examples of threefolds. An alternative obstruction to nonrationality, leading to many explicit examples in higher dimensions, relies on stable and unramified stable cohomology. Among these examples are quotient varieties, obtained from finite-dimensional linear representations of finite groups. In this case, the birational properties are tightly linked to group cohomology and its birational parts, i.e., stable cohomology. These are not easy to compute, in general. The article of Bogomolov and Böhning treats stable cohomology of finite groups obtained as iterated wreath products of cyclic groups of prime order. The main result is that all such elements are detected on elementary abelian subgroups. The article includes several results showing that stable cohomology of isoclinic groups exhibits similar properties. The structure of the Brauer group is a prime motive for Lieblich’s survey on twisted sheaves, i.e., sheaves indigenous to stacks associated with a variety. These shed new light on long-standing algebraic and arithmetic problems: the periodindex problem, the validity of the Brauer–Manin formalism for rational varieties over global fields, and the classification of K3 surfaces over small fields. After the translation to stacks, geometric techniques may be brought to bear, e.g., the

Preface

vii

birational triviality of moduli spaces of stable vector bundle with fixed determinant over a curve and the theory of rationally connected varieties. Várilly-Alvarado summarizes the current state of the art for the arithmetic of rational surfaces over global fields, with particular emphasis on counterexamples for small degree del Pezzo surfaces. Effective computations in the Brauer group play a central rôle. Liedtke offers a broad survey of the geometry of surfaces over fields of positive characteristic. Here again, much of what distinguishes positive characteristic is the unexpected behavior of rational curves, e.g., quasi-elliptic fibrations, the existence of unirational surfaces among supersingular K3 surfaces, and the appearance of “new” rational curves when a K3 surface is reduced modulo p. Liedtke shows how this rich geometry can be exploited to establish theorems even in characteristic zero, e.g., complex K3 surfaces with odd Picard rank admit infinitely many rational curves. Kebekus turns the emphasis of rational curves on its head: What can one say when there are very few rational curves? This has major implications for the rigidity of morphisms between such varieties. On the other hand, the criteria of Mori and Miyaoka for uniruledness, via positivity of the tangent bundle, suggest quantitative approaches to the structure of rational curves, expressed in the language of foliations. Debarre investigates the structure of varieties parametrizing smooth curves of a given geometric genus and degree in smooth hypersurfaces and discusses how they can be used to study the geometry of the hypersurface. When are these spaces of curves themselves uniruled, and what does it tell us about the position of the underlying variety in the classification scheme? New York, NY Houston, TX New York, NY

Fedor Bogomolov Brendan Hassett Yuri Tschinkel

Contents

Infinite Transitivity on Affine Varieties . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Ivan Arzhantsev, Hubert Flenner, Shulim Kaliman, Frank Kutzschebauch, and Mikhail Zaidenberg

1

The Birational Geometry of the Hilbert Scheme of Points on Surfaces . . . . Aaron Bertram and Izzet Coskun

15

Isoclinism and Stable Cohomology of Wreath Products .. . . . . . . . . . . . . . . . . . . . Fedor Bogomolov and Christian Böhning

57

Unirationality and Existence of Infinitely Transitive Models . . . . . . . . . . . . . . . Fedor Bogomolov, Ilya Karzhemanov, and Karine Kuyumzhiyan

77

Birational Geometry via Moduli Spaces . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Ivan Cheltsov, Ludmil Katzarkov, and Victor Przyjalkowski

93

Curves of Low Degrees on Fano Varieties .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 133 Olivier Debarre Uniruledness Criteria and Applications . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 147 Stefan Kebekus The Cone of Curves of K3 Surfaces Revisited . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 163 Sándor J. Kovács Around and Beyond the Canonical Class. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 171 Vladimir Lazić On the Ubiquity of Twisted Sheaves . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 205 Max Lieblich Algebraic Surfaces in Positive Characteristic .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 229 Christian Liedtke Arithmetic of Del Pezzo surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 293 Anthony Várilly-Alvarado ix

Infinite Transitivity on Affine Varieties Ivan Arzhantsev, Hubert Flenner, Shulim Kaliman, Frank Kutzschebauch, and Mikhail Zaidenberg

Mathematics Subject Classification codes (2000): 14R20, 14L30

1 Introduction An action of a group G on a set A is said to be m-transitive if for every two tuples of pairwise distinct points (a1 , . . . , am ) and (a1 , . . . , am ) in A there exists g ∈ G such that g · ai = ai for all i = 1, . . . , m. An action which is m-transitive for all m ∈ Z>0 will be called infinitely transitive.

I. Arzhantsev () Department of Higher Algebra, Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory 1, Moscow, 119991 Russia e-mail: [email protected] H. Flenner Fakultät für Mathematik, Ruhr Universität Bochum, Geb. NA 2/72, Universitätsstr. 150, 44780 Bochum, Germany e-mail: [email protected] S. Kaliman Department of Mathematics, University of Miami, Coral Gables, FL 33124, USA e-mail: [email protected] F. Kutzschebauch Mathematisches Institut, Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland e-mail: [email protected] M. Zaidenberg Université Grenoble I, Institut Fourier, UMR 5582 CNRS-UJF, BP 74, 38402 St. Martin d’Hères, France e-mail: [email protected] F. Bogomolov et al. (eds.), Birational Geometry, Rational Curves, and Arithmetic, DOI 10.1007/978-1-4614-6482-2__1, © Springer Science+Business Media New York 2013

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2

I. Arzhantsev et al.

Clearly, the group of all bijections of an infinite set A acts infinitely transitively on A. Infinite transitivity never occurs if G is a Lie group or algebraic group acting on a variety A. Indeed, m-transitivity implies that the map G → Am with g → (g.a1 , . . . , g.am ) is dominant for an m-tuple of pairwise distinct points (a1 , . . . , am ) ∈ Am . This shows that G cannot act on A m-transitively if dim G < m · dim A. According to A. Borel a much stronger result is valid: a real Lie group cannot act even 3-transitively on a simply connected, non-compact real manifold (see Theorems 5 and 6 in [5]). By a result of Knop [18], the most transitive action of an algebraic group over an algebraically closed field is the 3-transitive action of the group PSL2 on the projective line P1 . At the same time, the group Aut(An ) of all algebraic automorphisms of the affine space An over an infinite field acts infinitely transitively on An for n ≥ 2. To obtain this result, it suffices to use linear automorphisms and triangular automorphisms of the form (x1 , . . . , xn−1 , xn ) → (x1 , . . . , xn−1 , xn + P(x1 , . . . , xn−1 )), where P(x1 , . . . , xn−1 ) is an arbitrary polynomial. These automorphisms generate the tame automorphism group TAut(An ), which acts infinitely transitively on An for n ≥ 2. It was shown in [1] that for certain (infinite dimensional) groups of automorphisms of affine varieties transitivity implies infinite transitivity. We do not try to present here the results of [1] in full generality, but rather to concentrate on the most interesting features.

2 Main Results Let X be an algebraic variety over a field k. Unless we explicitly precise the opposite, we assume usually that k is algebraically closed of characteristic zero. Consider a regular action Ga × X → X of the additive group Ga = (k, +). The image, say, H of Ga in the automorphism group Aut(X) is a one-parameter unipotent subgroup of Aut(X). We let SAut(X) denote the subgroup of Aut(X) generated by all its one-parameter unipotent subgroups. Automorphisms from the group SAut(X) will be called special. Furthermore, SAut(X) is a normal subgroup of Aut(X). Denote by Xreg the smooth locus of an algebraic variety X. We say that a point x ∈ Xreg is flexible if the tangent space Tx X is spanned by the tangent vectors to the orbits H · x over all one-parameter unipotent subgroups H in Aut(X). The variety X is flexible if every point x ∈ Xreg is. Clearly, X is flexible if one point of Xreg is and the group Aut(X) acts transitively on Xreg . The following result conjectured in an earlier version of [2] is proven in [1, Theorem 0.1].

Infinite transitivity

3

Theorem 1. Let X be an irreducible affine variety of dimension ≥ 2. Then the following conditions are equivalent: 1. The group SAut(X) acts transitively on Xreg . 2. The group SAut(X) acts infinitely transitively on Xreg . 3. The variety X is flexible.

3 Examples of Flexible Varieties We are going to show that the equivalent conditions of Theorem 1 are satisfied for wide classes of affine varieties.

3.1 Suspensions Let X be an affine variety. Given a nonconstant regular function f ∈ k[X], we define a new affine variety Susp(X, f ) = {uv − f (x) = 0} ⊆ A2 × X called a suspension over X. It is shown in [2, Theorem 3.2] that a suspension over a flexible affine variety is again flexible. The case of suspensions over affine spaces was treated earlier in [14]. Iterating the construction of suspension yields new examples of flexible varieties. Flexibility and infinite transitivity of the action of SAut(X) is established in [2, Theorem 3.1] for a suspension X = {uv − f (x) = 0} over the affine line A1 under the assumption that f (k) = k, where k is an arbitrary field of characteristic zero. The same holds for suspensions over flexible real affine algebraic varieties with connected smooth loci [2, Theorem 3.3]. By [19], infinite transitivity holds on every connected component of the smooth loci of suspensions over flexible real affine varieties.

3.2 Affine Toric Varieties Recall that a normal algebraic variety X is toric if it admits a regular action of an algebraic torus T with an open orbit. In general, an affine toric variety does not need to be flexible. For instance, if X = T then the algebra k[X] is generated by invertible functions and hence the group SAut(X) is trivial.

4


We say that an affine toric variety X is nondegenerate if the only invertible regular functions on X are nonzero constants. Equivalently, X is nondegenerate if it is not isomorphic to X × (A1 \ {0}) for some toric variety X . By [2, Theorem 2.1] any nondegenerate affine toric variety is flexible. Considering affine toric surfaces, one obtains examples of affine varieties X such that Xreg is not a homogeneous space of an algebraic group, but the group SAut(X) acts on Xreg infinitely transitively; see [2, Example 2.2].

3.3 Homogeneous Spaces Let us consider (following [24]) the class of connected linear algebraic groups G generated by one-parameter unipotent subgroups. A connected linear algebraic group G belongs to this class if and only if G does not admit nontrivial characters or, equivalently, if a maximal reductive subgroup of G is semisimple. If such a group G acts on a variety X then the image of G in Aut(X) is contained in SAut(X). If G acts on Xreg transitively then X is flexible. As an example, consider a simple rational G-module V , where G is semisimple. The cone X of highest weight vectors in V consists of two G-orbits, namely, the open orbit X \ {0} and the origin {0} [25]. If X = V then G acts on Xreg transitively; hence the group SAut(X) is infinitely transitive on Xreg . Note that X may be considered as a (normal) affine cone over the flag variety G/P, where a parabolic subgroup P is the stabilizer of a point in the projectivization P(X) of the cone X in P(V ). In these terms infinite transitivity for X was proven in [2, Theorem 1.1]. Any affine homogeneous space G/H of dimension ≥ 2 satisfies the equivalent conditions of Theorem 1 provided that G does not admit nontrivial characters; see [1, Proposition 5.4]. In particular, for any semisimple group G and a reductive subgroup H ⊆ G, the homogeneous space X = G/H is flexible and the group SAut(X) is infinitely transitive on X. This applies as well to X = G.

3.4 Almost Homogeneous Varieties Suppose that a connected semisimple algebraic group G acts with an open orbit on an irreducible affine variety X. In this case we say that X is almost homogeneous. It turns out that under some additional assumptions this implies flexibility of X.

3.4.1 The Smooth Case Assume that an almost homogeneous affine variety X is smooth. Using Luna’s Étale Slice Theorem we show in [1, Theorem 5.6] that X is homogeneous under the action of a semidirect product G V , where V is a certain finite-dimensional G-module. In particular X is flexible.


5

3.4.2 SL2 -Embeddings Let the group SL2 = SL2 (k) act with an open orbit on a normal affine threefold X. All such SL2 -threefolds were classified in [23]. If X is smooth then it is flexible by the above argument. For a singular X the complement of the open SL2 -orbit consists of a two-dimensional orbit, say, O and a singular fixed point p ∈ X. It is shown in [3] that X can be obtained as the quotient of an affine hypersurface xb0 = x1 x4 − x2 x3 under an action of a one-dimensional diagonalizable group. Such a hypersurface is a suspension over A3 . Using this one can join a point in O with a point in the open SL2 -orbit by a Ga -orbit on X and thus to gain flexibility of X; see [1, Theorem 5.7] for details.

3.5 Vector Bundles Let π : E → X be a reduced, irreducible linear space over a flexible variety X, which is a vector bundle over Xreg . Assume that there is an action of the group SAut(X) on E such that the action of every one-parameter unipotent subgroup is algebraic and the morphism π is equivariant. It is shown in [1, Corollary 4.5] that the total space E is a flexible variety. In particular, the tangent bundle T X and all tensor bundles E = (T X)⊗a ⊗ (T ∗ X)⊗b are flexible.

3.6 Affine Cones Over Projective Varieties Let X be the affine cone over a projective variety Y polarized by a very ample divisor H. Then one can characterize flexibility of X in terms of certain geometric properties of the pair (Y, H) as follows (see [17, 22]). An open subset U ⊆ Y is called a cylinder if U ∼ = Z × A1 , where Z is a smooth affine variety (see [16, 17]). A cylinder U is called H-polar if U = Y \ Supp D for some effective Q-divisor D linearly equivalent to H. It is shown in [16, Theorem 3.9] that any H-polar cylinder U on Y gives rise to a Ga -action on the affine cone X over Y . A subset W ⊆ Y is called invariant with respect to a cylinder U ∼ = Z × A1 if −1 W ∩ U = π (π (W )), where π : U → Z is the first projection. In other words, W is invariant if every A1 -fiber of the cylinder is either contained in W or does not meet W . A variety Y is transversally covered by cylinders Ui , i = 1, . . . , s, if Y = i Ui and there is no proper subset W ⊆ Y invariant with respect to all the Ui . Theorem 2.5 in [22] states that if for some very ample divisor H on a normal projective variety Y there exists a transversal covering by H-polar cylinders then the corresponding affine cone X over Y is flexible. This criterion allows to establish that any affine cone over a del Pezzo surface of degree ≥ 5 is flexible. The same is true for certain affine cones over del Pezzo surfaces of degree 4, including the plurianticanonical ones. In contrast, the pluri-anticanonical cones over del Pezzo surfaces

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of degree 1 or 2 do not admit any nontrivial action of a unipotent algebraic group, neither any effective action of a two-dimensional connected algebraic group [17]. The case of cubic surfaces remains open.

3.7 Gizatullin Surfaces These are normal affine surfaces which admit a completion by a chain of smooth rational curves. It follows from Gizatullin’s Theorem ([12, Theorems 2 and 3], see also [7]) that a normal affine surface X different from A1 × (A1 \ {0}) is Gizatullin if and only if the special automorphism group SAut(X) has an open orbit; then this open orbit necessarily has a finite complement in X. It was conjectured in [12] that if the base field k has characteristic zero then the open SAut(X)-orbit coincides with Xreg , i.e., that every Gizatullin surface is flexible. This is definitely not true in a positive characteristic, where the automorphism group Aut(X) of a Gizatullin surface X can have fixed points that are smooth points of X [6]. We have seen in Sect. 3.1 that Gizatullin’s Conjecture is true for the Gizatullin surfaces given in A3 by equations xy − f (z) = 0, since these are suspensions over the affine line. Yet another class of flexible Gizatullin surfaces consists of the Danilov–Gizatullin surfaces; see [11]. Recently S. Kovalenko constructed a counterexample to the Gizatullin Conjecture over C (unpublished). We refer the reader to [9] and the references therein for a study of one-parameter groups acting on Gizatullin surfaces.

4 Technical Tools We do not try to expose the proof of Theorem 1 in detail. In this section we just present a couple of technical tools which play a crucial role in the proof. The first one is the well-known correspondence between regular Ga -actions on an affine variety X and locally nilpotent derivations of the algebra A = k[X] of regular functions on X.

4.1 Locally Nilpotent Derivations and their Replicas A derivation ∂ of an algebra A is called locally nilpotent if for any a ∈ A there exists m ∈ Z>0 such that ∂ m (a) = 0. If the group Ga acts on X = Spec A then the associated derivation ∂ of A is locally nilpotent. It is immediate that for every f ∈ ker ∂ the derivation f ∂ is again locally nilpotent. Conversely, given a locally nilpotent derivation ∂ : A → A and t ∈ k, the map exp(t ∂ ) : A → A is an automorphism of A. Furthermore for ∂ = 0, H = exp(t ∂ ) is a one-parameter unipotent subgroup of Aut(A). Via the isomorphism Aut(A) ∼ =


7

Aut(X) given by g → (g−1 )∗ this yields a one-parameter unipotent subgroup of Aut(X), which we denote by the same letter H. We refer to [10] for more details on locally nilpotent derivations. The algebra of invariants k[X]H = ker ∂ has transcendence degree dim X − 1 over k. Given an invariant f ∈ k[X]H the one-parameter unipotent subgroup H f = exp(k f ∂ ), called a replica of H, plays an important role in the sequel. The H f -action has the same general orbits as the H-action. However, the zero locus of f remains pointwise fixed under the H f -action. So given a finite set of points chosen on distinct general H-orbits one can find a replica H f of H that moves all the points but a given one. If we have at our disposal enough Ga -actions in transversal directions on X then by changing the velocity along the corresponding orbits as above, we can move the given ordered finite set in Xreg into a prescribed position. This gives the infinite transitivity of the SAut(X)-action on Xreg . Let us illustrate the notions of a replica and of a special automorphism in the case of an affine space An over k. The group SAut(An ) contains the one-parameter unipotent subgroup of translations in any given direction. The infinitesimal generator of such a subgroup is a directional partial derivative. Such a derivative defines a locally nilpotent derivation of the polynomial ring in n variables, whose phase flow is the group of translations in this direction. Its replicas are the one-parameter groups of shears in the same direction. As another example, consider the locally nilpotent derivation

∂ =X

∂ ∂ +Y ∂Y ∂Z

of the polynomial ring k[X,Y, Z] and an invariant function f = Y 2 − 2XZ ∈ ker ∂ . The corresponding replica H f contains in particular the famous Nagata automorphism H f (1) = exp( f · ∂ ) ∈ SAut(A3 ), which is known to be wild; see [30]. Notice that any automorphism α ∈ SAut(An ) preserves the usual volume form on An . Hence SAut(An ) ⊆ Gn , where Gn denotes the subgroup of Aut(An ) consisting of all automorphisms with Jacobian determinant 1. The problem whether the subgroup SAut(An ) coincides with Gn is widely open. Recall that this is the case in dimension 2 due to the Jung-van der Kulk Theorem.

4.2 Algebraically Generated Groups Our second tool is a technique to work with infinite dimensional groups. We say that a subgroup H of the automorphism group Aut(X) is algebraic if H has a structure of an algebraic group such that the natural action H × X → X is a morphism. A subgroup G of Aut(X) is called algebraically generated if it is generated as an abstract group by a family G of connected algebraic subgroups of Aut(X). Similar notions were studied in the literature earlier; see, e.g., [26, 29] and more recently [24].

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In [1] we extend some standard facts on finite-dimensional algebraic transformation groups to the case of algebraically generated groups. It is not difficult to show that for any point x ∈ X the orbit G · x is locally closed. What is more surprising, one can find (not necessarily distinct) subgroups H1 , . . . , Hs ∈ G such that G.x = (H1 · H2 · . . . · Hs ).x for any x ∈ X; see [1, Proposition 1.5]. In our setting we obtain the following version of Kleiman’s Transversality Theorem [1, Theorem 1.15]. Theorem 2. Let a subgroup G ⊆ Aut(X) be algebraically generated by a system G of connected algebraic subgroups closed under conjugation in G. Suppose that G acts with an open orbit O ⊆ X. Then there exist subgroups H1 , . . . , Hs ∈ G such that for any locally closed reduced subschemes Y and Z in O one can find a Zariski dense open subset U = U(Y, Z) ⊆ H1 × · · · × Hs such that every element (h1 , . . . , hs ) ∈ U satisfies the following: 1. The translate (h1 · . . . · hs ).Zreg meets Yreg transversally. 2. dim(Y ∩ (h1 · . . . · hs ).Z) ≤ dimY + dim Z − dim X. In particular Y ∩ (h1 · . . . · hs ).Z = 0/ if dimY + dimZ < dim X. The next generalization concerns the Rosenlicht Theorem on rational invariants. It turns out that for any algebraically generated subgroup G ⊆ Aut(X) there exists a finite collection of rational G-invariants on X which separate G-orbits in general position [1, Theorem 1.13]. In particular, the codimension of a general G-orbit in X equals the transcendence degree of the field k(X)G of rational G-invariants over k. The latter result has a useful corollary.

4.3 The Makar-Limanov Invariant Recall [10] that the Makar-Limanov invariant ML(X) of an affine algebraic variety X is the intersection of the kernels of all locally nilpotent derivations on k[X]. In other words, ML(X) is the subalgebra of all SAut(X)-invariants of the algebra k[X]. Similarly [21] the field Makar-Limanov invariant FML(X) is defined as the intersection of the kernels of extensions of all locally nilpotent derivations on k[X] to the field of fractions k(X). This is a subfield of k(X) which consists of all rational SAut(X)-invariants. If it is trivial, i.e., if FML(X) = k, then so is ML(X), while the converse is not true in general. Triviality of FML(X) is equivalent to the existence of a flexible point in Xreg and to the existence of an open SAut(X)-orbit in X. The question arises how these invariants are connected with rationality properties of the variety X. There are examples of non-unirational affine threefolds X with ML(X) = k birationally equivalent to C × A2 , where C is a curve of genus g ≥ 1; see [20, Example 4.2]. For such a threefold X the general SAut(X)-orbits have


9

dimension two, the field Makar-Limanov invariant FML(X) is nontrivial, and there is no flexible point in X. The next proposition confirms, in particular, Conjecture 5.3 in [21] (cf. also [4, 24]). Proposition 3 ([1, Proposition 5.1]). Let X be an irreducible affine variety. If the field Makar-Limanov invariant FML(X) is trivial then X is unirational. Indeed, the condition FML(X) = k implies that the group SAut(X) acts on X with an open orbit O. Thus there are Ga -subgroups H1 , . . . , Hs in SAut(X) and a point x ∈ X such that the image of the map H1 × · · · × Hs → X,

(h1 , . . . , hs ) → (h1 . . . hs ).x

coincides with O. Since H1 × · · · × Hs is isomorphic (as a variety) to the affine space As , this yields unirationality of X. Moreover, any two points in O are contained in the image of a morphism A1 → O. In particular, O is A1 -connected in the sense of [13, 6.2]. In general, flexibility implies neither rationality nor stable rationality. Indeed, there exists a finite subgroup F ⊂ SLn , where n ≥ 4, such that the smooth unirational affine variety X = SLn /F is not stably rational; see [24, Example 1.22]. However, by Sect. 3.3 the variety X is flexible and the group SAut(X) acts infinitely transitively on X. We expect further development of the invariant theory for algebraically generated groups.

5 Geometric Consequences Let us start with several results related to Theorem 1.

5.1 Collective Transitivity By a collective infinite transitivity we mean a possibility to move simultaneously (i.e., by the same automorphism) an arbitrary finite set of points along their orbits into a given position. We illustrate our general results in this direction on a concrete example from linear algebra, cf. [27, 28]. Let X = Mat(n, m) be the space of all n × m matrices over k. The subset Xr ⊆ X of matrices of rank r is well known to have dimension: mn − (m − r)(n − r).

10


In the following we always assume that this dimension is ≥ 2. The product SLn × SLm acts on X via the left-right multiplication preserving the strata Xr . For every k = l we let Ekl ∈ sln and E kl ∈ slm denote the nilpotent matrices with xkl = 1 and the other entries equal zero. Let further Hkl = In + kEkl ⊆ SLn and H kl = Im + kE kl ⊆ SLm be the corresponding one-parameter unipotent subgroups acting on the stratification X = r Xr , and let δkl and δ kl , respectively, be the corresponding locally nilpotent vector fields on X tangent to the strata. We call elementary the one-parameter unipotent subgroups Hkl , H kl , and all their replicas. In the following theorem we establish the collective infinite transitivity on the above stratification of the subgroup G of SAut(X) generated by the two sides elementary subgroups. By a well-known theorem of linear algebra, the subgroup SLn × SLm ⊆ G acts transitively on each stratum Xr (and so these strata are G-orbits) except for the open stratum Xn in the case where m = n. In the latter case the G-orbits contained in Xn are the level sets of the determinant. Theorem 4 ([1, Theorem 3.3]). Given two finite ordered collections B and B of distinct matrices in Mat(n, m) of the same cardinality, with the same sequence of ranks, and in the case where m = n with the same sequence of determinants, we can simultaneously transform B into B by means of an element g ∈ G, where G ⊆ SAut(Mat(n, m)) is the subgroup generated by all elementary one-parameter unipotent subgroups. See [1, Sect. 3.3] for similar results on symmetric and skew-symmetric matrices.

5.2 A1 -Richness Let X be a flexible affine variety of dimension ≥ 2, and let p1 , . . . , pk ∈ Xreg be a k-tuple. Fix a Ga -orbit C on X and some k-tuple of distinct points q1 , . . . , qk ∈ C. Due to infinite transitivity there is an element g ∈ SAut(X) such that g · q1 = p1 , . . . , g · qk = pk . So the translate g · C of C is a Ga -orbit on X passing through p1 , . . . , pk . This elementary observation can be strengthened in the following way. An affine variety X is called A1 -rich if for every finite subset Z and every algebraic subset Y of codimension ≥ 2 there is a curve in X isomorphic to the affine line A1 , which is disjoint with Y and passes through every point of Z [15]. The following result is immediate from the Transversality Theorem 2. Theorem 5 ([1, Corollary 4.18]). Let X be an affine variety. Suppose that the group SAut(X) acts with an open orbit O ⊆ X. Then for any finite subset Z ⊆ O and for any closed subset Y ⊆ X of codimension ≥ 2 with Z ∩Y = 0/ there is an orbit C∼ = A1 of a Ga -action on X which does not meet Y and passes through each point of Z. In the special case where X = AnC this also follows from the Gromov– Winkelmann Theorem [31] which says that the group Aut(An \Y ) acts transitively on An \ Y , combined with the equivalence of transitivity and infinite transitivity of


11

Theorem 1, which is valid in this setting as well. More generally, we also show that C as in the theorem can be chosen to have prescribed jets at the points of Z.

5.3 Prescribed Jets of Automorphisms Our results on infinite transitivity may be strengthened in the following way; see [1, Theorem 4.14 and Remark 4.16]. Theorem 6. Let X be a flexible affine variety of dimension n ≥ 2 equipped with an algebraic volume form1 ω . Then for any m ≥ 0 and for any finite subset Z ⊆ Xreg there exists an automorphism g ∈ SAut(X) with prescribed m-jets at the points p ∈ Z, provided these jets preserve ω and inject Z into Xreg . The same holds without the requirement that there is a global volume form on Xreg provided that for every p ∈ Z the corresponding jet fixes the point p and its linear part belongs to the group SL(Tp X).

5.4 The Oka–Grauert–Gromov Principle for Flexible Varieties Let us provide an important application of flexibility in analytic geometry; see [1, Theorem 6.2 and Proposition 6.3]. We address the reader to [1, Sect. 6] for more details and a survey. Theorem 7. Let π : X → B be a surjective submersion of smooth irreducible affine algebraic varieties over C such that for some algebraically generated subgroup G ⊆ Aut(X) the orbits of G coincide with the fibers of π . Then the Oka–Grauert–Gromov principle holds for π : X → B. That is, any continuous section of π is homotopic to a holomorphic one, and any two holomorphic sections of π that are homotopic via continuous sections are also homotopic via holomorphic ones.

6 Open Problems Let us finish this note with several open problems on flexible varieties. The examples from Sect. 3.4 motivate the following problem: Characterize flexible varieties among the normal almost homogeneous affine varieties. By the result described in Sect. 3.4.1, a smooth almost homogeneous variety is flexible. In fact, in all examples that we know, an almost homogeneous normal

1 By

this we mean a nowhere vanishing n-form defined on Xreg .

12


variety is flexible. For instance, one might hope for positive results in the class of spherical varieties. By definition, a G-variety X is spherical if a Borel subgroup B of G acts on X with an open orbit. An important particular case is the variety X = Spec k[G/U], where U is a maximal unipotent subgroup of a semisimple group G. To formulate the next problem we need to introduce some more notation. Let Y be a closed subvariety of an affine variety X. Denote by SAut(X)Y the subgroup generated by all one-parameter unipotent subgroups exp(k∂ ), where the locally nilpotent vector field ∂ vanishes on Y . Assume that the group SAut(X) acts on X with an open orbit O, and let Y ⊆ O be a closed subvariety of codimension ≥ 2. Is it true that the group SAut(X)Y acts on O \ Y transitively? In particular, is X \ Y flexible, if X is. Some positive results on this problem can be found in [1, Proposition 4.19]. Our last problem concerns exotic structures on the affine spaces. Does there exist a flexible exotic algebraic structure on an affine space, that is, a flexible smooth affine algebraic variety over C diffeomorphic but not isomorphic to an affine space AnC ? Notice that for all the exotic structures on AnC constructed so far, the MakarLimanov invariant is nontrivial, whereas for a flexible such structure, even the field Makar-Limanov invariant must be trivial (cf. however [8]).

References 1. ARZHANTSEV, I. V., FLENNER , H., KALIMAN , S., KUTZSCHEBAUCH , F., AND ZAIDENBERG , M., Flexible varieties and automorphism groups, Duke Math. J., 162 no. 4 (2013), 60p (to appear) arXiv:1011.5375, (2010). 2. ARZHANTSEV, I.V., KUYUMZHIYAN , K., AND ZAIDENBERG , M., Flag varieties, toric varieties, and suspensions: three instances of infinite transitivity, Sbornik: Math., 203, no. 7, 3–30, (2012). 3. BATYREV, V. AND HADDAD , F., On the geometry of SL(2)-equivariant flips, Moscow Math. J. 8, no. 4, 621–646, (2012). 4. BOGOMOLOV, F., KARZHEMANOV, I., AND KUYUMZHIYAN , K., Unirationality and existence of infinitely transitive models, this volume; see also arXiv:1204.0862, (2012). 5. BOREL, A., Les bouts des espaces homogènes de groupes de Lie, Ann. Math. (2) 58, 443–457, (1953). 6. DANILOV, V. I. AND GIZATULLIN , M. H., Examples of nonhomogeneous quasihomogeneous surfaces, Math. USSR Izv. 8, 43–60, (1974). 7. DUBOULOZ, A., Completions of normal affine surfaces with a trivial Makar-Limanov invariant, Michigan Math. J. 52, 289–308, (2004). 8. DUBOULOZ, A., The cylinder over the Koras-Russell cubic threefold has a trivial MakarLimanov invariant, Transform. Groups 14, 531–539, (2009). 9. FLENNER , H., KALIMAN , S., AND ZAIDENBERG , M., Smooth Affine Surfaces with NonUnique C∗ -Actions, J. Algebraic Geometry 20, 329–398, (2011). 10. FREUDENBURG , G., Algebraic Theory of Locally Nilpotent Derivations, Encyclopaedia Math. Sciences 136, Springer, Berlin, (2006). 11. GIZATULLIN , M. H., Affine surfaces that can be augmented by a nonsingular rational curve (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 34, 778–802, (1970).


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12. GIZATULLIN , M. H., Quasihomogeneous affine surfaces, Math. USSR Izv. 5, 1057–1081, (1971). 13. KALIMAN , S. AND KUTZSCHEBAUCH, F., Criteria for the density property of complex manifolds, Invent. Math. 172, 71–87, (2008). 14. KALIMAN , S. AND ZAIDENBERG , M., Affine modifications and affine hypersurfaces with a very transitive automorphism group, Transform. Groups 4, 53–95, (1999). 15. KALIMAN , S. AND ZAIDENBERG , M., Miyanishi’s characterization of the affine 3-space does not hold in higher dimensions, Ann. Inst. Fourier (Grenoble) 50, 1649–1669, (2000). 16. KISHIMOTO , T., PROKHOROV, YU ., AND ZAIDENBERG , M., Group actions on affine cones, CRM Proceedings and Lecture Notes 54, Amer. Math. Soc., 123–164, (2011). 17. KISHIMOTO , T., PROKHOROV, YU ., AND ZAIDENBERG , M., Unipotent group actions on del Pezzo cones, arXiv:1212.4479, (2012), 10p. 17a. KISHIMOTO , T., YU . PROKHOROV, AND ZAIDENBERG , M. Ga -actions on affine cones. arXiv:1212.4249, (2012), 14p. 18. KNOP, F., Mehrfach transitive Operationen algebraischer Gruppen, Arch. Math. 41, 438–446, (1983). 19. KUYUMZHIYAN , K. AND M ANGOLTE, F., Infinitely transitive actions on real affine suspensions, J. Pure Appl. Algebra 216, no. 10, 2106–2112, (2012). 20. LIENDO , A., Affine T -varieties of complexity one and locally nilpotent derivations, Transform. Groups 15, 389–425, (2010). 21. LIENDO , A., Ga -actions of fiber type on affine T -varieties, J. Algebra 324, 3653–3665, (2010). 22. PEREPECHKO , A. YU ., Flexibility of a ne cones over del Pezzo surfaces of degree 4 and 5, Funct. Anal. Appl. (to appear); arXiv:1108.5841, 6p. 23. POPOV, V. L., Quasihomogeneous affine algebraic varieties of the group SL(2), Math. USSR Izv. 7, 793–831, (1973). 24. POPOV, V. L., On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, CRM Proceedings and Lecture Notes 54, Amer. Math. Soc., 289–311, (2011). 25. POPOV, V. L. AND VINBERG , E. B., On a certain class of quasihomogeneous affine varieties, Math. USSR Izv. 6, 743–758, (1972). 26. RAMANUJAM, C. P., A note on automorphism groups of algebraic varieties, Math. Ann. 156, 25–33, (1964). 27. REICHSTEIN , Z., On automorphisms of matrix invariants, Trans. Amer. Math. Soc. 340, 353–371, (1993). 28. REICHSTEIN , Z., On automorphisms of matrix invariants induced from the trace ring, Linear Algebra Appl. 193, 51–74, (1993). 29. SHAFAREVICH , I. R., On some infinite-dimensional groups, Rend. Mat. Appl. (5) 25, no. 1–2, 208–212, (1966). 30. SHESTAKOV, I.P. AND UMIRBAEV, U. U., The tame and the wild automorphisms of polynomial rings in three variables, J. Amer. Math. Soc. 17, 197–227, (2004). 31. W INKELMANN , J., On automorphisms of complements of analytic subsets in Cn , Math. Z. 204, 117–127, (1990).

The Birational Geometry of the Hilbert Scheme of Points on Surfaces Aaron Bertram and Izzet Coskun

To Jim Simons, with gratitude Mathematics Subject Classification codes (2000): 14E30, 14C05, 14D20, 14D23

1 Introduction Bridgeland stability brings a new perspective to the study of the birational geometry of moduli spaces of sheaves on surfaces allowing one to construct flips explicitly and to improve classical bounds on nef and effective cones (see, e.g., [2, 3, 5]). In this paper, we study the relation between the stable base locus decomposition of the effective cone and the chamber decomposition of the stability manifold for the Hilbert scheme of points on a smooth, projective surface. We primarily concentrate on rational surfaces such as Hirzebruch and del Pezzo surfaces. The paper [3] describes a one-to-one correspondence between the Mori walls Mt and Bridgeland walls Wx=t− 3 for P2[n] and proves the correspondence for n ≤ 9 in 2

complete generality and for all n in the region t ≤ − n−1 2 . Precisely, a scheme Z is in the stable base locus of the divisors Dt = H[n] + 2tB for t < t0 if and only if the ideal sheaf IZ is destabilized at the Bridgeland wall Wx=t0 − 3 . Using the correspondence, 2 one can determine the base loci of linear systems on Hilbert schemes that are not apparent from a purely classical point of view. The correspondence is also useful in A. Bertram Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA e-mail: [email protected] I. Coskun () Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA e-mail: [email protected] F. Bogomolov et al. (eds.), Birational Geometry, Rational Curves, and Arithmetic, DOI 10.1007/978-1-4614-6482-2__2, © Springer Science+Business Media New York 2013

15

16

A. Bertram and I. Coskun

the other direction allowing classical geometry constructions to determine walls in the stability manifold. In this paper, we extend the correspondence to other surfaces such as P1 × P1 and F1 . Let X be a smooth, projective surface over the complex numbers. Let X [n] denote the Hilbert scheme parameterizing zero-dimensional schemes of length n. Let X (n) = X n /Sn denote the nth symmetric product of X. There is a natural morphism h : X [n] → X (n) , called the Hilbert-Chow morphism, that maps a zerodimensional scheme Z of length n to its support weighted by multiplicity. In [18], Fogarty proved that if X is a smooth projective surface, then X [n] is a smooth, projective, irreducible variety of dimension 2n. The locus of n distinct, unordered points is a Zariski-dense open subset of X [n] . Furthermore, the HilbertChow morphism h : X [n] → X (n) is a crepant resolution of singularities. In this paper, we will study the ample and effective cones of X [n] in the NéronSeveri space of X [n] . For simplicity, we always assume that the irregularity of the surface q(X) vanishes. In [19], Fogarty determined the Picard group of X [n] in terms of the Picard group of X. A line bundle L on X naturally determines a line bundle L[n] on X [n] as follows. The line bundle L on X gives rise to a line bundle L · · · L on X n , which is invariant under the action of the symmetric group Sn . Therefore, L · · · L descends to a line bundle LX (n) on the symmetric product X (n) under the natural quotient map π : X n → X (n) . The pullback L[n] = h∗ LX (n) under the HilbertChow morphism gives the desired line bundle on X [n] . Let B denote the class of the exceptional divisor of the Hilbert-Chow morphism. Geometrically, the exceptional divisor is the divisor parameterizing non-reduced schemes in X [n] . Since X [n] is smooth, we obtain an additional line bundle OX [n] (B) on X [n] . The element OX [n] (B) is divisible by two in the Picard group. However, since OX [n] ( B2 ) has no global sections, it is often easier to work with OX [n] (B). In [19], Fogarty proves that if the irregularity q(X) = 0, then Pic(X [n] ) ∼ = Pic(X) × Z B2 . [n] In particular, the Néron-Severi space of X is spanned by the Néron-Severi space of X and the divisor class B. In Sect. 2, we discuss the ample cone of X [n] . If L is an ample line bundle on X, then L[n] is a nef line bundle on X [n] . Hence, knowing the ample cone of X allows one to determine the part of the nef cone in the subspace where the coefficient of B is zero. Results of Beltrametti, Sommese, Catanese, and Göttsche (see [7, 9, 13]) allow one to construct further nef divisors on X [n] from (n − 1)-very ample line bundles on X. There are good criteria for verifying that a line bundle is (n − 1)-very ample on a surface. For a large class of surfaces, these criteria allow one to classify the (n − 1)-very ample line bundles. We will show that for simple surfaces such as P2 , Hirzebruch surfaces, and del Pezzo surfaces with K 2 ≥ 2, these constructions suffice to determine the nef cone of X [n] . For example, in Theorem 1, we will prove the following: • The nef cone of (P2 )[n] is the cone spanned by H[n] and (n − 1)H[n] − B2 , where H is the hyperplane class in P2 [28].

Hilbert Schemes of Points on Surfaces

17

• The nef cone of (P1 × P1 )[n] is the cone spanned by H1 [n], H2 [n], and (n − 1)H1 [n] + (n − 1)H2 [n] − B2 , where H1 and H2 are the classes of the two fibers in P1 × P1. [n] • Let Fr denote the surface P(OP1 ⊕ OP1 (r)), r ≥ 1. Then the nef cone of Fr is the cone spanned by E[n] + rF[n], F[n] and (n − 1)(E[n] + rF[n]) + (n − 1)F[n] − B2 , where E is the class of the exceptional curve and F is the class of a fiber in Fr . As a consequence of Theorem 1, we prove that if X is a Fano surface, then X [n] is log Fano and, in particular, a Mori dream space. In Sect. 3, we discuss the effective cone of X [n] . The effective cone of X [n] is much more subtle and depends on arithmetic properties of n. In many instances, the extremal rays of the cone can be constructed as loci of subschemes in X [n] that fail to impose independent conditions on sections of a given vector bundle on X. Showing that these loci are divisors, in general, is a difficult problem. Recently, Huizenga has made some progress when X = P2 (see [25]). Dually, we will construct moving curves to give upper bounds on the effective cone. In Sect. 4, we will compute the stable base locus decomposition of X [n] for X = P1 × P1 and 2 ≤ n ≤ 5 and X = F1 and 2 ≤ n ≤ 4 in full detail. A quick glance at Figs. 4 and 7, will convince the reader that these decompositions become complicated very quickly. These examples were chosen because they have fewer than 30 chambers. As we will discuss below, there is a rich interplay between the Mori chamber decomposition and the Bridgeland chamber decomposition. We hope that these examples will allow readers to explore connections that we will not discuss in this paper. On the other hand, if one restricts one’s attention to chambers that are not too far from the nef cone, then it is possible to describe the stable base locus decomposition completely. We will do so for P1 × P1 in Theorem 16 and for Fr in Theorem 24. In Sect. 5, we will recall the definition of Bridgeland stability conditions. We will be interested in very specific stability conditions on X. Let H be an ample line bundle on X; then it is possible to find an abelian subcategory As,t of the bounded derived category Db (coh(X)) of coherent sheaves on X such that when endowed with the central charge Zs,t (E) = −

X

e−(s+it)H · ch(E),

the pair (As,t , Zs,t ) is a Bridgeland stability condition [2, 12]. We will pick a basis of the Néron-Severi space of X and consider the slice of the stability manifold corresponding to these Bridgeland stability conditions. Fix a numerical class ν . Then the space of stability conditions Stab(X) has a chamber decomposition into regions where the set of semi-stable objects with class ν in As,t remains constant [5, 12]. In Sect. 5, we will derive the basic properties of the chamber structure for the Hilbert scheme of points. When ν is the numerical

18


class of an ideal sheaf of a zero-dimensional scheme of length n, the walls in the second quadrant are nested semicircles. We will study the walls in detail when X is a Hirzebruch surface or a del Pezzo surface. In Sect. 6, we will describe a precise correspondence between the Bridgeland walls and the Mori walls for P1 × P1 and F1 . The correspondence is cleanest to state when H is a multiple of the anti-canonical bundle; however, it seems to hold much more generally. When X = P1 × P1 , set H = − 14 KP1 ×P1 . When X = F1 , set H = − 16 KF1 . Consider the divisor Dt = H[n] + 2tB for t < 0 on X [n] . Then we find the following conjectural correspondence. The divisor Dt intersects the Mori wall Mt0 corresponding to a subscheme Z if and only if IZ is destabilized at the wall Wt0 −c(X) , where c(X) is a constant depending on the surface (c(P1 × P1 ) = −2 and c(F1 ) = −3). As in the case of P2 , we can prove the correspondence for small values of n in complete generality and for all n assuming that t is bounded above by a function depending on n and X. The Hilbert scheme represents the Hilbert functor. When one studies the birational models of a moduli or parameter space, it is natural to ask whether the other birational models also have modular interpretations (see [22, 23] for a discussion in the case of moduli spaces of curves and [14, 15] for the case of Kontsevich moduli spaces). Classically, it is not clear how to vary the Hilbert functor to get alternative moduli spaces. The key idea is to represent the Hilbert scheme X [n] as a moduli space of Bridgeland-stable objects and then to vary the Bridgeland stability. As we will see in Sect. 6, in many cases, one thus obtains modular interpretations of log canonical models of X [n] . Combining the results of Abramovich and Polishchuk [1], Lieblich [2], and Toda [31], one may construct moduli spaces of Bridgeland semi-stable objects parameterizing isomorphism classes of Zs,t -semi-stable objects in As,t as Artin stacks. There are many open questions about the geometry of these moduli spaces. For example, it is not in general known whether they are projective. When X = P2 , there are at least two ways of showing that the moduli spaces of Bridgelandstable objects are projective. These moduli spaces are projective because they are isomorphic to moduli spaces of quiver representations and can be constructed by GIT (see [3, 26]). For the surfaces we consider, we do not in general know a GIT construction of the moduli space. Alternatively, Bayer and Macrì have constructed nef divisors on the moduli space of Bridgeland-stable objects [5]. Given a stability condition σ = (Z, A), a choice of numerical invariants ν and a fine moduli space M parameterizing Bridgeland-stable objects in A with numerical invariants ν , Bayer and Macrì define a nef divisor on M by specifying its intersection number with every curve. Let E ∈ Db (X × M) be a universal family. They define D ·C as Z(ΦE (OC )) I − , Z(ν ) where I denotes the imaginary part. They prove that the divisor class is nef and if a curve has zero intersection with this divisor class, then for any two closed points, x, x on the curve Ex and Ex are S-equivalent. In some cases, it can be shown that


19

the divisors they construct are ample, giving a more general and better proof of the projectivity of the moduli space. This allows one to obtain modular interpretations of certain log canonical models of the Hilbert scheme in terms of moduli spaces of Bridgeland-stable objects. The organization of this paper is as follows: In Sect. 2, we discuss the nef cones of X [n] . In Sect. 3, we give constructions of effective divisors and moving curves. In Sect. 4, we discuss general features of the stable base locus decomposition of [n] (P1 × P1 )[n] and Fr . We also calculate the complete decomposition for small n. In Sect. 5, we recall preliminaries about Bridgeland stability conditions. Finally, in Sect. 6, we study the correspondence between Bridgeland walls and Mori walls.

2 The Ample Cone of the Hilbert Scheme In this section, we survey the description of the nef cone of X [n] for surfaces such as Hirzebruch and del Pezzo surfaces with K 2 ≥ 2. In many cases, one can give a complete description of the nef cone of X [n] using the theory of k-very ample line bundles developed by Beltrametti, Sommese, Catanese, Göttsche, and others. For more details about k-very ample line bundles, we refer the reader to [7, 9, 13]. We begin by giving two constructions of nef divisors on X [n] . Construction 1. Let L be an ample line bundle on X. Then Ln = L · · · L is an ample line bundle on X n invariant under the action of the symmetric group Sn . Consequently, Ln descends to an ample line bundle L(n) on the symmetric product X (n) . Since the Hilbert-Chow morphism is birational, the induced line bundle L[n] is big and nef on X [n] . However, since L[n] has degree zero on the fibers of the HilbertChow morphism, L[n] is not ample. Hence, the line bundle L[n] lies on the boundary of the nef cone of X [n] . Construction 2. Given a line bundle L on X, the short exact sequence 0 −→ L ⊗ IZ −→ L −→ L ⊗ OZ −→ 0

(1)

induces an inclusion H 0 (X, L ⊗ IZ ) ⊂ H 0 (X, L). Suppose that H 0 (X, L) = N > n, then this inclusion induces a rational map:

φL : X [n] G(N − n, N). Let DL (n) = φL∗ (OG(N−n,N) (1)) denote the pullback of OG(N−n,N) (1) under the rational map φL . A straightforward calculation using the Grothendieck-RiemannRoch Theorem shows that the class of DL (n) is B DL (n) = L[n] − . 2

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Since OG(N−n,N) (1) is very ample, the base locus of DL is contained in the indeterminacy locus of φL . In particular, if φL is a morphism, then DL (n) is basepoint-free and, consequently, nef. A line bundle L on X is called k-very ample if the restriction map H 0 (X, L) → H 0 (X, L ⊗ OZ ) is a surjection for every zero-dimensional scheme Z of length at most k + 1. In [9], the authors give a useful characterization of k-very ampleness for adjoint bundles: Let L be a nef and big line bundle on a surface S such that L2 ≥ 4k + 5. Then either KS + L is k-very ample or there exists an effective divisor D such that L − 2D is Q-effective, D contains a zero-dimensional subscheme of degree at most k + 1 for which k-very ampleness fails, and L·D−k−1≤ D·D
0 and for the ideal sheaf IZ associated to every zero-dimensional scheme Z ∈ X [n] . Let

Ξn ⊂ X [n] × X be the universal family and let π1 and π2 denote the natural projections. By cohomology and base change, π1∗ (π2∗ L ⊗ IΞn ) is a vector bundle of rank N − n on X [n] . Hence, by the universal property of the Grassmannian,

φL : X [n] → G(N − n, N) is a morphism. Therefore, we conclude that DL (n) = L[n] − 12 B is base-point-free. After introducing some notation, we will show that the nef divisors defined in Constructions 1 and 2 suffice to describe the nef cone of X [n] for surfaces like P2 , del Pezzo surfaces with K 2 ≥ 2, and Hirzebruch surfaces. Notation 1. The Picard group of P2 is generated by the hyperplane class denoted by H. The Picard group of P1 × P1 is generated by H1 = π1∗ (OP1 (1)) and H2 = π2∗ (OP1 (1)), where πi are the two projections. Let Fr , r ≥ 1, denote the Hirzebruch surface P(OP1 ⊕ OP1 (r)). The Picard group of Fr is the free abelian group generated by E and F, where E is the class of the section with self-intersection −r and F is the class of a fiber. We have the intersection numbers: E 2 = −r, E · F = 1, F 2 = 0. Let Dn denote the del Pezzo surface of degree n. The surface Dn is the blowup of P2 at 9 − n general points. The Picard group is the free abelian group generated by H and E1 , . . . , E9−n , where H is the pullback of the hyperplane class from P2 and Ei are the exceptional divisors of the blowup. We have the intersection numbers:


21

H 2 = 1, H · Ei = 0, Ei · E j = −δi, j , where δi, j denotes the Kronecker delta function. We summarize the nef cone of X [n] for various rational surfaces in the following theorem. Theorem 1. The nef cone X [n] , when X is a minimal rational surface or a del Pezzo surface with K 2 ≥ 2, is as follows: 1. [28, 3.12], [3, 3.1] The nef cone of P2[n] is the closed cone bounded by 1 (n − 1)H[n] − B and H[n]. 2 2. The nef cone of (P1 × P1 )[n] is the cone α H1 [n] + β H2 [n] + γ B2 satisfying the inequalities

γ ≤ 0, α + (n − 1)γ ≥ 0, and β + (n − 1)γ ≥ 0. [n]

3. The nef cone of the Hilbert scheme Fr of n points on the Hirzebruch surface Fr is the cone α (E[n] + rF[n]) + β F[n] + γ B2 satisfying the inequalities

γ ≤ 0, α + (n − 1)γ ≥ 0, and β + (n − 1)γ ≥ 0. [n]

4. The nef cone of the Hilbert scheme D9−r of n points on a del Pezzo surface D9−r of degree 9 − r is the cone aH[n] − b1 E1 [n] − · · · − br Er [n] + c B2 satisfying the following inequalities: • If 2 ≤ r ≤ 7, then c ≤ 0, bi + (n − 1)c ≥ 0, and a + (n − 1)c ≥ bi + b j for 1 ≤ i = j ≤ r. • If 5 ≤ r ≤ 7, then we require the additional inequality: 2a + (n − 1)c ≥

5

∑ bi j .

j=1

• If r = 7, we require the additional inequality: 6

3a + (n − 1)c ≥ 2bi + ∑ b jt , jt = i. t=1

Proof. By Construction 1, if L is an ample line bundle on X, then L[n] is nef on X [n] . By Construction 2, if L is an (n − 1)-very ample line bundle on X with vanishing higher cohomology, then L[n] − B2 is nef on X [n] . The proof has two parts. We first show that these two constructions generate the cones defined by the inequalities in

22


the theorem. Hence, the nef cone of X [n] contains the cone described in the theorem. We then exhibit curves in X [n] that realize each of the inequalities in the theorem. Since a nef divisor intersects every curve nonnegatively, this shows that the nef cone has to be contained in the cone defined by the inequalities in the theorem. Let R be a general fiber of the Hilbert-Chow morphism over the singular locus of X (n) . The curve R has the intersection numbers R · B = −2 and R · L[n] = 0 for any line bundle L on X. Consequently, the coefficient of B in any nef line bundle on X [n] has to be nonpositive. All other curves in X [n] that we exhibit have a very specific form. Let C be a curve in X that admits a g1n . The morphism f : C → P1 defined by the g1n induces a curve C(n) in X [n] . For the surfaces described in the theorem, the curve classes dual to the other faces of the nef cone will be of this form. We now carry out the analysis for each of the surfaces. If X = P2 , then OP2 (n − 1) is (n − 1)-ample [9, 28]. One may deduce this either using the criterion of Beltrametti and Sommese or using minimal resolutions directly. Let Z ∈ P2[n] . Then IZ admits a resolution: 0 −→

r

OP2 (−ai ) −→

i=1

r+1

OP2 (−b j ) −→ IZ −→ 0,

j=1

where 0 < ai ≤ n + 1 [17, Proposition 3.8]. Let d ≥ n − 1 and tensor the sequence by OP2 (d). The associated long exact sequence of cohomology implies that H 1 (P2 , IZ (d)) ∼ =

r

H 2 (P2 , OP2 (d − ai )) = 0

i=1

since d − ai ≥ −2. We conclude that OP2 (d) is (n − 1)-ample if d ≥ n − 1. Consequently, the nef cone contains the closed cone spanned by H[n] and (n − 1) H[n] − B2 . On the other hand, let C be a line on P2 ; then the induced curve C(n) satisfies the intersection numbers: C(n) · H[n] = 1, C(n) ·

B = n − 1. 2

Therefore, a nef divisor aH[n]+b B2 satisfies b ≤ 0 and a+(n−1)b ≥ 0. We conclude that the nef cone is the cone spanned by H[n] and (n − 1)H[n] − B2 . If X = P1 × P1 , using the criterion of Beltrametti and Sommese, it is easy to see that OP1 ×P1 (a, b) is (n − 1)-ample if a, b ≥ n − 1. We conclude that the nef cone of (P1 × P1 )[n] contains the closed cone spanned by B H1 [n], H2 [n] and (n − 1)H1[n] + (n − 1)H2[n] − . 2 On the other hand, let Fi be a fiber of the projection πi . Then the induced curve Fi (n) has intersection numbers:


23

Fi (n) · H j [n] = 1 − δi, j , Fi (n) ·

B = n − 1. 2

We conclude that a nef divisor a1 H1 [n]+ a2H2 [n]+ b B2 satisfies the inequalities b ≤ 0 and ai + (n − 1)b ≥ 0. Since any class satisfying these properties is a non-negative linear combination of H1 [n], H2 [n], and (n−1)H1 [n]+(n−1)H2 [n]− B2 , we conclude that the nef cone of (P1 × P1)[n] is the cone spanned by these classes. By the work of Beltrametti and Sommese ([8], [30, Lemma 10]), it is well-known that M = (n − 1)E + (r + 1)(n − 1)F is (n − 1)-very ample on Fr . Since the nef cone of Fr is generated by E + rF and F, we conclude that the cone spanned by E[n] + rF[n], F[n] and M[n] = (n − 1)(E[n] + rF[n]) + (n − 1)F[n] −

B 2

[n]

is contained in the nef cone of Fr . Consider the curves E(n) and F(n) induced in [n] Fr by a g1n on E and F, respectively. Then we have the intersection numbers: E(n) · E[n] = −r, E(n) · F[n] = 1, E(n) ·

B = n − 1, 2

B = n − 1. 2 Consequently, the nef cone is spanned by E[n] + rF[n], F[n], and M[n]. The strategy for the surfaces D9−r is identical. On D9−r , there are finitely many (−1)-curves whose classes can be listed explicitly (see [21, V.4] or [16]). Moreover, the (n − 1)-very ample line bundles on D9−r have been classified in [16]. Since D9 is isomorphic to P2 and D8 is isomorphic to F1 , we may assume that 2 ≤ r ≤ 8. When 2 ≤ r ≤ 4, then the (n − 1)-ample line bundles on D9−r are those with class aH − ∑ri=1 bi Ei such that bi ≥ n − 1 and a ≥ bi + b j + (n − 1) [16]. Hence, given a line bundle satisfying these inequalities, aH[n] − ∑ri=1 bi Ei [n] − B2 is nef on F(n) · E[n] = 1, F(n) · F[n] = 0, F(n) ·

[n]

Dr . In particular, −(n − 1)K − B2 = 3(n − 1)H[n] − (n − 1) ∑ri=1 Ei [n] − B2 is nef on [n]

D9−r . Similarly, setting n = 1, any line bundle satisfying bi ≥ 0 and a ≥ bi + b j [n]

is nef on D9−r . Hence, the divisors aH[n] − ∑ri=1 bi Ei [n] are nef on D9−r if bi ≥ 0 and a ≥ bi + b j . Since every divisor satisfying the inequalities in the theorem is a nonnegative linear combination of −(n − 1)K − B2 and L[n], where L is nef on D9−r , [n]

we conclude that the nef cone of D9−r contains the cone described in the theorem. Conversely, let R be a (−1)-curve. When 2 ≤ r ≤ 4, then the only possible classes for R are Ei or H − Ei − E j . If R = Ei , we have the intersection condition:

r

B R(n) · aH[n] − ∑ bi Ei [n] + c 2 i=1

= bi + (n − 1)c ≥ 0.

24


If R = H − Ei − E j , we have the intersection condition:

r

B R(n) · aH[n] − ∑ bi Ei [n] + c 2 i=1

= a − bi − b j + (n − 1)c ≥ 0.

We conclude that the nef cone is precisely the cone determined by the inequalities in the theorem. When 5 ≤ r ≤ 6, the condition that a line bundle is (n − 1)-ample requires the additional inequality 2a ≥ bi1 + · · · + bi5 + n − 1. Hence, −(n − 1)K − B2 is nef on [n]

D9−r and every divisor in the cone described in the theorem is a nonnegative linear combination of −(n − 1)K − B2 and L[n] for a nef line bundle L on D9−r . Conversely, there are new (−1)-curves R with class 2H − Ei1 − · · · − Ei5 . Intersecting a divisor class with R(n), we see that

r

B R(n) · aH[n] − ∑ bi Ei [n] + c 2 i=1

= a − bi1 − · · · − bi5 + (n − 1)c ≥ 0.

Hence, the nef cone is the cone determined by the inequalities in the theorem. When r = 7, assume first that n > 2. Then the same argument as in the previous two paragraphs determines the nef cone. The class −(n − 1)K − B2 is nef and the classes satisfying the inequalities in the theorem can be expressed as a nonnegative linear combination of −(n − 1)K − B2 and L[n] for a nef classes L on D9−r . The new inequalities come from the new types of (−1)-curves. The argument breaks down for n = 2 because −KD2 is not very ample; hence [2] one needs to argue that −K − B2 is nef on D2 . The linear system | − KD2 | defines a 2 two-to-one map f : D2 → P . Consequently, schemes of length two fail to impose independent conditions on sections of −KD2 if and only if they are fibers of the [2] map f . Hence, the base locus of the linear system | − K − B2 | is the P2 = Y ⊂ D2 parameterizing the fibers of f . We claim that the restriction of −K − B2 to Y is trivial. Since a line bundle on 2 P is determined by its degree, it suffices to show that the restriction of −K − B2 to Y has degree zero. The map f restricted to the zero locus C of a general section of −KD2 gives a two-to-one map from the elliptic curve C onto a line in P2 . Hence, a line on Y parameterizes the fibers of a hyperelliptic map on a section of −KD2 . Since (−KD2 )2 = 2 and the hyperelliptic map on an elliptic curve has four branch points, we conclude that the intersection of a line in Y with −K − B2 is 0. Therefore, the restriction of −K − B2 to Y is trivial. Consider the line bundle ε H[2] − K − B2 . Since H[2] is base-point-free, the base locus of this line bundle is contained in Y . On the other hand, this line bundle restricts to an ample line bundle on Y , hence is semi-ample by the Theorem of Fujita and Zariski [27, 2.1.32]. Since this is true for every ε > 0, we conclude that −K − B2 is nef. Now the rest of the argument follows as before. This concludes the proof of the theorem.


25

[n]

Remark 2. The nef cone of D1 is more complicated and highlights a shortcoming [n] of the method for producing nef divisors described so far. Writing a divisor on D1 as aH[n] − b1E1 [n] − · · · − b8 E8 [n] + c B2 we see that the nef cone has to satisfy the inequalities: c ≤ 0, bi + (n − 1)c ≥ 0, a + (n − 1)c ≥ bi + b j for 1 ≤ i = j ≤ r, 2a + (n − 1)c ≥

5

∑ bi j ,

j=1

4a + (n − 1)c ≥

3

5

t=1

t=1

6

3a + (n − 1)c ≥ 2bi + ∑ b jt , t=1

∑ 2bit + ∑ b jt ,

6

5a + (n − 1)c ≥ bi1 + bi2 + ∑ 2b jt , t=1

7

8

t=1

i=1

6a + (n − 1)c ≥ 3bi1 + ∑ 2b jt and 3a + nc ≥ ∑ bi . All the inequalities but the last one arise from curves of type C(n), where C is a (−1)-curve on the surface. The last inequality arises from −KD1 (n). The line bundles −(n − 1)KD1 and −nKD1 are no longer (n − 1)-ample. In fact, the class [n] −(n − 1)K − B2 is not nef on D1 since it does not satisfy the last inequality. The class −nK − B2 is nef, even though −nKD1 is not (n − 1)-ample. The base locus of the linear system −nK − B2 consists of the locus of n points that fail to impose independent conditions on the linear system | − nKD1 |. Let p be the base-point of the linear system | − KD1 |. A scheme Z of length n fails to impose independent conditions on the linear system | − nKD1 | if and only if Z is contained in a member C of the linear system | − KD1 | and is linearly equivalent to np on C. In other words, the base locus of the linear system −nK − B2 is an n-dimensional scroll Y over P1 . Since −KD1 (n) · (−nK − B2 ) = 0, the restriction of −nK − B2 to Y is equivalent to a multiple (in fact, n 2−n ) of the class of a fiber. Hence ε H[n] − nK − B2 is semi-ample for every ε > 0 by the theorem of Fujita and Zariski [27, 2.1.32]. Hence, −nK − B2 is nef. By [16], we conclude that any class of the form L[n] or L[n] − B2 , where L is 2

[n]

a line bundle on D1 , satisfying the inequalities is nef on D1 . Unfortunately, the cone defined by the inequalities is larger than the cone generated by such classes. There is a further source of nef divisors we have not explored in this paper. A vector bundle E of rank r on a surface X is called kvery ample if the map H 0 (X, E) → H 0 (X, E ⊗ OZ ) is surjective for every scheme Z of length k + 1. As in the case of line bundles, if E is an (n − 1)-very ample vector bundle of rank r on X with h0 (X, E) = N, we get a morphism φE : X [n] → G = G(N − rn, N). Then φ ∗ OG (1) is a base-point-free line bundle with class c1 (E)[n] − 2r B. To generate the entire cone defined by the inequalities one would have to use this improved construction; however, we will not pursue this here any further. For the next corollary, we simply note that, by [13], −mK − B2 is very ample [n]

on D1 for m ≥ n + 3.

26


Corollary 3. Let X be surface with ample anti-canonical bundle. Then X [n] is a log Fano variety and a Mori dream space. Proof. Recall that the surfaces with ample anti-canonical bundle are P2 , P1 × P1 and the del Pezzo surfaces Dr . By [10], a log Fano variety is a Mori dream space. Therefore, it suffices to check that X [n] is a log Fano variety when X has ample anti-canonical bundle. By Fogarty’s Theorem [18], the Hilbert-Chow morphism is a crepant resolution. Hence, −KX [n] = −KX [n]. Since −KX is ample on X, −KX [n] is big and nef. Hence, −(KX [n] + ε B) is ample for 1 >> ε > 0. Let l be the log canonical threshold of B. As long as l > ε , the pair (X, ε B) is klt. We conclude that X [n] is log Fano when X has ample anti-canonical bundle. The Minimal Model Program for a Mori dream space M is especially nice. The effective cone of M decomposes into finitely many Mori chambers. Moreover, the decomposition of the movable cone of M into Mori chambers is given by the ample cones of the birational models of M obtained via birational contractions. Furthermore, the nef divisors on each of these models are semi-ample. We refer the reader to [24] for further details. Hence, one can ask for a complete description of the decomposition of the effective cone of X [n] into Mori chambers and for the corresponding birational models when X is a del Pezzo surface. Remark 4. It would be interesting to compute the Cox ring of X [n] when X is a surface with ample anti-canonical bundle.

3 The Effective Cone of the Hilbert Scheme In this section, we study the effective cone of X [n] . The effective cone of X [n] can be very subtle and often depends on the existence of higher-rank vector bundles satisfying interpolation. We give several constructions of effective divisors and a construction of moving curves. In the next section, we show that for small n these constructions determine the entire effective cone. The Néron-Severi space of X [n] may be identified with the vector space spanned by the Néron-Severi space of X and the divisor class B. Under this identification, we can view the nef and pseudo-effective cones of X [n] for different n as cones in the same abstract vector space. Often the effective cones are easier to determine when n satisfies certain arithmetic properties. For example, the effective cone of P2[n] is r(r+1) easy to describe when n = 2 + j with −1 ≤ j ≤ 1, j = r − 1, or j = 2r for even r (see [3, Remark 4.6]). For other n, it is harder to construct the extremal effective ray. The following lemma then gives a way of bounding the effective cone in cases when it is not so easy to determine the cone. Lemma 5. Eff (X [n+1] ) ⊂ Eff (X [n] ). Proof. Let p be a point of X. Then there is a rational map ρ p : X [n] X [n+1] defined by mapping a zero-dimensional scheme Z to Z ∪ p. The map ρ p is well-defined


27

provided that p is not in the support of Z. The rational map ρ p induces a map ρ p∗ : Pic(X [n+1] ) → Pic(X [n] ) on the Picard groups. The map ρ p∗ maps L[n + 1] to L[n] and B to B; consequently, it induces the identity on the Néron-Severi spaces and is independent of p. Let D be an effective divisor on X [n+1]. Let Z ∈ D be a union of n + 1 distinct points p1 , . . . , pn+1 . Then the pullback of D by ρ pn+1 is an effective divisor on X [n] with the same class as D. We conclude that Eff (X [n+1] ) ⊂ Eff (X [n] ). Express a divisor class on X [n] as aL[n] + bB. In the region of the effective cone lying in the half-space b > 0, the stable base locus contains B as a divisorial component. Proposition 6. Let D = aL[n] + bB be an effective divisor with b > 0. Then the stable base locus of D contains the divisor of non-reduced schemes. If L is ample on X, then the model corresponding to D is the symmetric product X (n) and the induced map is the Hilbert-Chow morphism. Proof. Let R be the fiber of the Hilbert-Chow morphism over a general point of the diagonal of X (n) . Since the intersection numbers are R · L[n] = 0, R · B = −2, we conclude that R · D < 0 for any divisor D = aL[n] + bB with b > 0. Since curves in the class R cover the divisor B of non-reduced schemes, we conclude that B is in the base locus of D. Hence, the model corresponding to D is the same as the model corresponding to aL[n]. If L is ample on X, then L[n] is the pullback to X [n] of the ample line bundle L(n) on the symmetric product X (n) by the Hilbert-Chow morphism. Consequently, the birational model corresponding to L[n] is X (n) [27, 2.1.B]. The geometry of the surface plays a critical role in the half-space b ≤ 0. We first recall the construction of a large family of effective divisors on X [n] depending on vector bundles on X. Definition 7. Let E be a rank r vector bundle on X such that h0 (X, E) ≥ rn. Let W ⊂ H 0 (X, E) be an rn-dimensional subspace. We say that W satisfies interpolation for n points if W ∩ H 0 (X, E ⊗ IZ ) = 0 for a general Z ∈ X [n] . We say that E satisfies interpolation for n points if h0 (X, E) = rn and h0 (X, E ⊗ IZ ) = 0 for a general Z ∈ X [n] . Construction 3. Let W be a subset of H 0 (X, E) that satisfies interpolation for n points. Let E have rank r and let c1 (E) = D. Let Ξn ⊂ X [n] × X be the universal family with projections π1 and π2 . Then the locus where W ⊗ OX [n] → π1∗ (π2∗ (E) ⊗ OΞn ) fails to be an isomorphism is a determinantal subscheme of codimension one in X [n] . Hence, we obtain an effective divisor DE,W (n) on X [n] . By the GrothendieckRiemann-Roch Theorem, the class of DE,W (n) is r D[n] − B. 2

28


In particular, by finding vector bundles on X that satisfy interpolation, we generate a subcone of the effective cone. More generally, if W ⊂ H 0 (X, E) satisfies interpolation for n points and n ≥ n, the locus DE,W (n ) of schemes Z ∈ X [n ] that have a subscheme Z of length n such that W ∩ H 0 (X, E ⊗ IZ ) = 0 is a divisor in X [n ] . If W = H 0 (X, E), we omit it from the notation. Example 8. Line bundles satisfy interpolation. Hence, if L is a line bundle on X with h0 (X, L) = n, then L[n] − B2 is an effective divisor on X [n] . However, classifying vector bundles on X that satisfy interpolation for n points is a hard problem. Jack Huizenga has made progress in classifying Steiner bundles on P2 that satisfy interpolation. Let

0 1 3 8 −1 , , , ,··· , Φ = {x | x ≥ φ } ∪ 1 2 5 13 √

where φ = 1+2 5 is the golden ratio and the fractions are consecutive ratios of + s, s ≥ 0. Consider a general vector bundle Fibonacci numbers. Let n = r(r+1) 2 E given by the resolution: 0 → OP2 (r − 2)⊕ks → OP2 (r − 1)⊕k(s+r) → E → 0. Huizenga [25, Theorem 4.1] proves that for sufficiently large k, E is a vector bundle that satisfies interpolation for n points if and only if sr ∈ Φ . Similarly, let F be a general vector bundle given by the resolution 0 → F → OP2 (r)⊕k(2r−s+3) → OP2 (r + 1)⊕k(r−s+1) → 0. s+1 ∈ For sufficiently large k, F has interpolation for n points if and only if 1 − r+2 s r 2 2[n] Φ . We conclude that if r ∈ Φ , then (r − r + s)H[n] − 2 B is effective on P . If s+1 2[n] . ∈ Φ and s ≥ 1, then (r2 + r + s − 1)H[n] − r+2 1 − r+2 2 B is effective on P

Construction 4. A moving curve class C on a variety Y is a curve class whose representatives cover a Zariski-dense subset of Y . If D is an effective divisor on Y and C is a moving curve class, then C · D ≥ 0. Hence, each moving curve class on X [n] gives a bound on the effective cone of X [n] . Let L be a very ample line bundle on X such that h0 (X, L) > n. Suppose that a general section of L is a smooth curve of genus g < n. Then we obtain a moving curve in X [n] as follows. Since h0 (X, L) > n, by Bertini’s Theorem, a general scheme Z of length n is contained in a smooth curve δ ∈ |L|. The scheme Z defines a divisor on the curve δ . By the Riemann-Roch Theorem, h0 (δ , Z) ≥ 2, so there exists a map f : δ → P1 such that Z is a fiber of this map. The map f induces a curve C(n) in the Hilbert scheme X [n] . Since C(n) passes through a general point Z ∈ X [n] , we conclude that C(n) is a moving curve in X [n] . If D = M[n] + b B2 is an effective divisor, then we have the inequality C(n) · D = δ · M + b(g − 1 + n) ≥ 0, where the intersection number C(n) · B2 is computed using the Riemann-Hurwitz formula.


29

More generally, given a moving curve C in X [n] , we obtain a moving curve C in for n ≥ n by taking the unions of the schemes parameterized by C with a fixed general scheme Z of length n − n. X [n ]

s+1 Example 9. Let n = r(r+1) + s, 0 ≤ s ≤ r. If rs ∈ Φ , let L = rH, and if 1 − r+2 ∈ Φ, 2 let L = (r + 2)H in Construction 4. Then combining Constructions 3 and 4, we obtain the following description of the effective cone of (P2 )[n] .

Theorem 10 ([3, Theorem 4.5]). Let n = 1. If

s r

r(r+1) 2

+ s, 0 ≤ s ≤ r.

∈ Φ , then the effective cone of P2[n] is the closed cone bounded by the rays: H[n] −

r 2(r2 − r + s)

B and B.

2[n] is the closed cone 2. If 1 − s+1 r+2 ∈ Φ and s ≥ 1, then the effective cone of P bounded by the rays

H[n] −

r+2 2(r2 + r + s − 1)

B and B.

Construction 4 gives bounds on the ratio of the first Chern class and the rank of a vector bundle E on X that can satisfy interpolation for n points. If L[n] − α B2 is not effective, then a vector bundle E with c1 (E) = aL and rank r cannot satisfy interpolation for n points if ar ≥ α . Example 11. Let X = P1 × P1 and let a, b ∈ Z such that n = (a + 1)(b + 1). Then, by Construction 3, the locus of schemes Z ∈ (P1 × P1 )[n] that are contained in a curve of type (a, b) is an effective divisor with class aH1 [n] + bH2[n] − B2 . Hence, the effective cone contains the cone generated by these divisors and B. On the other hand, if (a + 1)(b + 1) > n > (a − 1)(b − 1), then every scheme of length n on P1 × P1 is contained in a curve of type (a, b). By Construction 4, any effective divisor α H1 [n] + β H2 [n] − γ2 B has to satisfy aβ + bα − γ (ab − a− b + n) ≥ 0. Example 12. Let X = Fr . Let a, b be integers such that b ≥ ar ≥ 0. Let n = (a + 1) (b + 1 − ra 2 ). To simplify notation, set D(i) = (a − i)E + (b − ri)F. Using induction on i for 0 ≤ i ≤ a − 1 and the exact sequence 0 → OFr (D(i + 1)) → OFr (D(i)) → OP1 (b − ri) → 0, we see that the higher cohomology of aE + bF vanishes. Hence, by the RiemannRoch formula, h0 (Fr , aE + bF) = n. Then, by Construction 3, the locus of schemes [n] Z ∈ Fr that are contained in a curve of class aE + bF is an effective divisor with class aE[n] + bF[n] − B2 . Hence, the effective cone contains the cone generated by these divisors and B.

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4 Stable Base Locus Decomposition of the Effective Cone of X [n] In this section, we describe the stable base locus decompositions of X [n] for small n when X is P1 × P1 or F1 . Even when n and the Picard rank of X are small, the stable base locus decomposition of X [n] can be very complicated. Moreover, the number of chambers grows very rapidly with n. We begin by a construction that helps determine the stable base locus. Construction 5. Construction 4 can be generalized to study the stable base loci of linear systems on X [n] . Let R be a smooth curve on X of genus g and suppose that a general scheme Z0 of length m ≤ n contained in R satisfies h0 (R, Z0 ) ≥ 2. Then we obtain a curve R(m, n) on X [n] by considering Zt = Zt ∪ Z , where Z is a fixed scheme of length n − m not supported on R and Zt are the fibers of a map f : R → P1 containing Z0 . The locus of schemes Z that have a subscheme of length m contained in a curve of type R is in the base locus of any linear system D = L[n] + b B2 such that L · R + b(g − 1 + m) < 0. Curves in the class R(m, n) sweep out this locus. Hence, any effective divisor that has negative intersection with R(m, n) has to contain this locus. Example 13. For example, schemes Z ∈ P2[n] that have a linear subscheme of length m are contained in the base locus of linear systems aH[n] − B2 if a < m − 1. Schemes Z ∈ P2[n] that have a subscheme of length m in a conic are contained in the base locus of linear systems aH[n] − B2 if 2a < m − 1. More generally, schemes Z ∈ P2[n] that + 1 ≤ m ≤ n contained in a curve of degree have a subscheme of length (d−1)(d−2) 2 − 1 + m. d are in the base locus of linear systems aH[n] − B2 if da < (d−1)(d−2) 2 These observations suffice to describe a large portion of the stable base locus decomposition of P2[n] . These decompositions have been described in detail in [3], so we will turn our attention to other surfaces. Example 14. Let n ≥ m > (a − 1)(b − 1). Then the locus of Z ∈ (P1 × P1)[n] that have a subscheme of length m contained on a curve of type (a, b) is in the base locus of a divisor α H1 [n] + β H2[n] − γ B2 if aβ + bα − γ (ab − a − b + m) < 0. The stable base locus decomposition of (P1 × P1 )[n] . In this subsection, we will compute the stable base locus decomposition of (P1 × P1 )[n] for 2 ≤ n ≤ 5 in full detail and discuss some aspects of the decomposition for general n. By Construction 3, the locus of schemes that have a subscheme of length two in a fiber with class Hi is the divisor DHi (n) on (P1 × P1 )[n] with class (n − 1)Hi [n] − B2 . To compute the class of DHi (n), let Fj , j = i, be a fiber with class H j and let R be a curve of type (1, 1). Since Fj · Hi = R · Hi = 1, the curves Fj (n, n) and R(n, n) defined in Construction 5 have intersection number zero with DHi (n). This determines the class of DHi (n) up to a constant. The constant can be determined by intersecting with Fj (1, n).


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Let D = a1 H1 [n] + a2 H2 [n] + c B2 . The moving curve Fi (1, n) has intersection number zero with both B and (n − 1)Hi [n] − B2 . Hence, the intersection of the effective cone of (P1 × P1 )[n] with the half-space a1 + a2 ≥ n − 1 is the cone generated by (n − 1)H1 [n] − B2 , (n − 1)H2 [n] − B2 , and B. The stable base locus decomposition is easy to understand in this subcone. Lemma 15. (1) Let n ≤ a + b + 1 and let Z ∈ (P1 × P1 )[n] . If Z does not have a subscheme of length b + 2 contained in a fiber with class H1 or a subscheme of length a + 2 contained in a fiber with class H2 , then Z imposes independent conditions on sections of OP1 ×P1 (a, b). (2) Let n = a + b + 2 and let Z ∈ (P1 × P1 )[n] . If Z is not contained in a curve of type (1, 1) and does not have a subscheme of length b + 2 contained in a fiber with class H1 or a subscheme of length a + 2 contained in a fiber with class H2 , then Z imposes independent conditions on sections of OP1 ×P1 (a, b). Proof. The lemma follows by induction on a and b and residuation. Consider the exact sequences: 0 → IZ1 (a − 1, b) → IZ (a, b) → IZ∩F1 ⊂F1 (a, b) → 0 0 → IZ2 (a, b − 1) → IZ (a, b) → IZ∩F2 ⊂F2 (a, b) → 0, where Zi are the residual schemes defined by the ideals (IZ : IFi ). By our assumption that none of the fibers with class H1 (respectively, H2 ) contain a subscheme of length b + 2 (a + 2), H 1 (IZ∩Fi ⊂Fi (a, b)) = 0. If there exists a fiber Fi with class Hi that contains a subscheme of length b + 1 when i = 1 or a + 1 when i = 2, then consider the residuation sequence with respect to Fi . Otherwise, consider the residuation sequence with respect to a fiber that contains the maximal length subscheme of Z. For concreteness, let us say that the fiber is F1 . By our choice of F1 , Z1 does not have a subscheme of length b + 2 in a fiber with class H1 and cannot have a subscheme of length a + 1 in a fiber with class H2 (otherwise the length of Z would be at least a + b + 2 > n). Hence, by induction H 1 (IZ (a − 1, b)) = 0. The long exact sequence of cohomology implies that H 1 (IZ (a, b)) = 0, proving (1). The proof of (2) is almost identical. If there are fibers that contain a subscheme of Z of length greater than one, then using the residuation sequence for the fibers, the proof of part (2) reduces to the proof of part (1). Otherwise, the residuation sequence 0 → IZ (a − 1, b − 1) → IZ (a, b) → IZ∩R⊂R (a, b) → 0 applied to a curve R of type (1, 1) containing a maximal length subscheme of Z and induction proves (2). Notation 2. Let Z(a, b; j) denote the locus of schemes in X [n] that have a subscheme of length j supported on a curve of type (a, b). For i, j ∈ Z, let Xi, j denote the divisor class iH1 [n] + jH2[n] − B2 .

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Theorem 16. (1) Let i, j ∈ Z be such that i + j > n − 1 and n − 2 ≥ i, j > 0. Then the cone generated by Xi, j , Xi+1, j , Xi+1, j+1 , Xi, j+1 is a chamber of the stable base locus decomposition, where the stable base locus consists of Z(1, 0; j + 2) ∪Z(0, 1; i + 2). (2) If j = n − 1 (respectively, i = n − 1) and n − 2 ≥ i ≥ 0 (respectively, n − 2 ≥ j ≥ 0), then the cone generated by Xi,n−1 , Xi+1,n−1 and H2 [n] (respectively, Xn−1, j , Xn−1, j+1 , H1 [n] ) is a chamber of the stable base locus decomposition, where the stable base locus is Z(0, 1; i + 2) (respectively, Z(1, 0; j + 2)). (3) If i + j = n − 1 and i, j > 0, then the cone generated by Xi, j , Xi−1, j+1, and Xi+1, j+1 is a chamber of the stable base locus decomposition, where the stable base locus consists of Z(1, 0; j + 2) ∪ Z(0, 1; i + 1). Proof. By Construction 2, if n > (i+1)( j +1), the linear systems |OP1 ×P1 (i, j)| give rise to rational maps to Grassmannians. If i + j ≥ n − 1, then (i + 1)( j + 1) = i j + i + j + 1 > n unless i + j = n − 1 and one of i or j is zero. Hence, Xi, j is an effective divisor with base locus equal to schemes that fail to impose independent conditions on the linear system |OP1 ×P1 (i, j)|. The divisors DH1 (n) and DH2 (n) have classes Xn−1,0 and X0,n−1 , respectively. Hence, their base locus is contained in DH1 (n) = Z(1, 0; 2) and DH2 (n) = Z(0, 1; 2), respectively. By Lemma 15, a scheme Z of length n imposes independent conditions on |OP1 ×P1 (i, j)| as long as Z is not contained in the locus Z(1, 0; j + 2) ∪ Z(0, 1; i + 2). Since every divisor in the cone generated by Xi, j , Xi+1, j , Xi+1, j+1 , Xi, j+1 is a nonnegative linear combination of Xi, j and the base-point-free divisors H1 [n] and H2 [n], we conclude that the stable base locus in this cone is contained in the locus Z(1, 0; j + 2)∪Z(0, 1; i+ 2). By the same argument, for divisors in the plane spanned by Xi+1, j and Xi+1, j+1 (respectively, Xi, j+1 and Xi+1, j+1), the stable base locus is contained in the locus of schemes Z(1, 0; j + 2) ∪ Z(0, 1; i + 3) (respectively, Z(1, 0; j + 3) ∪ Z(0, 1; i + 2)). Conversely, using the curves F1 ( j + 2, n) and F2 (i + 2, n) defined in Construction 5, where F1 and F2 are fibers with classes H1 and H2 , respectively, we see that the locus Z(1, 0; j + 2) ∪ Z(0, 1; i + 2) is contained in the base locus of every divisor a1 H1 [n] + a2H2 [n] + b B2 if a1 + (i + 1)b < 0 and a2 + ( j + 1)b < 0. We conclude that the cone generated by Xi, j , Xi+1, j , Xi+1, j+1 , Xi, j+1 is a chamber of the stable base locus decomposition, where the base locus is exactly Z(1, 0; j + 2) ∪ Z(0, 1; i + 2). This concludes the proof of part (1). The same argument shows that when i = n − 1, then in the cone generated by Xn−1, j , Xn−1, j+1 and H1 [n], the stable base locus is equal to Z(1, 0; j + 2). By the symmetry exchanging the two fibers, this proves part (2). When i + j = n − 1, the argument shows that in the cone generated by Xi, j , Xi−1, j+1 and Xi+1, j+1 the stable base locus is equal to Z(1, 0; j + 2) ∪ Z(0, 1; i + 1). To conclude that this cone is a chamber of the stable base locus decomposition, we use Construction 5 for a curve R of type (1, 1). Then R(n, n) has intersection number a1 + a2 + (n − 1)b with the divisor a1 H1 [n] + a2 H2 [n] + b B2 . Hence, if a1 + a2 + (n − 1)b < 0, then the locus Z(1, 1; n) is in the base locus of D. We


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conclude that the cones generated by Xi, j , Xi−1, j+1 , and Xi+1, j+1 with i + j = n − 1 form chambers of the stable base locus decomposition. This concludes the proof of part (3). Remark 17. Combining Theorems 16, and 1, and Proposition 6, we obtain the complete stable base locus decomposition of the cone generated by (n − 1)H1 [n] − B B 2 , (n − 1)H2 [n] − 2 , and B. We now turn to the explicit decomposition of the effective cone for 2 ≤ n ≤ 5. The automorphism of P1 × P1 exchanging the factors gives rise to a symmetry exchanging H1 [n] and H2 [n]. Hence, all the decompositions are symmetric with respect to the vertical axis. We explicitly explain one half of the diagrams and leave it to the reader to exchange H1 and H2 to obtain the rest. In each case, we will draw a cross-section of the cone and label the important rays by a meaningful divisor on that ray rather than the point that is contained in the cross-section. In order to avoid cluttering the diagrams, we will write Hi instead of Hi [n]. Example 18. Figure 1 shows the stable base locus decomposition of (P1 × P1 )[2] . Recall our convention that Xi, j = iH1 [2] + jH2 [2] − B2 . The chambers in this decomposition have the following descriptions: B

H2

H1 H1 + H2 − H1 −

B 2

B 2 H2 −

B 2

Fig. 1 The stable base locus decomposition of (P1 × P1 )[2]

1. The effective cone is the closed cone spanned by B, X1,0 , and X0,1 . 2. The base-point-free, nef, and moving cones coincide and are equal to the closed cone spanned by H1 [2], H2 [2], and X1,1 . 3. In the cone spanned by B, H1 [2], and H2 [2], the base locus is equal to B. 4. In the cone spanned by H1 [2], X1,0 , and X1,1 , the base locus is equal to Z(1, 0; 2). 5. Finally, in the cone spanned by X1,0 , X0,1 , and X1,1 , the base locus is equal to Z(1, 0; 2) ∪ Z(0, 1; 2). Proof. Theorem 1 (2) describes the base-point-free and nef cones of (P1 × P1 )[n] in general. Proposition 6 proves that the base locus in the cone spanned by H1 [2], H2 [2], and B contains the divisor B. Since Hi [n] are base-point-free, part (3) follows. By Theorem 16, the cone generated by X1,0 , X1,1 , and H1 [2] is a chamber of the

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stable base locus decomposition with base locus equal to DH1 (2) = Z(1, 0; 2). By symmetry, the base locus in the cone spanned by X0,1 , X1,1 , and H2 [2] is the divisor DH2 (2) = Z(0, 1; 2). By Theorem 16, in the cone generated by X0,1 , X1,0 , and X1,1 , the base locus is the union of the divisors Z(1, 0; 2) ∪ Z(0, 1; 2). Hence, the moving cone of (P1 × P1)[2] is equal to the base-point-free cone in this case. To complete the proof there remains to show that the effective cone is equal to the cone spanned by X1,0 , X0,1 , and B. We already know that these divisors are effective, so it suffices to give moving curves dual to each face of the cone. The moving curves F1 (1, 2) and F2 (1, 2) defined in Construction 5 are dual to the faces spanned by X1,0 , B, and X0,1 , B, respectively. Similarly, let R be a curve of type (1, 1). The moving curve R(2) defined in Construction 4 is dual to the face spanned by X1,0 and X0,1 . This concludes the discussion of this example. Example 19. Figure 2 shows the stable base locus decomposition of (P1 × P1 )[3] . The chambers have the following description B

H1

H2

X2,1

2H1 −

B 2

X2,2

H1 + H2 −

X1,2 B 2

2H2 −

B 2


1. 2. 3. 4.

The effective cone is the closed cone spanned by B, X2,0 , and X0,2 . The base-point-free cone is the closed cone generated by H1 [3], H2 [3], and X2,2 . The moving cone is the closed cone generated by H1 [3], H2 [3], and X1,1 . In the cone generated by B, H1 [3], and H2 [3], the base locus is divisorial equal to B. 5. For 0 ≤ i ≤ 1, in the cone generated by H1 [3], X2,i , and X2,i+1 , the base locus is Z(1, 0; i + 2). 6. In the cone generated by X2,0 , X1,1 , and X2,1 , the base locus is Z(1, 0; 2) ∪ Z(0, 1; 3). 7. Finally, in the cone generated by X1,1 , X2,1 , X2,2 , and X1,2 , the base locus is Z(1, 0; 3) ∪ Z(0, 1; 3). Proof. Part (2) follows from Theorem 1. Part (4) follows from Proposition 6. Parts (5), (6), and (7) follow from Theorem 16. Since B, Z(1, 0; 2), and Z(0, 1; 2) are divisors and Z(1, 0; 3) and Z(0, 1; 3) have codimension 2; part (3) follows from parts (4), (5), (6), and (7). Hence, there remains to prove part (1). The divisors X2,0 , X0,2 ,


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and B are effective. To show that the effective cone is equal to the cone generated by them, we exhibit dual moving curves. The moving curves F1 (1, 3) and F2 (1, 3) are dual to the faces spanned by X2,0 , B and X0,2 , B, respectively. Let R be a curve of type (1, 1). Then, the moving curve R(3) is dual to the face spanned by X2,0 and X0,2 . This concludes the discussion of this example. Example 20. Figure 3 shows the stable base locus decomposition of (P1 × P1 )[4] . The chambers in the decomposition have the following descriptions B H1

H2 X3,3 X3,1

3H1 −

B 2

X1,3

X2,2

X2,1

X1,2

H1 + H2 −

3H2 −

B 2

B 2


1. 2. 3. 4. 5. 6.

7. 8. 9.

The effective cone is the closed cone generated by B, X3,0 , X1,1 , and X0,3 . The base-point-free cone is the closed cone spanned by H1 [4], H2 [4], and X3,3 . The moving cone is the closed cone spanned by H1 [4], H2 [4], X1,2 , and X2,1 . In the cone spanned by H1 [4], H2 [4], and B, the base locus is divisorial equal to B. The stable base locus contains Z(1, 0; 2) (respectively, Z(0, 1; 2)) exactly in the cone spanned by H1 [4], X3,0 , and X1,1 (respectively, H2 [4], X0,3 , and X1,1 ). The stable base locus contains the divisor Z(1, 1; 4) exactly in the cone spanned by X3,0 , X0,3 , and X1,1 . Hence, to give a complete description of the stable base locus decomposition, it suffices to give the stable base locus decomposition of the moving cone. In the cone spanned by X2,1 , X1,2 , and X2,2 , the stable base locus is Z(1, 0; 3) ∪ Z(0, 1; 3). In the cone spanned by Xi, j , Xi+1, j , Xi+1, j+1, Xi, j+1 , for i + j ≥ 3 and 2 ≥ i, j > 0, the stable base locus is Z(1, 0; j + 2) ∪ Z(0, 1; i + 2). In the cone spanned by H1 [4], X3,i , and X3,i+1 , for 1 ≤ i ≤ 2, the base locus is Z(1, 0; i + 2).

Proof. Parts (2) and (4) follow from Theorem 1 and Proposition 6. Parts (7), (8), (9) follow from Theorem 16. Parts (4)–(9) imply (3). Since H1 [4] and H2 [4] are basepoint-free, in the cone spanned by X1,1 , H1 [4], and H2 [4], the base locus is contained in the divisor Z(1, 1; 4). Similarly, in the cone spanned by X3,0 , X0,3 , and H1 [4] and

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H2 [4], the base locus is contained in the union of divisors Z(1, 0; 2) ∪ Z(0, 1; 2). The curves F1 (2, 4) and F2 (2, 4) defined in Construction 5 are dual to the faces H1 [4], X1,1 and H2 [4], X1,1 . Therefore, the divisors Z(1, 0; 2) and Z(0, 1; 2) are contained in the base locus precisely in the cones generated by H1 [4], X3,0 , X1,1 and H2 [4], X0,3 , X1,1 , respectively, proving (5). Let Ri, j denote a curve of type (i, j). The curve R1,1 (4, 4) defined in Construction 5 is dual to the face spanned by X3,0 and X0,3 . Therefore, the divisor Z(1, 1; 4) is in the stable base locus exactly in the cone spanned by X1,1 , X3,0 , and X0,3 , proving (6). Finally, to prove (1), note that the moving curves F1 (1, 4), R2,1 (4), R1,2 (4), and F2 (1, 4) defined in Constructions 4 and 5 are dual to the faces [B, X3,0 ], [X3,0 , X1,1 ], [X1,1 , X0,3 ], and [X0,3 , B], respectively. Example 21. Figure 4 shows the stable base locus decomposition for (P1 × P1 )[5] . The locus Z(1, 1; 4) of schemes that have a subscheme of length 4 contained in a curve of type (1, 1) is a divisor with class 4H1 [5]+4H2[5]− 32 B. It is easy to calculate this class by intersecting with test curves. Let R be an irreducible curve of type (1, 2) or (2, 1). Since a curve of type (1, 1) has intersection number 3 with R, as long as five points vary on R, the scheme they determine does not lie in Z(1, 1; 4). Hence, the curves R(5) defined in Construction 4 are dual to Z(1, 1; 4) and determine its class up to a multiple, which can easily be determined by pairing with another curve. Since these curves are also dual to the faces X4,0 , Z(1, 1; 4) and X0,4 , Z(1, 1; 4), we conclude that the effective cone of (P1 × P1 )[5] is the cone spanned by B, X4,0 , Z(1, 1; 4), and X0,4 . The chambers of the stable base locus decomposition have the following description: 1. 2. 3. 4. 5.

6.

7. 8. 9. 10. 11.

The effective cone is the closed cone spanned by B, X4,0 , Z(1, 1; 4) and X0,4 . The base-point-free cone is the closed cone spanned by H1 [5], H2 [5], and X4,4 . The moving cone is the closed cone spanned by H1 [5], H2 [5], X1,2 , and X2,1 . In the cone generated by B, H1 [5], and H2 [5], the base locus is the divisor B. The divisor Z(1, 0; 2) (respectively, Z(0, 1; 2)) is in the stable base locus precisely in the cone generated by H1 [5], X4,0 , and X2,1 (respectively, H2 [5], X0,4 , and X1,2 ). The divisor Z(1, 1; 4) is contained in the base locus precisely in the cone spanned by X1,2 , X2,1 , and Z(1, 1; 4). Hence, it suffices to describe the stable base locus in the moving cone to get a complete description of the stable base locus. Let i + j > 3, 3 ≥ i, j > 0. In the cone spanned by Xi, j , Xi+1, j , Xi+1, j+1 , and Xi, j+1 , the base locus is Z(1, 0; j + 2) ∪ Z(0, 1; i + 2). Let 0 ≤ i ≤ 3. In the cone generated by X4,i , X4,i+1 , and H1 [5], the base locus is Z(1, 0; i + 2). In the cone generated by X3,1 , X2,2 , and X3,2 , the base locus is Z(1, 0; 3) ∪ Z(0, 1; 4). In the cone generated by X2,1 , X3,1 , and X2,2 , the base locus is Z(1, 0; 3) ∪ Z(0, 1; 4) ∪ Z(1, 1; 5). Finally, in the cone spanned by X2,1 , X1,2 , and X2,2 , the base locus is Z(1, 0; 3) ∪ Z(0, 1; 3) ∪ Z(1, 1; 5).


37 B

H1

H2 X4,3

X4,1

X4,2

X4,4

X3,4

X3,3

X1,4

X2,3

X3,2 X3,1

X2,4

X2,2

X1,3

B 4H1 − 2

4H2 − X2.1

4H1 + 4H2 −

B 2

X1,2

3 B 2


Proof. We proved (1) before stating the decomposition. Parts (2), (4), (7), (8), and (9) follow from Theorems 1, 16 and Proposition 6. The curves F1 (2, 5) and F2 (2, 5) are dual to the faces spanned by X2,1 H1 [5] and X1,2 H2 [5], respectively. Consequently, the divisors Z(1, 0; 2) and Z(0, 1; 2) are in the base loci in the cones spanned by X2,1 , X4,0 , and H1 [5] and X1,2 , X0,4 and H2 [5], respectively. Similarly, R1,1 (4, 5) is dual to the face spanned by X2,1 and X1,2 , so the divisor Z(1, 1; 4) is in the base locus in the cone generated by Z(1, 1; 4), X1,2 , and X2,1 . On the other hand, the divisors X1,2 and X2,1 are pullbacks of O(1) from the Grassmannian via the rational map induced by the linear systems |OP1 ×P1 (1, 2)| and |OP1 ×P1 (2, 1)|, respectively. By Lemma 15, a scheme Z imposes independent conditions on these linear systems unless Z is contained in Z(1, 1; 5) ∪ Z(1, 0; 3) ∪ Z(0, 1; 3). Since H1 [5] and H2 [5] are base-point-free, it follows that the moving cone is equal to the cone generated by X2,1 , X1,2 , H2 [5], and H1 [5]. This proves parts (3), (5), and (6). Finally, to conclude the proof, note that the curve R1,1 (5, 5) is dual to the face generated by X4,0 and X0,4 . Hence, Z(1, 1; 5) is contained in the base locus of any divisor in the cone generated by Z(1, 1; 4), X4,0 , and X0,4 . Parts (10), (11) follow. This concludes the discussion of this example. Remark 22. We will refrain from listing the explicit cone decompositions for n > 5. However, the reader should have no trouble determining these decompositions for the next few cases. The reason for listing these decompositions in such great detail will be apparent when we match these to the Bridgeland walls in the last section.

38

A. Bertram and I. Coskun [n]

The Stable Base Locus Decomposition of Fr . In this subsection, we discuss [n] some general features of the stable base locus decomposition of Fr . We also [n] compute the complete decomposition for F1 , when 2 ≤ n ≤ 4. [n] Denote divisors on Fr by aE[n] + bF[n] + c B2 . By Construction 3, the locus of schemes that have a subscheme of length two contained in a fiber is an effective [n] divisor in Fr with class (n − 1)F[n] − B2 . Consequently, the effective cone contains the cone spanned by E[n], (n − 1)F[n] − B2 , and B. Moreover, the intersection of the effective cone with the half-space b + (n − 1)c ≥ 0 equals this cone. Let R be a curve of class E + rF. Then, applying Construction 4, we conclude that the face generated by E[n] and B is dual to R(1), which is a moving curve. Similarly, the moving curve F(1) is dual to the face generated by (n − 1)F[n] − B2 and B. The stable base locus decomposition in the subcone generated by E[n] + rF[n], (n − 1)F[n] − B2 , and B is easy to describe. Lemma 23. Let |aE + (ar + b)F| be a linear system on Fr such that 0 ≤ a, b. 1. If Z is a scheme of length n ≤ a + b + 1, then Z imposes independent conditions on the linear system unless Z has a subscheme of length a + 2 contained in a fiber or a subscheme of length b + 2 contained in the exceptional curve. 2. If Z is a scheme of length n = a + b + 2, then Z imposes independent conditions on the linear system unless Z is contained in a curve with class E + rF or has a subscheme of length a + 2 contained in a fiber or a subscheme of length b + 2 contained in the exceptional curve. Proof. The lemma follows by the residuation and induction on a and b. To simplify notation, set D(a, b) = aE + (ar + b)F. Consider the three exact sequences, 0 → IZF (D(a, b) − F) → IZ (D(a, b)) → IZ∩F⊂F (D(a, b)) → 0, 0 → IZE (D(a, b) − E) → IZ (D(a, b)) → IZ∩E⊂E (D(a, b)) → 0, and 0 → IZR ((a − 1)(E + rF)) → IZ (a(E + rF)) → IZ∩R⊂R (a(E + rF)) → 0. If Z has a subscheme of length a + 1 contained in a fiber F or of length b + 1 contained in E, we use the first and second exact sequences, respectively, and induction to conclude that H 1 (IZ (aE + (ar + b)F)) = 0. Otherwise, if b > 0, we use the first exact sequence, where F is a fiber containing a maximal length subscheme of Z, and induction to conclude that H 1 (IZ (aE + (ar + b)F)) = 0. If b = 0, we use the third exact sequence, where R is a curve of class E + rF containing a maximal length subscheme of Z, and induction to conclude that H 1 (IZ (aE + (ar + b)F)) = 0. [n]

Notation 3. Let Z(a, b; m) denote the locus of Fr parameterizing schemes that have a subscheme of length m contained in a curve with class aE + bF. Let Xi, j denote the divisor i(E[n] + rF[n]) + jF[n] − B2 . Xi, j is effective if and only if i ≥ 0 and j ≥ −ir.


39

Theorem 24. (1) Let n − 2 ≥ i, j > 0 and let i + j > n − 1. The cone generated by Xi, j , Xi+1, j , Xi+1, j+1 , and Xi, j+1 is a chamber of the stable base locus decomposition, where the base locus is Z(1, 0; j + 2) ∪ Z(0, 1; i + 2). (2) Let i = n − 1 and n − 2 ≥ j ≥ 0 (respectively, j = n − 1 and n − 2 ≥ i ≥ 0). The cone generated by E[n] + rF[n], Xn−1, j , and Xn−1, j+1 (respectively, F[n], Xi,n−1 , and Xi+1,n−1) is a chamber of the stable base locus decomposition, where the stable base locus is Z(1, 0; j + 2) (respectively, Z(0, 1; i + 2)). (3) The locus Z(1, r; n) is contained in the stable base locus of a divisor aE[n] + bF[n] + c B2 if and only if b + (n − 1)c < 0. (4) If −ir ≤ j < 0, then the stable base locus contains the divisor E[n] = Z(1, 0; 1). Proof. The divisor Xi, j is the pullback of O(1) from the Grassmannian by the rational map induced by the linear system |i(E + rF) + jF|. By Lemma 23, if n ≤ i + j + 1, the map is a morphism along the locus of schemes Z that are not contained in Z(1, 0; j + 2) ∪ Z(0, 1; i + 2). Hence, the stable base locus of Xi, j is contained in Z(1, 0; j +2)∪Z(0, 1; i+2). Since E[n]+rF[n] and F[n] are base-pointfree, we conclude that the stable base locus of the divisors contained in the cone generated by Xi, j , Xi+1, j , Xi+1, j+1, Xi, j+1 is contained in Z(1, 0; j + 2) ∪ Z(0, 1; i + 2). On the other hand, consider the curves E( j + 2, n) and F(i + 2, n) defined in Construction 5. These curves have intersection number zero with divisors along the face generated by Xi, j+1 Xi+1, j+1 and Xi+1, j Xi+1, j+1 , respectively. Consequently, the base locus in the cone generated by Xi, j , Xi+1, j , Xi+1, j+1 , Xi, j+1 contains Z(1, 0; j + 2) ∪Z(0, 1; i + 2). Part (1) of the theorem follows. A similar argument proves (2). To finish the proof of (3), we observe that the curve R(n, n) defined in Construction 5, where R has the class E + rF, is dual to the face spanned by E[n] and (n − 1)F[n] − B2 . We conclude that the locus Z(1, r; n) is contained in the base locus of any divisor with b + (n − 1)c < 0. If n ≤ r + 1, then every scheme of length n is [n] contained in a curve of class E + rF and we conclude that the effective cone of Fr B is the cone E[n], (n − 1)F[n] − 2 , and B. We may assume that n > r + 1. Since the base locus of any divisor with b + (n − 1)c ≥ 0 is contained in the union of the base loci of E[n], B, and (n − 1)F − B2 , which is equal to Z(1, 0; 1) ∪ B ∪ Z(0, 1; 2), we conclude that Z(1, r; n) is not contained in the stable base locus of such a divisor. Finally, since the curve E(1, n), whose deformations cover the divisor E[n], has negative intersection number with any Xi, j such that −ir ≤ j < 0, E[n] is in the base locus. This concludes the proof of the theorem. [2]

Example 25. Figure 5 shows the stable base locus decomposition of F1 . The chambers have the following description: 1. The effective cone is the closed cone spanned by B, E[2], and X0,1 . 2. The base-point-free cone is the closed cone spanned by E[2] + F[2], F[2], and X1,1 . 3. The moving cone is the closed cone spanned by X1,0 , E[2] + F[2], and F[2]. 4. In the cone spanned by X1,0 , X1,1 , and E[2] + F[2], the stable base locus is Z(1, 0; 2).

40

A. Bertram and I. Coskun B

E

E+ F

F X1,1

X1,0 F−

B 2 [2]

Fig. 5 The stable base locus decomposition of Fr

5. 6. 7. 8. 9.

In the cone spanned by B, E[2] + F[2], and F[2], the base locus is B. In the cone spanned by E[2], E[2] + F[2], and B, the base locus is B ∪ Z(1, 0; 1). In the cone spanned by E[2], E[2] + F[2], and X1,0 , the base locus is Z(1, 0; 1). In the cone spanned by F[2], X1,1 and X0,1 , the base locus is Z(0, 1; 2). In the cone spanned by X1,0 , X1,1 , and X0,1 , the base locus is Z(1, 0; 2) ∪ Z(0, 1; 2).

Proof. Theorems 1, and 24, and Proposition 6 imply (2), (4), (5), (6), (7), and (8). The effective cone contains the cone spanned by B, E[2] and X0,1 . In view of the discussion preceding Theorem 24, to prove (1), it suffices to exhibit a moving curve dual to the face spanned by E[2], X0,1 . Let R be a curve in the class E + F. Then the curve R(2) defined in Construction 4 is the required moving curve. Since the base loci described in parts (5)–(9) all contain a fixed divisor and the base locus in (4) is not divisorial; parts (4)–(9) imply (3). The curves E(1, 2), E(2, 2) and F(2, 2) defined in Construction 5 are dual to the faces spanned by [B, E[2] + F[2]], [E[2] + F[2], X0,1], and [F[2], X1,1 ], respectively. Part (9) follows. [3]

Example 26. Figure 6 shows the stable base locus decomposition of F1 . The chambers have the following descriptions: 1. The effective cone is the closed cone generated by E[3], B, X0,2 , and X1,0 . 2. The base-point-free cone is the closed cone generated by E[3] + F[3], F[3], and X2,2 . 3. The moving cone is the closed cone generated by E[3] + F[3], F[3], X1,1 , and X2,0 . 4. The divisor B is in the base locus precisely in the cone spanned by E[3], F[3], and B. 5. The divisor Z(1, 0; 1) is in the base locus precisely in the cone spanned by E[3], B, and X1,0 .


41

B

F

E +F

X2,2

X1,2

X2,1 E

X2,0

X1,1 2F −

E +F−

B 2

B 2 [3]

Fig. 6 The stable base locus decomposition of F1

6. The divisor Z(1, 1; 3) is in the base locus precisely in the cone spanned by E[3], X0,2 , and X1,0 . 7. The divisor Z(0, 1; 2) is the base locus precisely in the cone spanned by F[3], X1,0 , and X0,2 . We are thus reduced to describing the stable base locus decomposition of the moving cone. 8. In the cone spanned by E[3] + F[3], X2,1 , and X2,2 , the stable base locus is Z(1, 0; 3). 9. In the cone spanned by F[3], X1,2 , and X2,2 , the stable base locus is Z(0, 1; 3). 10. In the cone spanned by X1,1 , X1,2 , X2,2 , and X2,1 , the stable base locus is Z(1, 0; 3) ∪ Z(0, 1; 3). 11. In the cone spanned by X2,0 , E[3] + F[3], and X2,1 , the stable base locus is Z(1, 0; 2). 12. In the cone spanned by X2,0 , X1,1 , and X2,1 , the stable base locus is Z(0, 1; 3) ∪ Z(1, 0; 2). [2]

Since the proof is analogous to the cases of F1 and (P1 × P1 )[3] , we leave it to the reader.

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Example 27. We complete our discussion of the stable base locus decomposition of [n] [4] Fr by describing the stable base locus decomposition of F1 . Figure 7 shows the decomposition. The chambers have the following interpretations: B

E

X3,0

E +F X3,3

F X2,3

X2,1 X2,0

X1,3

X1,2 X0,3

X1,1 3E + 3F − B [4]

Fig. 7 The stable base locus decomposition of F1

1. The effective cone is the closed cone spanned by B, E[4], 3E[4] + 3F[4] − B, and X0,3 . 2. The base-point-free cone is the closed cone spanned by E[4] + F[4], F[4], and X3,3 . 3. The moving cone is the closed cone spanned by X2,0 , X1,1 , F[4], and E[4] + F[4]. 4. The divisor B is in the base locus in the cone generated by B, E[4], and F[4]. The divisor Z(1, 0; 1) is in the base locus in the cone generated by B, E[4], and 3E[4]+ 3F[4]− B. The divisor Z(1, 1; 3) is in the base locus in the cone generated by X1,1 , E[4], and 3E[4] + 3F[4] − B. Finally, the divisor Z(0, 1; 2) is in the base locus in the cone generated by X1,1 , X0,3 , and F[4]. Hence, it suffices to describe the decomposition of the moving cone. 5. The decomposition in the cone spanned by X3,0 , X0,3 , F[4], and E[4] + F[4] is as in Theorem 24. 6. In the cone spanned by X2,0 , X2,1 , X3,0 , the stable base locus is Z(1, 1; 4) ∪ Z(1, 0; 2) ∪ Z(0, 1; 4). 7. In the cone spanned by X2,0 , X1,1 , X2,1 , the stable base locus is Z(1, 1; 4) ∪ Z(1, 0; 2) ∪ Z(0, 1; 3). 8. Finally, in the cone spanned by X1,1 , X1,2 , X2,1 , the stable base locus is Z(1, 1; 4) ∪ Z(1, 0; 3) ∪ Z(0, 1; 3).


43

Proof. The divisor Z(1, 1; 3) of schemes that have a subscheme of length 3 contained in a curve with class E + F has class 3E[4] + 3F[4] − B. The proof of parts (1)–(5) is now analogous to the previous cases. By Lemma 23, the base loci in the cones described in parts (6)–(8) are contained in the claimed loci. The curves E(m, 4), F(m, 4), and R(4, 4), where R is a curve of class E + F on F1 , defined in Construction 5 show that the claimed loci are contained in the stable base locus. This completes the proof.

5 Preliminaries on Bridgeland Stability and Bridgeland Walls In this section, we recall preliminaries concerning Bridgeland stability. We refer the reader to [1, 2, 11] for more detailed information. We then determine the general features of Bridgeland walls. Bridgeland Stability Conditions. Let X be a smooth projective variety. Let Db (X) denote the bounded derived category of coherent sheaves on X. A Bridgeland stability condition σ on X consists of a pair σ = (A, Z) such that A is the heart of a bounded t-structure on Db (X) and Z : K(Db (X)) → C is a homomorphism satisfying the following properties: 1. (Positivity) For every nonzero object E of A, Z(E) lies in the semi-closed upper half-plane: Z(E) = reiπθ , where r > 0, 0 < θ ≤ 1. Writing Z = −d(E) + ir(E), one may view this condition as two separate positivity conditions requiring r(E) ≥ 0, and if r(E) = 0, then d(E) > 0. 2. (Harder-Narasimhan property) For an object E of A, let the Z-slope of E be defined by setting μ (E) = d(E)/r(E) with the understanding that μ (E) = ∞ if r(E) = 0. An object E is called Z-(semi)-stable, if for every proper subobject F, μ (F)(≤) < μ (E). The pair (A, Z) is required to satisfy the Harder-Narasimhan property. Namely, every object of A has a finite filtration 0 = E0 → E1 → · · · → En = E such that Fi = Ei /Ei−1 is Z-semi-stable and μ (Fi ) > μ (Fi+1 ) for all i. One also imposes a technical support property, which we will not mention here. The set of stability conditions on Db (X) satisfying these three properties is called the stability manifold of X and is denoted by Stab(X). In [11], Bridgeland proves the following theorem. Theorem 28 (Bridgeland). The map (A, Z) → Z is a local homeomorphism onto an open set in a linear subspace of Hom(K(Db (X)), C). In particular, the space Stab(X) of stability conditions on X is a complex manifold.

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When dim(X) > 2, we do not know in general whether the stability manifolds Stab(X) are nonempty (see [6, 32] for a discussion and references). When X is a surface, Bridgeland [12] and Arcara and Bertram [2] have constructed stability conditions. We will only use the region in the stability manifold corresponding to these special stability conditions. Example 29. If X is a curve, then setting A to be the category of coherent sheaves on X and Z(E) = − deg(E) + i rk(E), one obtains a Bridgeland stability condition. If X is a surface and H is an ample line bundle on X, one can still define A to be the category of coherent sheaves on X and Z(E) = − degH (E) + i rk(E), where the degree is measured with respect to the ample line bundle H. However, this is not a Bridgeland stability condition because Z is zero on sheaves supported on points. The idea of Bridgeland, Arcara, and Bertram is to fix this problem by tilting the category. Bridgeland Stability Conditions for Surfaces. For the remainder of this section, let X be a smooth, projective surface and let H be an ample line bundle. Mumford stability with respect to H gives rise to a Harder-Narasimhan filtration. Definition 30. Given s ∈ R, define full subcategories Qs and Fs of coh(X) by the following conditions on their objects: • Q ∈ Qs if Q is torsion or if each μi > sH 2 in the Harder-Narasimhan filtration of Q. • F ∈ Fs if F is torsion-free, and each μi ≤ sH 2 in the Harder-Narasimhan filtration of F. By [12, Lemma 6.1], each pair (Fs , Qs ) of full subcategories satisfies the two properties: (a) For all F ∈ Fs and Q ∈ Qs , Hom(Q, F) = 0. (b) Every coherent sheaf E fits in a short exact sequence 0 → Q → E → F → 0, where Q ∈ Qs , F ∈ Fs , and the extension class are uniquely determined up to isomorphism. A pair of full subcategories (F , Q) of an abelian category A satisfying conditions (a) and (b) is called a torsion pair. A torsion pair (F , Q) defines a t-structure on Db (A) [20] with D≥0 = {complexes E | H−1 (E) ∈ F and Hi (E) = 0 for i < −1} D≤0 = {complexes E | H0 (E) ∈ Q and Hi (E) = 0 for i > 0} The heart of the t-structure defined by a torsion pair consists of {E | H−1 (E) ∈ F , H0 (E) ∈ Q, and Hi (E) = 0 otherwise}.


45

The natural exact sequence 0 → H−1 (E)[1] → E → H0 (E) → 0 for such an object of Db (A) implies that the objects of the heart are all given by pairs of objects F ∈ F and Q ∈ Q together with an extension class in Ext2A (Q, F) [20]. Definition 31. Let As be the heart of the t-structure on Db (coh(X)) obtained from the torsion pair (Fs , Qs ) in Definition 30. Define the central charge by setting Zs,t (E) = −

e−(s+it)H ch(E).

X

With this definition, (As , Zs,t ) is a Bridgeland stability condition. Theorem 32 (Bridgeland [12], Arcara–Bertram [2], Bayer–Macrì [4]). For each s ∈ R and t > 0, the pair (As , Zs,t ) defines Bridgeland stability conditions on Db (coh(X)). Bridgeland Walls. Fix a class ν in the numerical Grothendieck group. Then there exists a locally finite set of walls in Stab(X), depending only on ν , such that as the stability condition σ varies in a chamber, the set of σ -(semi)-stable objects of class ν does not change [4, 5, 12]. We will call these walls Bridgeland walls. We are interested in calculating the Bridgeland walls in the case of an ideal sheaf IZ of n points on X. We will record the numerical invariant by (ch0 , ch1 , ch2 ). If Z ∈ X [n] , then the corresponding invariant is (1, 0, −n). It has been worked out in several contexts [3, 29] that the potential walls are lines or nonintersecting nested semicircles. Let (s,t) be a point of a Bridgeland wall. Then there exists an object E destabilizing an object of the category As with invariant (1, 0, −n). Hence, the Zs,t -slope of E has to equal the Zs,t -slope of an object with invariant (1, 0, −n); 2 ℜ(Zs,t (E)) ch2 (E) − s ch1 (E) · H + s −t 2 ch0 (E)H = μs,t (E) = − . ℑ(Zs,t (E)) t ch1 (E) · H − st ch0 (E)H 2 2

2

In particular, 2 n − s −t 2 H . 2 st H Equating the two slopes and assuming that t > 0 and s < 0, we get the equation of a semicircle: 2

2

μs,t (IZ ) =

(s − x)2 + t 2 = r2 , where the center is (x, 0) with n ch0 (E) + ch2 (E) , and the radius r = x= ch1 (E) · H

x2 −

2n . H2

46


Observe that two distinct semicircles do not intersect. We will index the Bridgeland walls by their centers and denote them by Wx . Example 33. When X = P2 and H is the hyperplane class, then the potential Bridgeland walls have center (x, 0) with n ch0 (E) + ch2 (E) x= and radius r = x2 − 2n. ch1 (E) Example 34. When X = P1 × P1 and H is the ample class aH1 + bH2 , then the potential Bridgeland wall corresponding to E with c1 (E) = α H1 + β H2 has center (x, 0) with n ch0 (E) + ch2 (E) n x= and radius r = x2 − . aβ + bα ab Example 35. When X = Fr is a Hirzebruch surface and H is the ample class aE + bF with b > ra, then the potential Bridgeland wall corresponding to E with c1 (E) = α E + β F has center (x, 0) with 2n n ch0 (E) + ch2 (E) . and radius r = x2 − x= −aα r + aβ + bα −a2r + 2ab Example 36. When X = D9−r is a del Pezzo surface and H is an ample class aH − ∑ri=1 bi Ei , then the potential Bridgeland wall corresponding to E with c1 (E) = α H − ∑ri=1 βi Ei has center (x, 0) with

2n n ch0 (E) + ch2 (E) and radius r = x2 − 2 . x= aα − ∑ri=1 bi βi a − ∑ri=1 b2i Rank One Walls. The key problem is to determine which of these walls are actual Bridgeland walls in Stab(X). By [3, Proposition 6.2(d)] line bundles L are stable objects of As for L · H > s and all t > 0. More generally, any destabilizing subsheaf of IZ of rank one has the form IZ ⊗ L ⊂ IZ for some ideal sheaf IZ and some line bundle L on X. Any such subsheaf is a subobject in the category As as long as s < L · H. Hence, these sheaves give rise to rank one walls Wx with 2 n + L2 − l(Z ) , x= L·H where l(Z ) denotes the length of Z . Example 37. Taking X = P2 , H the hyperplane class, and L = OP2 (−k), we get the rank one walls with center: n k l(Z ) . x=− − + k 2 k


47

Example 38. Taking X = P1 × P1, H = aH1 + bH2 , and L = OP1 ×P1 (−α , −β ), we get the rank one walls with center: x=

−n − αβ + l(Z ) . aβ + bα

Example 39. Taking X = Fr , H = aE + bF and L = OFr (−α E − β F), we get the rank one walls with center −n + α2 r − αβ + l(Z ) . −aα r + bα + aβ 2

x=

Higher Rank Walls. The geometry of the moduli spaces of Bridgeland semi-stable objects becomes harder to understand once we cross a higher-rank wall. Hence, it is important to bound the centers of higher rank walls that can occur. We use an observation from [3] to get the desired bound. Suppose that F → IZ is a destabilizing subsheaf of rank at least two giving rise to a Bridgeland wall Wx . Let 0 → K → F → IZ be the kernel of the morphism. Then, by [3, Corollary 6.4], we have that both F and K[1] have to be contained in all the categories As along the wall Wx . Even though Proposition 6.2, Lemma 6.3, and Corollary 6.4 of [3] are stated for P2 , the proofs do not use the fact that the surface is P2 , but only use categorical properties and the fact that walls are nested semicircles. We conclude that x−r ≥

c1 (K) · H c1 (F ) · H , x+r ≤ . rk(K) rk(F )

Combining this with rk(K) = rk(F ) − 1 and c1 (K) · H ≥ c1 (F ) · H, we obtain that x+r ≤

c1 (K) · H rk(K) rk(K) c1 (F ) · H ≤ ≤ (x − r) . rk(F ) rk(F ) rk(K) rk(F )

Hence, we obtain the following inequality on the centers of potential walls of higherrank: rk(F )(x + r) ≤ c1 (F ) · H ≤ (rk(F ) − 1)(x − r). In particular, one obtains the bound x2 ≤

(2rk(F ) − 1)2 n . 2H 2 rk(F )(rk(F ) − 1)

Example 40. When X = P2 and H is the hyperplane class, the inequality translates to n(2rk(F ) − 1)2 . x2 ≤ 2rk(F )(rk(F ) − 1)

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Example 41. When X = P1 × P1 and H is aH1 + bH2, the inequality translates to x2 ≤

n (2rk(F ) − 1)2 . 4ab rk(F )(rk(F ) − 1)

Example 42. When X = Fr and H = aE + bF, the inequality translates to x2 ≤

n (2rk(F ) − 1)2 . 2(−a2r + 2ab) rk(F )(rk(F ) − 1)

We call the Bridgeland wall where all ideal sheaves are destabilized the collapsing wall. We remark that the inequalities become strictly sharper as the rank of F increases. In particular, if the inequalities force the centers of potential walls of rank r to be larger than that of a collapsing wall, then for every r > r, the centers of the potential walls are larger than that of the collapsing wall. This observation will help us eliminate potential higher-rank walls.

6 The Correspondence Between Bridgeland Walls and Mori Walls In this section, we calculate the Bridgeland walls for the examples we discussed in Sect. 4 and find that there is a precise correspondence between the Bridgeland walls and the Mori walls. The correspondence is cleanest when H is a multiple of the anti-canonical bundle. The most interesting aspect of this correspondence is that it does not depend on the number of points, making it a powerful tool for studying base loci decompositions. However, the correspondence appears to be much more general: Traversing the Bridgeland walls for a specific H corresponds to running a log minimal model program along a face [−K, D] in the Néron-Severi space of X [n] , where there is a precise relation between H and D . When H is a multiple of −K, then the corresponding face is [−K, B]. The Bridgeland Walls for (P1 × P1 )[n] . We let H = 12 H1 + 12 H2 . Since the HilbertChow morphism is a crepant resolution, we see that H[n] = − 14 K(P1 ×P1 )[n] . The coefficient of 14 is a normalization chosen so that the transformation is a nice integer. √ We will denote a Bridgeland wall with center x by Wx . The radius of Wx is x2 − 4n. We now calculate the Bridgeland walls for n = 2, 3 and all the walls with sufficiently small center for all n. Example 43. When n = 2, the Bridgeland walls are as follows: 1. The wall W−4 given by the destabilizing objects OP1 ×P1 (0, −1) and OP1 ×P1 (−1, 0) 2. The collapsing wall W−3 given by the destabilizing object OP1 ×P1 (−1, −1)


49

Proof. Since every sheaf of length two on P1 × P1 is contained in a curve of type (1, 1), there exists a map OP1 ×P1 (−1, −1) → IZ destabilizing every ideal sheaf. Hence, W−3 is a collapsing wall. We have already described the rank one walls in Example 38. It suffices to show that there are no higher-rank walls. By Example 41, the center of any higher rank wall has to satisfy the inequality x2 ≤ 9 = (−3)2 . Hence, any potential higher-rank wall either coincides or is contained in W−3 . Consider the divisors Dt = H[2] + 2tB , t < 0, in Fig. 1. Then the divisor crosses Mori walls at t = −2 and −1. At a Mori wall Mt , the divisor Dt picks up as base locus the sheaves that are destabilized at the Bridgeland wall Wx=t−2 . We will see that this picture persists for all n ≥ 2. Example 44. When n = 3, the Bridgeland walls are as follows: 1. The wall W−6 given by the destabilizing objects OP1 ×P1 (0, −1) and OP1 ×P1 (−1, 0) 2. The wall W−4 given by the destabilizing objects OP1 ×P1 (−1, −1), I p (0, −1), and I p (−1, 0) Proof. Since every scheme of length 3 is contained in a curve of type (1, 1), W−4 is a collapsing wall. Using Example 41, we see that any higher-rank wall satisfies x2 ≤ 27 2 < 16. Hence, all the walls are rank one walls. The description follows from Example 38. As in the previous example, setting Dt = H[2] + 2tB , t < 0, in Fig. 2, we see that the Mori walls occur at t = −4, −2. At the Mori wall Mt , the divisor Dt picks up the locus of sheaves destabilized at Wx=t−2 in its base locus. Unfortunately, we do not know how to compute all the Bridgeland walls as n increases. Two difficulties arise. We do not know how to control higher-rank walls in general. We also do not know how to control walls where none of the objects destabilized are ideal sheaves. In particular, we do not know whether walls of the latter kind exist. However, if we bound x from above (depending on n), then we can compute all the Bridgeland walls and show that the correspondence persists for all n. We have the following proposition. Proposition 45. Let Dt = 12 H1 [n] + 12 H2 [n] + 2tB , for t < 0, be a divisor on (P1 × P1 )[n] . Assume that 1 − n ≥ t. Then there is a one-to-one correspondence between the Mori walls Mti and the Bridgeland walls Wxi =ti −2 when 1 − n ≥ t and −1 − n ≥ x. An ideal sheaf IZ is in the base locus of Dt if and only if IZ is destabilized at Wxi =ti −2 for t > ti . Proof. Since ti ≤ 1 − n, xi = ti − 2 ≤ −1 − n. By Example 41, we have that the 9n 2 centers of higher-rank walls satisfy x2 ≤ 9n 2 . Since (n + 1) ≥ 2 for all n ≥ 2, we conclude that there cannot be any higher-rank walls in this range. Consequently, all the Bridgeland walls are rank one walls, which have been determined in Example 38. We see that the Bridgeland walls occur at W−2n ,W−2n+2, · · · ,W−n−1 and correspond to destabilizing objects IZ (0, −1) or IZ (−1, 0), where the length

50


of Z giving the wall W−2n+2 j has length j. Finally, W−n−1 also corresponds to the destabilizing object OP1 ×P1 (−1, −1). On the other hand, by Theorem 16, the Mori walls occur at t = −2i for n−1 2 ≤ i ≤ n − 1. When −2i ≤ t < −2i − 2, the stable base locus consists of schemes that have subschemes of length at least i + 2 contained in a fiber. When Dt crosses the value t = −n + 1, then Dt contains the locus of schemes contained in a curve of type (1, 1). This concludes the proof of the proposition. Remark 46. The correspondence seems to work in greater generality and can be proved for a larger part of the cone than covered in Proposition 45. For example, when n = 4, the rank one Bridgeland walls are W−8 ,W−6 ,W−5 ,W−4 . These walls correspond to destabilizing objects OP1 ×P1 (0, −1), I p (0, −1), OP1 ×P1 (−1, −1), and OP1 ×P1 (−1, −2) or IZ (0, −1) with l(Z ) = 2, respectively. The sheaves obtained by switching the two fiber classes also give rise to the same walls. These correspond precisely to the Mori walls in Figure 3, which occur at t = −6, −4, −3, −2. Similarly, when n = 5, the rank one Bridgeland walls occur at W−10 ,W−8 ,W−6 ,W−5 ,W− 14 . These correspond precisely to the Mori walls in Fig. 4, 3

which occur at t = −8, −6, −4, −3, and − 83 . One may conjecture that there is always a one-to-one correspondence between Bridgeland walls and Mori walls given by the relation x = t − 2. Even when one does not a priori know this correspondence, it is still very useful for guessing base loci of linear systems on (P1 × P1)[n] . Remark 47. Recently, Bayer and Macrì have constructed nef divisors on the moduli spaces of Bridgeland semi-stable objects. For the moduli spaces discussed here, their arguments show that their nef divisor is ample. Hence, these moduli spaces are projective. One may thus obtain a modular interpretation of the log anti-canonical models of (P1 × P1 )[n] with respect to the boundary divisor B, at least in the ranges covered by Proposition 45. [n]

The Bridgeland Walls for F1 . We describe the correspondence between Bridge[n]

land walls and Mori walls for the Hilbert schemes F1 . Let H = 13 E + 12 F. Observe that H = − 16 KF1 . In this subsection, we will see that if we set Dt = − 16 K [n] + 2tB for F1

t < 0, then there is a correspondence between the Mori wall Mt and the Bridgeland wall Wx=t−3 –at least in certain ranges. The coefficient 16 is chosen to make the correspondence be given by an integer. We begin by listing the Bridgeland walls for n = 2 and n = 4. Example 48. The Bridgeland walls when n = 2 are as follows: 1. The wall W−9 corresponding to the destabilizing object OF1 (−E) 2. The wall W−6 corresponding to the destabilizing object OF1 (−F) 3. The wall W−5 corresponding to the destabilizing object OF1 (−E − F) Proof. Every scheme of length two is contained in a curve of class E + F. Hence, W−5 is a collapsing wall. By Example 42 any higher-rank wall Wx satisfies x2 ≤ 81 4 < 25. Hence, the only walls are rank one walls and have been described in Example 39.


51

The Mori walls corresponding to the divisor Dt = − 16 K

[2]

F1

+

B 2t

occur at t =

−6, −3, −2 as can be seen from Fig. 5. Using the given descriptions of the base loci, the reader can easily check that the divisor Dt picks up a subscheme in its base locus at t if and only if the corresponding ideal sheaf is destabilized at the Bridgeland wall Wt−3 . Example 49. The Bridgeland walls when n = 4 are as follows: The wall W−21 corresponding to the destabilizing object OF1 (−E) The wall W−15 corresponding to the destabilizing object I p (−E) The wall W−12 corresponding to the destabilizing object OF1 (−F) The wall W−9 corresponding to the destabilizing objects IZ (−E), where l(Z ) = 2 and I p (−F) 5. The wall W−7 corresponding to the destabilizing object OF1 (−E − F) 6. The wall W− 33 corresponding to the destabilizing object OF1 (−E − 2F) 1. 2. 3. 4.

5

Proof. Every scheme of length 4 is contained in a curve with class E + 2F. Hence, W− 33 is a collapsing wall. By Example 42 any higher-rank wall Wx satisfies x2 ≤ 5 33 2 81 < . Hence, all the Bridgeland walls are rank one walls and are described in 2 5 Example 39. The reader can compare this to Fig. 7. The divisor Dt = − 16 K

[2]

F1

+

B 2t

crosses

walls precisely when t = −18, −12, −9, −6, −4, − 18 5 . From the description of the base loci, we see that the divisor Dt picks up a subscheme in its base locus at t if and only if the corresponding ideal sheaf is destabilized at the Bridgeland wall Wt−3 . As in the case of P2 and P1 × P1 , while we do not know how to prove this correspondence for every wall, we can prove it for walls in certain ranges. [n]

Proposition 50. Let Dt = 13 E[n]+ 12 F[n]+ 2tB , for t < 0, be a divisor on F1 . Assume that 2 − 2n ≥ t. Then there is a one-to-one correspondence between the Mori walls Mti and the Bridgeland walls Wxi =ti −3 when 2 − 2n ≥ t and −1 − 2n ≥ x. An ideal sheaf IZ is in the base locus of Dt if and only if IZ is destabilized at Wxi =ti −3 for t > ti . Proof. The proof of this proposition is analogous to the proof of Proposition 45. Since (−2n − 1)2 > 81n 8 for n ≥ 2, in this range, there are only rank one Bridgeland walls, which have been described in Example 39. We see that they occur at x = −6n + 3 + 6l(Z ), when α = 1, β = 0, or at x = −3n + 3l(Z ), when α = 0, β = 1, or at x = −2n − 2 when α = β = 1. On the other hand, in this range, the stable base locus decomposition is described by Theorem 24. The divisor Dt intersects the Mori wall spanned by Xi,n−1−i and E[n] + F[n] when t = −6n + 6 + 6i. Similarly, Dt intersects the Mori wall spanned by Xn−i−1,i and F[n] at t = −3n + 3 + 3i. Finally, Dt intersects the wall spanned by Xn−1,0 and X0,n−1 at t = −2n + 1. One obtains the proposition by matching the two descriptions.

52


Remark 51. One can speculate that the relation x = t − 3 gives a one-to-one correspondence between Bridgeland and Mori walls in general. As in the case of P1 × P1 , at least in the cases covered by Proposition 50, one obtains a modular interpretation of the log canonical models Proj(R(−K [n] − cB)), where R(−K [n] − F1

cB) is the log canonical ring associated to the divisor −K

[n] F1

− cB.

F1

The Correspondence for Other Slices. The correspondence between the two sets of walls extends beyond the slice we have studied so far. We can decompose the ample cone of X into chambers such that for ampleclasses in a chamber the Bridgeland walls with respect to the central charge − X e−(s+it)H ch(E) have the same order with respect to the same destabilizing objects. Example 52. For (P1 × P1 )[2] the rank one Bridgeland walls with respect to aH1 + bH2 are • W− 2 ,W− 1 corresponding to OP1 ×P1 (0, −1) and I p (0, −1) a a • W− 2 ,W− 1 corresponding to OP1 ×P1 (−1, 0) and I p (−1, 0) b b • W− 3 corresponding to OP1 ×P1 (−1, −1) a+b

Correspondingly, the ample cone of P1 × P1 decomposes into regions separated by hyperplanes a = b2 , a = b, and a = 2b. When 2b > a > b, then the rank one walls are W− 2 ,W− 2 ,W− 3 ordered by increasing centers and W− 3 is a collapsing wall. a b a+b a+b Whereas, when a > 2b, the relevant walls are W− 2 ,W− 1 , where W− 1 is a collapsing b b b wall. The threshold value a = 2b is the value where the three walls W− 2 , W− 3 , a a+b and W− 1 become equal. Similarly, at the threshold value a = b, the walls W− 2 and a b W− 2 become equal. b

We can similarly divide the ample cone of X [n] into regions, where in a fixed region, the face spanned by −K and D intersects the same Mori walls in the same order. Example 53. For (P1 × P1 )2 , these regions are separated by the planes spanned by [−K, X1,2 ], [−K, X2,2 ], and [−K, X2,1 ]. Note that if D is in the region bounded by [−K, X2,2 ] and [−K, X1,2 ], then it intersects the walls [X1,0 , H2 [2]], [X0,1 , H1 [2]], and [X1,0 , X0,1 ] in order (see Fig. 1). These precisely correspond to the (rank one) Bridgeland walls in the region 2b > a > b, where a scheme Z defines a Mori wall if and only if the ideal sheaf IZ is destabilized at the corresponding Bridgeland wall. If D is in the region bounded by [−K, X1,2 ] and [−K, H2 [2]], then [−K, D] intersects the walls [X1,0 , H2 [2]] and [X0,1 , H2 [2]]. These precisely correspond to the (rank one) Bridgeland walls in the region a > 2b. Example 54. As a final example, we work out the two decompositions for (P1 × P1 )[3] . The rank one Bridgeland walls with respect to aH1 + bH2 are the walls W− 3 ,W− 2 ,W− 1 corresponding to IZ (0, −1) with length of Z = 0, 1, or 2, the a a a walls W− 3 ,W− 2 ,W− 1 obtained by symmetry, and the wall W− 4 corresponding b

b

b

a+b


53

to OP1 ×P1 (−1, −1). The ample cone of P1 × P1 decomposes into chambers where the boundaries are given by the hyperplanes a = b, 2a = 3b, a = 3b, 3a = 2b, and 3a = b. If 32 b > a > b, then the rank one Bridgeland walls are W− 3 ,W− 3 ,W− 2 , a b b and W− 4 listed in the order of increasing centers. When 2a = 3b, the walls W− 3 a

a+b

and W− 2 coincide. When 3b > a > 32 b, then the rank one Bridgeland walls are b W− 3 ,W− 2 ,W− 3 , and W− 4 listed in the order of increasing centers. When a = 3b, a b b a+b the three walls W− 3 , W− 1 , and W− 4 coincide. If a > 3b, then the walls are a b a+b W− 3 ,W− 2 ,W− 1 . b

b

b

Correspondingly, the ample cone of (P1 × P1 )[3] decomposes into regions bounded by the faces [−K, X4,2 ], [−K, X3,2 ], [−K, X2,2 ], [−K, X2,3 ], and [−K, X2,4 ]. By symmetry, let us assume that a > b. If D is in the region bounded by [−K, X2,2 ] and [−K, X2,3 ], then [−K, D] intersects the walls [X2,0 , H2 [3]], [X0,2 , H1 [3]], and [X2,0 , X0,2 ] in order (see Fig. 2). These correspond precisely to the (rank one) Bridgeland walls in the region 32 b > a > b. If D is in the region bounded by [−K, X2,3 ] and [−K, X2,4 ], then [−K, D] intersect the Mori walls that correspond to the Bridgeland walls in the region 3b > a > 32 b. Similarly, if D is in the region bounded by [−K, X2,4 ] and [−K, H2 [3]], then the Mori walls correspond to the walls in the region a > 3b. Remark 55. One can conjecture that there is always a one-to-one correspondence between the walls decomposing the ample cone of P1 × P1 into chambers and the walls decomposing the ample cone of (P1 × P1 )[n] . Since the decompositions are symmetric about a = b, we may assume that a ≥ b. Based on the examples, one can predict that the critical ratio −n − i1 j1 + m1 −n − i2 j2 + m2 = , j1 a + i1b j2 a + i2b where two rank one walls become equal corresponds to the wall [−K, D], where D = α H1 [n] + β H2[n] − γ B2 with (α , β , γ ) satisfying the two equations js α + is β − (is js + n − ms − is − js )γ = 0,

for s = 1, 2. For example, the walls (n − k)a = nb correspond to the walls [−K, D], where D = (n − 1)H1[n] + (n − 1 + k)H2[n] − B2 . Using Fig. 3 and Example 38, the reader can check that for (P1 × P1 )[4] , the two sets of walls occur at a = b, 3a = 4b, 2a = 3b, a = 2b, and a = 4b and at [−K, 3H1 [4] + (3 + k)H2 [4] − B2 ] with k = 0, 1, 32 , 2, 3. Similarly, by Fig. 4 and Example 38, for (P1 × P1 )[5] , the two sets of walls occur at a = b, 4a = 5b, 3a = 4b, 3a = 5b, 2a = 5b, and a = 5b and at [−K, 4H1 [4] + (4 + k)H2 [4] − B2 ] with k = 0, 1, 2, 52 , 3, 4. Suppose that an ample divisor D is contained in a chamber determined by two walls [−K, D1 ] and [−K, D2 ] in the ample cone of (P1 × P1 )[n] . Suppose that a = kn1 b

54


and a = kn2 b are the corresponding chambers in the ample cone of P1 × P1 . Then, in the examples, there is a one-to-one correspondence between the Mori walls that intersect [−K, D] and the (rank one) Bridgeland walls that occur for an ample divisor aH1 + bH2 satisfying kn1 b < a < kn2 b. One can speculate that there is a one-to-one correspondence between Mori walls intersecting [−K, D] and Bridgeland walls in full generality. Furthermore, one can expect that running the log minimal model program in the face [−K, D] corresponds to the birational transformations that take place as one crosses the Bridgeland walls, giving modular interpretations to all the models. [n] We leave it to the reader to check that a similar story holds for F1 for 2 ≤ n ≤ 4 using Sect. 4. It would be interesting to explore the connection between the Bridgeland walls and Mori walls for other smooth, projective surfaces, especially those with ample canonical bundle. Acknowledgements During the preparation of this paper the first author was partially supported by the NSF grant DMS-0901128 and the second author was partially supported by the NSF CAREER grant DMS-0950951535 and an Alfred P. Sloan Foundation Fellowship. The second author would like to thank the Simons Foundation and the organizers of the Simons Symposium on Rational Points over Non-algebraically Closed Fields, Fyodor Bogomolov, Brendan Hassett, and Yuri Tschinkel, for a very productive and enlightening conference. It is a pleasure to thank Daniele Arcara, Arend Bayer, Jack Huizenga, and Emanuele Macrì for discussions about Bridgeland stability and the birational geometry of Hilbert schemes.

References 1. ABRAMOVICH , D. AND POLISHCHUK , A., Sheaves of t-structures and valuative criteria for stable complexes, J. Reine Angew. Math. 590, 89–130, (2006). 2. ARCARA , D. AND BERTRAM, A., Bridgeland-stable moduli spaces for K-trivial surfaces, with an appendix by Max Lieblich, J. Euro. Math. Soc. 15(1), 1–38, (2013). 3. ARCARA , D., BERTRAM, A., COSKUN , I., AND HUIZENGA , J., The birational geometry of the Hilbert scheme of points on P2 and Bridgeland stability, arxiv:1203.0316, Adv. Math. 235, 580–626, (2013). 4. BAYER , A. AND M ACRÌ , E., The space of stability conditions on the local projective plane, Duke Math. J. 160, 263–322, (2011). 5. BAYER , A. AND M ACRÌ , E., Projectivity and birational geometry of Bridgeland moduli spaces, arxiv:1203.4613, (2012). 6. BAYER , A., M ACRÌ , E., AND TODA , Y., Bridgeland stability conditions on 3-folds I: Bogomolov-Gieseker type inequalities, arxiv:1103.5010, (2011). 7. BELTRAMETTI , M., FRANCIA , P., AND SOMMESE, A., On Reider’s method and higher order embedding, Duke Math. J. 58, 425–439, (1989). 8. BELTRAMETTI M. AND SOMMESE, A., On k-spannedness for projective surfaces, LNM 1417, 24–51, (1988). 9. BELTRAMETTI M. AND SOMMESE, A., Zero cycles and kth order embeddings of smooth projective surfaces, with an appendix by L. Göttsche. Sympos. Math., XXXII Problems in the theory of surfaces and their classification (Cortona 1988), 33–48, Academic Press, London, 1991.


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10. BIRKAR , C., CASCINI , P., HACON , C.D., AND M C KERNAN , J., Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23, no. 2, 405–468, (2010). 11. BRIDGELAND , T., Stability conditions on triangulated categories. Ann. of Math. (2) 166, no. 2, 317–345, (2007). 12. BRIDGELAND , T., Stability conditions on K3 surfaces. Duke Math. J. 141, no. 2, 241–291, (2008). 13. CATANESE, F. AND GÖTTSCHE, L., d-very-ample line bundles and embeddings of Hilbert schemes of 0-cycles. Manuscripta Math. 68, no.3, 337–341, (1990). 14. CHEN , D. AND COSKUN , I., Stable base locus decompositions of the Kontsevich moduli spaces. Michigan Math. J. , 59, no.2, 435–466, (2010). 15. CHEN , D. AND COSKUN , I., Towards the Mori program for Kontsevich moduli spaces, with an appendix by Charley Crissman. Amer. J. Math. 133, no.5, 1389–1419, (2011). 16. DI ROCCO , S., k-very ample line bundles on del Pezzo surfaces, Math. Nachrichten 179, 47–56, (1996). 17. EISENBUD , D., The geometry of syzygies, Springer, 2005. 18. FOGARTY, J., Algebraic families on an algebraic surface, Amer. J. Math. 90, 511–521, (1968). 19. FOGARTY, J., Algebraic families on an algebraic surface II: The Picard scheme of the punctual Hilbert scheme. Amer. J. Math. 95, 660–687, (1973). 20. HAPPEL, D., REITEN , I., AND SMALO , S., Tilting in abelian categories and quasitilted algebras, Mem. of the Am. Math. Soc., 120, no. 575, (1996). 21. HARTSHORNE, R., Algebraic Geometry, Springer, 1977. 22. HASSETT, B. AND HYEON , D., Log minimal model program for the moduli space of curves: The first divisorial contraction, Trans. Amer. Math. Soc., 361, 4471–4489, (2009). 23. HASSETT, B. AND HYEON , D., Log minimal model program for the moduli space of curves: The first flip, Ann. Math. 177(3), 911–968, (2013). 24. HU , Y. AND KEEL, S., Mori dream spaces and GIT, Michigan Math. J. 48, 331–348, (2000). 25. HUIZENGA , J., Restrictions of Steiner bundles and divisors on the Hilbert scheme of points in the plane, Harvard University, thesis, 2012. 26. KING , A., Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxford 45, no. 2, 515–530, (1994). 27. LAZARSFELD , R., Positivity in Algebraic Geometry I, Springer-Verlag, 2004. 28. LI , W.P., QIN , Z., AND ZHANG , Q., Curves in the Hilbert schemes of points on surfaces. Contemp. Math. 322, 89–96, (2003). 29. M ACIOCIA , A., Computing the walls associated to Bridgeland stability conditions on projective surfaces, arxiv:1202.4587, (2012). ´ , H., General blow-ups of ruled surfaces, Abh. Math. 30. SZEMBERG , T. AND TUTAJ-GASI NSKA Sem. Univ. Hamburg 70, 93–103, (2000). 31. TODA , Y., Moduli stacks and invariants of semi-stable objects on K3 surfaces, Adv. Math. 217 no. 6, 2736–2781, (2008). 32. TODA , Y., Stability conditions and extremal contractions, arxiv:1204.0602, (2012).

Isoclinism and Stable Cohomology of Wreath Products Fedor Bogomolov and Christian Böhning

Mathematics Subject Classification codes (2000): 14E08, 14F43

1 Introduction Unless otherwise stated, the base field in this paper will be C, the field of complex numbers. Recall that for a prime p and finite group G, the stable cohomology Hs∗ (G, Z/p) is defined as the quotient H ∗ (G, Z/p)/NG,p where, for some generically free G-representation V (defined over C) with open part V L with free G-action, NG,p is the kernel of the map H ∗ (G, Z/p) → lim H i (U/G, Z/p) , −→ U

the direct limit running over all nonempty G-invariant Zariski open subsets U ⊂ V L . In fact, NG,p is independent of the choice of V . In Hs∗ (G; Z/p) we have the ∗ (G, Z/p); these play a vital role in the study subring of unramified elements Hnr of birational properties of generically free linear quotients V /G and varieties X in general (see [3] for definitions and background). In particular, there is an analogous notion of stable cohomology for any variety X: Hs∗ (X, Z/p) is the image of the (singular, say) cohomology of X in the Galois cohomology of the function field of X.

F. Bogomolov Courant Institute of Mathematical Sciences, 251 Mercer St., New York, NY 10012, USA Laboratory of Algebraic Geometry, GU-HSE, 7 Vavilova Str., Moscow, 117312, Russia e-mail: [email protected] C. Böhning () Fachbereich Mathematik der Universität Hamburg, Bundesstraße 55, Hamburg, 20146, Germany e-mail: [email protected] F. Bogomolov et al. (eds.), Birational Geometry, Rational Curves, and Arithmetic, DOI 10.1007/978-1-4614-6482-2__3, © Springer Science+Business Media New York 2013

57

58

F. Bogomolov and C. Böhning

∗ (G, Z/p)) amenable The objective of this paper is to make Hs∗ (G, Z/p) (and Hnr to effective computation for rather important and large classes of groups. The development of the theory formally parallels that in the ordinary cohomology of finite groups: iterated wreath products of finite cyclic p-groups play an important part because they occur as building blocks of Sylow subgroups of a variety of classes of finite groups, in particular many finite groups of Lie type. In ordinary group cohomology, systematic studies along these lines were made in the famous papers by Quillen [17–19]. Our treatment uses the notion of isoclinism of finite p-groups introduced by Hall in the paper [11]. It turns out that generically free linear quotients by isoclinic groups are stably birational (see Theorem 6); this answers a question raised in [12] (Question 1.11) in the affirmative. A partial result in this direction has been obtained previously in [15] where it is proven that isoclinic groups have isomorphic unramified cohomology in degree 2. Moreover, the stable cohomology of isoclinic groups shares some important properties (see Proposition 3), namely, if G1 and G2 are isoclinic, then the stable cohomology of G1 is detected by abelian subgroups if and only if the same is true for G2 . Here is a further outline of the contents of this paper: in Sects. 2 and 3 we prove the aforementioned results for isoclinic groups and, as an ingredient of the proof, we show that the notions of isoclinism and being toroidally related coincide for group extensions. Section 4 then contains the application of the results on isoclinic groups to the computation of the stable cohomology of iterated wreath products of groups Z/p. The main results are Theorem 11 and its Corollary 12 saying that for the stable cohomology of these groups we have detection by abelian subgroups. Theorem 13 shows that for Gn = Z/p · · · Z/p (n-factors) we even get detection by elementary abelian p-subgroups. This enables us to determine the structure of the stable cohomology algebra of Gn in Theorem 15 completely. In Sect. 5 we give some applications of the preceding results to the computation of the unramified and stable cohomology of some finite groups of Lie type. A more intensive treatment will be given elsewhere. In particular, we recover (and extend) results from [6] by this method. Here are a few references to background material and texts which an interested reader might consult for foundational results regarding the notions explored in the present work. For stable and unramified cohomology, [3, 4, 9, 10] are sources which may be useful. For the connection to the birational anabelian geometry program, one may turn to [3] and the recent [8]. Good references on group cohomology in general are [1, 2, 20], and the articles by Quillen [18, 19].

2 Toroidally Related Extensions Definition 1. Let G be a finite group and let A1 , A2 be finite abelian groups. If e1 : 1 → A1 → G1 → G → 1 and e2 : 1 → A2 → G2 → G → 1 are two central extensions of G, we call them (resp. G1 and G2 ) toroidally related if there is an

Stable Cohomology

59

algebraic torus T (C∗ )r together with embeddings ik : Ak → T , k = 1, 2, such that the images of e1 ∈ H 2 (G, A1 ) and e2 ∈ H 2 (G, A2 ) in the cohomology H 2 (G, T ) coincide. Example 2. (1) If G is abelian, then one knows for the group homology H2 (G, Z) = Λ 2 G, and the universal coefficient sequence for an arbitrary G-module M reads c

0 → Ext1 (G, M) → H 2 (G, M) → Hom(Λ 2 G, M) → 0 . For M = Q/Z, we have Ext1 (G, Q/Z) = 0 as Q/Z is an injective Z-module; hence, associated to e, one gets a homomorphism ∼

c : H 2 (G, Q/Z) → Hom(Λ 2 G, Q/Z) . More concretely, for any M = A with trivial action, the map c : H 2 (G, A) → Hom(Λ 2 G, A) can be described as follows: to a central extension ι π e : 1 → A → G˜ → G → 1

one associates the A-valued skew-form c(e) on G given by the commutator: ˜ = h. ι (c(e)(g, h)) = g˜h˜ g˜−1 h˜ −1 , where π (g) ˜ = g, π (h) The kernel Ext1 (G, A) of c in this more general setup can be identified with the abelian extensions Eab (G, A) of G by A. From the short exact sequence of coefficients 0 → Q/Z → C∗ → C∗ /(Q/Z) → 1 we get H 2 (G, Q/Z) H 2 (G, C∗ ) because C∗ /(Q/Z) is an infinite-dimensional vector space over Q (with “vector addition” = multiplication and “scalar multiplication”=exponentiation; this uses the algebraic closedness of C), whence H j (G, C∗ /(Q/Z)) = 0 for j > 0. If i : A → (C∗ )r is an embedding, the diagram H 2 (G, A) ⏐ ⏐ i∗

c

−−−−→

Hom(Λ 2 G, A) ⏐ ⏐ i◦(−)

c

H 2 (G, (C∗ )r ) −−−−→ Hom(Λ 2 G, (C∗ )r ) . Thus, for fixed abelian G and A, two central extensions e1 and e2 of G by A are toroidally related if and only if they give the same skew form in Hom(Λ 2 G, A), or equivalently, their difference e1 − e2 ∈ H 2 (G, A) represents the class of an abelian extension. If A and G are elementary abelian p-groups, this is the same as saying that e1 − e2 lies in the subspace of H 2 (G, A) spanned by the Bocksteins. Here by the term Bocksteins we mean, in the case A = Z/p, the images of elements coming from H 1 (G, Z/p) in H 2 (G, Z/p) under the connecting homomorphism arising from the short exact sequence 0 → Z/p → Z/p2 → Z/p → 0 of coefficients. If

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A (Z/p)r , then H 2 (G, A) H 2 (G, Z/p)⊕r , and a Bockstein in this situation is a sum of Bocksteins in the single factors in this decomposition. (2) In general, when G is not necessarily abelian, we still have a universal coefficient sequence 0 → Ext1Z (H1 (G, Z), A) → H 2 (G, A) → Hom(H2 (G, Z), A) → 0 and, for an embedding i : A → (C∗ )r a diagram H 2 (G, A) ⏐ ⏐ i∗

c

−−−−→

Hom(H2 (G, Z), A) ⏐ ⏐ i◦(−)

c

H 2 (G, (C∗ )r ) −−−−→ Hom(H2 (G, Z), (C∗ )r ) , where the bottom horizontal arrow is again an isomorphism. Thus extensions e1 and e2 are toroidally related in this case if and only if their difference e1 − e2 is in the subspace Ext1Z (H1 (G, Z), A). Now H1 (G, Z) = Gab and the map Ext1Z (Gab , A) → H 2 (G, A) factors p∗

Ext1Z (Gab , A) → H 2 (Gab , A) → H 2 (G, A), where p : G → Gab is the projection and the arrow Ext1Z (Gab , A) → H 2 (Gab , A) is the one in the universal coefficient sequence c

0 → Ext1 (Gab , A) → H 2 (Gab , A) → Hom(Λ 2 Gab , A) → 0 . This means that if e1 and e2 are toroidally related, then their difference e1 − e2 corresponds to an extension G˜ of G by A induced from an abelian extension G˜ ab of Gab by A: 1 −−−−→ A −−−−→ G˜ −−−−→ G −−−−→ 1 ⏐ ⏐ ⏐ ⏐ 1 −−−−→ A −−−−→ G˜ ab −−−−→ Gab −−−−→ 1 . The next Proposition shows that if G1 and G2 are toroidally related, then their stable cohomologies have important properties in common. Below we say that the stable cohomology of a group G is detected by a class of subgroups H j , j ∈ J, if the product of restriction maps Hs∗ (G, Z/p) → ∏ j∈J Hs∗ (H j , Z/p) is injective. Moreover, we say that an extension is toroidally trivial if it is toroidally related to a trivial extension. Proposition 3. Let G1 and G2 be toroidally related as in Definition 1. Then we have ∗ (G , Z/p) H ∗ (G , Z/p). (1) Hnr 1 2 nr

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61

(2) For toroidally related p-groups G1 and G2 one has that if Hs∗ (G1 , Z/p) is detected by abelian subgroups, so is Hs∗ (G2 , Z/p). Proof. (1) Let Gc be the central extension of G by T = (C∗ )r determined by G1 and G2 . Then Gc is an extension 1 → Gi → Gc → T /Ai → 1. Then a generically free representation V for Gc gives a generically free representation for both Gi , and V /Gi → V /Gc is a torus principal bundle, hence locally trivial. In particular, V /Gi , i = 1, 2, are stably birationally isomorphic and have the same unramified cohomology. (2) Look at the fiber product of G1 and G2 over G: π

G12 = G1 ×G G2 −−−1−→ ⏐ ⏐ π2 G2

G1 ⏐ ⏐

−−−−→ G .

Then G12 is a toroidally trivial extension of both G1 and G2 . We subdivide the proof into two auxiliary steps: (a) If the stable cohomology of G1 is detected by abelian subgroups, the same holds for the stable cohomology of G12 . (b) Hs∗ (G2 , Z/p) injects into Hs∗ (G12 , Z/p). Given (a) and (b), one may conclude as follows: A nontrivial element α ∈ Hs∗ (G2 , Z/p) is still nonzero in Hs∗ (G12 , Z/p), hence is nonzero on some abelian subgroup A ⊂ G12 by (a). Hence α will be nontrivial on π2 (A) ⊂ G2 . To prove (b) note that G12 , being a toroidally trivial extension of G1 , is induced ab from an abelian extension G˜ ab 1 of G1 , 1 −−−−→ Z2 −−−−→

G12 −−−−→ ⏐ ⏐

G1 −−−−→ 1 ⏐ ⏐

ab 1 −−−−→ Z2 −−−−→ G˜ ab 1 −−−−→ G1 −−−−→ 1 .

Now G˜ ab 1 splits as a direct product A2 + A1 of abelian groups, where A1 is an abelian ab ab extension of Gab 1 of the same p-rank as G1 , i.e., dimF p (A1 /p) = dimF p (G1 /p), and A2 + A1 is then a trivial abelian extension of A1 . It follows that G12 = A2 × G1 , where G1 is the preimage of A1 in G12 under the map G12 → G˜ ab 1 . There is a natural map G1 → G1 . We claim that the induced map Hs∗ (G1 , Z/p) → Hs∗ (G1 , Z/p) is injective.

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We prove that G1 and G1 have a faithful representation W with the same quotient. Indeed, let V be a faithful representation of G1 and let W = V ⊕ ∏ j (C)χ j where ∏ j (C)χ j gives an embedding of the abelian group A1 into a torus via characters χ j . This W is then a faithful representation of G1 and G1 . Let g j be generators of A1 ij

L dual to the characters χ j , and h j = g pj generators of Gab 1 . Let V be the open part of V , where the action is free, and then the map

V L × ∏(C∗ )χ j → V L × ∏(C∗ )χ j , j

j

ij

which is the identity on V L and maps the coordinate z j in (C∗ )χ j to z pj , induces an isomorphism (V L × ∏(C∗ )χ j )/G1 → (V L × ∏(C∗ )χ j )/G1 . j

j

This finishes the proof of (b). Let us now prove (a), i.e., we assume that Hs∗ (G1 , Z/p) is detected by abelian subgroups and we want to show the same for G12 which is a toroidally trivial extension of G1 . It suffices to treat the case where we are in the situation 1 −−−−→ Z/p −−−−→

G12 ⏐ ⏐

−−−−→ G1 −−−−→ 1 ⏐ ⏐

1 −−−−→ Z/p −−−−→ Z/pi+1 −−−−→ Z/pi −−−−→ 1 . To see that we can indeed reduce to this picture, notice that in general G12 is obtained ab from some abelian extension A → Gab 12 → G1 . We can get the latter as a succession ab of Z/p-extensions of G1 . In each intermediate step we get an induced toroidally trivial extension of G1 . Now either the extension of Gab 1 by Z/p (in one fixed step of ab the construction) does not increase the number of generators of Gab 1 , i.e., dim G1 /p (then the corresponding extension of G1 is induced from an extension as in the above diagram) or it increases the number of generators of Gab 1 —then it is a trivial extension and hence the corresponding extension of G1 is a product with Z/p as well and the result holds trivially. Thus let Gc12 be the C∗ -extension induced by G12 → G1 resp. G12 → G2 . Then c G12 is trivial: Gc12 Gi × C∗ . Consider a generically free linear representation V of G1 and a linear representation V ⊕ CM of G12 where the quotient Z/pi+1 acts in CM via multiplication by a pi+1 st root of unity. Let V 0 be the open part of V where the G1 -action is free. Note that likewise Gc12 acts naturally in V ⊕ CM (here C∗ acts in CM via homotheties). We have natural maps

Stable Cohomology

63 p

L× := (V 0 ⊕ (CM − {0}))/G12 −−−−→ P := (V 0 ⊕ (CM − {0}))/Gc12 ⏐ ⏐ q B := V 0 /G1 . Here q makes P into a (trivial) projective bundle over B, and L× is a line bundle over P with zero section removed. Note that if we replace V by a direct sum V ⊕V ⊕ · · · ⊕V with sufficiently many copies of V and make M large, we can assume that the group actions on the respective spaces are free in high codimension; hence the Z/pcohomology of L× , P resp. B agrees with the group cohomology H ∗ (G12 , Z/p), H ∗ (Gc12 , Z/p) resp. H ∗ (G1 , Z/p) up to arbitrarily high degree. Now Hs∗ (Gc12 , Z/p) Hs∗ (G1 , Z/p) via the pullback map q (the stable cohomology of the algebraic group C∗ is trivial; see [3] or [4] for the extension of the concept of stable cohomology to linear algebraic groups and varieties). Moreover, for the usual cohomology, it clearly holds H ∗ (P, Z/p) H ∗ (B, Z/p)[b], where b is a generator of H 2 ((CM − {0})/C∗, Z/p). Now we have Hs∗ (L× , Z/p) → Hs∗ (P, Z/p) ⊗ Hs∗(C∗ , Z/p) , where Hs∗ (C∗ , Z/p) is the stable cohomology of the algebraic variety C∗ . To see this it is sufficient to consider the spectral sequence of the fibration p H i (P, H j (C∗ , Z/p)) =⇒ H i+ j (L× , Z/p) to show that every cohomology class α ∈ H ∗ (L× , Z/p) can be written as β + γ ⊗ t with β , γ ∈ H ∗ (P, Z/p) and t a generator of H 1 (C∗ , Z/p), and then pass to the stabilized cohomology. Put differently, we have maps

Γ ⏐ ⏐ s

r

−−−−→ G12

G1 × Z/p, where Γ is a profinite group, the decomposition group of the valuation correspondˆ and r∗ resp. s∗ stabilize the ing to the zero section of L× , which splits as Γ ⊕ Z, cohomology of G12 resp. G1 × Z/p. Note that there is some profinite abelian group Aˆ in Γ lying over every abelian subgroup A in G1 × Z/p. This follows from the proof of Lemma 4 below (a variant of this appears also in [4], Lemmas 7.5, and 7.7).

64


Then Aˆ stabilizes the cohomology of A. Now given a cohomology class in G12 , when pulled back to Γ , it is in the image of the cohomology of G1 × Z/p, hence ˆ Hence the original class will be nontrivial in the stable nontrivial on some such A. cohomology of the image of Aˆ under r in G12 . Lemma 4. Assume that A is a finite abelian group acting generically freely on a smooth variety X with an invariant point z. Then Hsi (A, Z/p) ⊂ H i (X L /A, Z/p) (X L is the open part of X, where the action of A is free). Proof. The action is locally linear and diagonalizable at z and there is a complete flag of A-invariant subvarieties passing through z: {z} ⊂ X 1 ⊂ X 2 ⊂ · · · ⊂ X n−1 ⊂ X n = X. This sequence defines an A-invariant Zn -valuation on C(X). The inertia subgroup Iν =: Aˆ of the induced valuation ν of C(X)A is isomorphic to Zˆ n , surjects onto A and injects into Gal(C(X/A)/C(X/A)). Hence the result.

3 Comparison to the Notion of Isoclinism In [11], Hall introduced the notion of isoclinism of finite groups which morally speaking means that the two groups have the same commutator function. Definition 5. Two finite groups G1 and G2 with centers Z1 and Z2 are said to be isoclinic if there are isomorphisms i : G1 /Z1 → G2 /Z2 , such that

j : [G1 , G1 ] → [G2 , G2 ] (i, i)

G1 /Z1 × G1 /Z1 −−−−→ G2 /Z2 × G2/Z2 ⏐ ⏐ ⏐ ⏐ [·,·] [·,·] [G1 , G1 ]

j

−−−−→

[G2 , G2 ]

commutes. As was remarked above, if G1 and G2 are toroidally related extensions of the group G, then G12 is a toroidally trivial extension of both G1 and G2 , which means ab that it is an extension induced from an abelian extension of Gab 1 resp. G2 . Hence G1 and G2 have the same commutator function (that of G12 ) and are isoclinic. Suppose conversely that G1 and G2 are isoclinic. Then naturally G = G1 /Z1 G2 /Z2 and we want to show that e1 : 1 → Z1 → G1 → G → 1 , e2 : 1 → Z2 → G2 → G → 1

Stable Cohomology

65

are toroidally related. We have to show that e1 and e2 map to the same element under ιi ◦ αi where αi is the map in the sequence α

i 0 → Ext1Z (H1 (G, Z), Zi ) → H 2 (G, Zi ) −−−− → Hom(H2 (G, Z), Zi ) → 0

and ιi is the map

ιi : Hom(H2 (G, Z), Zi ) → Hom(H2 (G, Z), (C∗ )r ) induced by an appropriately chosen embedding Zi → (C∗ )r . To do this we use an interpretation of the Schur multiplier H2 (G, Z) from [14], Sect. 2.6, in terms of commutator relations. Let G, G be the free group generated by all pairs x, y with x, y ∈ G together with its natural map c : G, G → [G, G] with c(x, y) = [x, y] = xyx−1 y−1 . The kernel of p, denoted by C(G), consists of relations among commutators in G. Moreover, there are the following universal commutator relations valid in any group G: x, x, x, yy, x, y, zx x, zxy, z−1 , y, zx y, z−1 x, [y, z]−1 , where for x, y, z ∈ G, y, zx = xyx−1 , xzx−1 . The smallest normal subgroup in C(G) containing all these universal relations is denoted by B(G). Let H(G) = C(G)/B(G) be the quotient. The universal relations are the ones that hold in a free group. We have now, by Theorem 2.6.6 of [14], that naturally H(G) H2 (G, Z), the Schur multiplier, for a finite group G. In fact this is a consequence of Hopf’s formula for H2 (G, Z) which says that if F G is a free presentation of G with subgroup of relations R, then H2 (G, Z) (R ∩ [F, F])/[F, R]. Now we want to reinterpret the maps

ιi ◦ αi : H 2 (G, Zi ) → Hom(H(G), Zi ) → Hom(H(G), (C∗ )r ) (see also the proof of Theorem 2.6.6 in [14]): if A is one of Z1 or Z2 , α one of αi , then to a central extension e ∈ H 2 (G, A) given by 1 → A → G˜ → G → 1, we first associate the homomorphism G, G → G˜ which maps x, y to [x, ¯ y], ¯ x, ¯ y¯ ˜ This homomorphism maps C(G) into A (in fact onto A ∩ [G, ˜ G]), ˜ lifts of x, y in G. and it maps B(G) to {1}; hence, when associated to e, one gets a homomorphism ˜ G] ˜ ⊂ A. ψe : H(G) → A ∩ [G,

66


Then α (e) = ψe . It is obvious that if e1 and e2 have the same commutator function, then we can choose embeddings Z1 → (C∗ )r , Z2 → (C∗ )r which agree on Z1 ∩ [G1 , G1 ] Z2 ∩ [G2 , G2 ] and such that then ψe1 = ψe2 , viewed as maps into (C∗ )r . Hence we have proven Theorem 6. The notions of being toroidally related and isoclinic coincide. In particular, if G1 and G2 are isoclinic, then generically free linear quotients for G1 and G2 are stably equivalent. This answers Question (1.11) of [12] in the affirmative; the partial result that G1 and G2 have isomorphic second unramified cohomology groups has been proven in [15]. We give an additional elementary argument for the implication “G1 and G2 isoclinic” =⇒ “the extensions e1 and e2 are toroidally related.” Suppose that G1 and G2 are isoclinic, and consider the fiber product over G as above: π

G12 = G1 ×G G2 −−−1−→ ⏐ ⏐ π2 G2

G1 ⏐ ⏐

−−−−→ G .

We want to show that then the extensions 1 → Z2 → G12 → G1 → 1 and 1 → Z1 → G12 → G2 → 1 are toroidally trivial. Note that the preimages of [G1 , G1 ] ⊂ G1 and [G2 , G2 ] ⊂ G2 coincide with [G12 , G12 ] and this group equals [G12 , G12 ] = ([g1 , h1 ], j([g1 , h1 ]))

g 1 , h 1 ∈ G1 ,

which is a “diagonal subgroup” of G12 which intersects both Z1 {(z1 , 1) | z1 ∈ Z1 } ⊂ G12 and Z2 {(1, z2 ) | z2 ∈ Z2 } trivially. In other words, Z1 maps isomorphiab cally to G2 /[G2 , G2 ] = Gab 2 and Z2 isomorphically to G1 /[G1 , G1 ] = G1 . Therefore, for example, the extension G12 of G2 by Z1 is induced by an abelian extension 1 → Z1 → G12 /[G12 , G12 ] → G2 /[G2 , G2 ] → 1 . The same holds for the extension G12 of G1 . Hence these two are toroidally trivial, and hence e1 and e2 are toroidally related. The following remark is sometimes useful and summarizes some compatibilities of isoclinism with passing to subgroups or quotients. It can be found already in [11]. Remark 7. An isoclinism between G1 and G2 sets up a bijective correspondence between subgroups of G1 containing Z1 and subgroups of G2 containing Z2 , and the corresponding subgroups are isoclinic. In particular, a centralizer in the group G2 is isoclinic to a centralizer in G1 .

Stable Cohomology

67

Moreover, an isoclinism also gives a bijective correspondence between quotient groups G1 /K1 and G2 /K2 , where K1 ⊂ [G1 , G1 ] and K2 ⊂ [G2 , G2 ], and the corresponding quotient groups are isoclinic.

4 Stable Cohomology of Wreath Products Now we want to use Proposition 3 to compute the stable cohomology of the iterated wreath product Gn = Z/p Z/p · · · Z/p of groups Z/p (there are n factors Z/p). Note that here, strictly speaking, Gn = (((. . . (Z/p Z/p) Z/p) Z/p) . . .) Z/p, but the brackets are not important because of the associativity (Z/p Z/p) Z/p Z/p (Z/p Z/p) which, however, has to be suitably interpreted: on the right-hand side we view Z/p 2 Z/p as a subgroup of S p2 and define the action on the normal subgroup (Z/p) p via this inclusion. We first define a class of groups which will turn out to be stable under the operation of taking iterated centralizers, provided one identifies groups which are isoclinic/toroidally related and contains Gn . Definition 8. Let D be a class of groups with the property that (1) every group in D has stably rational generically free linear quotients, and (2) the centralizer of any element in a group in D again belongs to D. Then we define a group to belong to the class C p (D) if it can be reached starting from a group in D by a finite number of the following operations, which successively enlarge the set of groups in C p (D) already constructed: (a) Taking a wreath product with a group Z/p, i.e., passing from H to H Z/p (b) Taking a finite direct product (c) Passing from H to an isoclinic group H It follows that all the groups in C p (D) have trivial higher unramified cohomology; in fact, they all have stably rational quotients. The fact that operation (a) does not take us outside groups with stably rational generically free quotients that follows from Lemma 2.4 of [7], and for operation (b) it is obvious (if G1 , G2 are two groups, look at V1 × V2 , where V1 is a generically free representation of G1 and V2 a generically free representation of G2 ). Moreover, clearly, Gn belongs to C p if we take D to consist only of the group Z/p. The key result will be Proposition 10 below. We precede it with a Lemma on isoclinism types of centralizers in a wreath product H = H Z/p which will be used in the proof of Proposition 10.

68


Lemma 9. Let H be a finite group, denote (H ) p = H × · · · × H (p factors), and let H = H Z/p be its wreath product with Z/p. Let x ∈ H be some element, and ZH (x) be its centralizer. Then one of the following is true: (a) The element x is contained in (H ) p ⊂ H, ZH (x) does not surject onto the quotient Z/p under the natural projection H Z/p, and the centralizer ZH (x) is a product ZH (x1 ) × · · · × ZH (x p ) of the centralizers ZH (xi ) of the components xi of x with respect to the product H × . . . × H (p factors) in H. (b) The cyclic subgroup x generated by x in H surjects onto the quotient Z/p under the natural projection H Z/p, and if x p = ((a, a, . . . , a), id), then ZH (x) is isoclinic to ZH (a) × Z/p . (c) The element x is contained in (H ) p and ZH (x) surjects onto Z/p. Then x = (x , . . . , x ) ∈ (H ) p and ZH (x) is isomorphic to ZH (x ) Z/p . Proof. The cases enumerated in (a), (b), and (c) obviously cover all the possibilities and are mutually exclusive. We deal with them one by one. Case (a) The centralizer ZH (x) is contained in (H ) p . Then ZH (x) is obviously the product of the centralizers of components. Case (b) Suppose that x surjects onto the quotient Z/p of H = H Z/p. After conjugating by an element in (H ) p we may assume x = ((a, id, . . . , id), σ ) for some a ∈ H and σ ∈ Z/p a generator. Indeed, if a priori x = ((x1 , . . . , x p ), σ ), it is sufficient for this to solve the equations = 1 for i = 1 ci xi c−1 σ −1 (i) in elements ci of H , which is always possible successively, and conjugate by c = ((c1 , . . . , c p ), id). To see that it is indeed possible successively, put, e.g., cσ −1 (1) = 1, cσ −2 (1) = xσ −1 (1) , cσ −3 (1) = xσ −1 (1) xσ −2 (1) , . . . , cσ −p (1) = c1 = xσ −1 (1) . . . xσ −(p−1) (1) , a = xσ −1 (1) . . . xσ −(p−1) (1) x1 = ∏i xi . We will assume therefore now that x = ((a, id, . . . , id), σ ) with σ a generator of Z/p for simplicity. Clearly, ZH (x) is generated by x and those y = ((y1 , . . . , y p ), id) which commute with x. These elements y will also commute with all powers of x, hence with ((a, a, . . . , a), id). That is, it is necessary that a commutes with every y j . The condition ((y1 , . . . , y p ), 1)((a, 1, . . . , 1), σ ) = ((a, 1, . . . , 1), σ )((y1 , . . . , y p ), 1)

Stable Cohomology

69

reads y1 a = ayσ −1 (1) y2 = yσ −1 (2) , .. . y p = yσ −1 (p) . This implies yσ −1 (1) = yσ −2 (1) = · · · = yσ −(p−1) (1) = y1 , so the elements y which commute with x are precisely those such that y1 = · · · = y p and all of them commute with a. In this case it follows that ZH (x) is an extension 0 → ZH (a) → ZH (x) → Z/p → 0 which will in general be nontrivial; however, we claim that ZH (x) is toroidally related to the product ZH (a) × Z/p. More precisely, ZH (x) is a central extension 1 → a → ZH (x) → ZH (x)/a ZH (a)/a × Z/p → 1 of ZH (a)/a × Z/p by the cyclic group generated by a. We claim that this extension is toroidally related to the extension 1 → a → ZH (a) × Z/p → ZH (a)/a × Z/p → 1 . Indeed, look at the extension Z H (x) of ZH (a)/a × Z/p by Q/Z induced by ZH (x): 1

ZH (x)

1

Z H (x)

Q/Z ·N

Q/Z

1

ZH (a)/a × Z/p

1

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As Q/Z is divisible, the element a ∈ ZH (a) ⊂ ZH (x) is a p-th power of a central element A in Z H (x). In particular, ZH (x) contains also ZH (a) × Z/p; we can map a generator of Z/p to ((a, 1, . . . , 1), σ )A−1 in ZH (x). Case (c) We have x ∈ (H ) p , but there is an element g ∈ ZH (x) such that the subgroup generated by it surjects onto Z/p. The element g is then conjugated to ((a, id, . . . , id), σ ) for some a ∈ H and σ ∈ Z/p a generator. We can assume then that g = ((a, id, . . . , id), σ ). Now g p = ((a, a, . . . , a), id) ∈ Z(H ) p (x), and as this is a product of the centralizers of the components of x, also b = ((a, id, . . . , id), id) ∈ ZH (x). So b−1 · g = ((id, . . . , id), σ ) is also always in ZH (x) in this case. Hence x = (x , . . . , x ) here, and ZH (x) = ZH (x ) Z/p. Proposition 10. Suppose G is a group in C p (D) and let h ∈ G be some element. Then the centralizer ZG (h) is again a group in C p (D). Proof. If we denote =⇒ one of the construction steps in Definition 8 and by G0 =⇒ G1 , G0 , G1 ∈ C p (D), the fact that G1 is gotten from G0 applying one construction step, then we call G0 =⇒ G1 =⇒ . . . =⇒ Gs with G0 in D a chain of length s and Gs the end of the chain. Let C p (D)≤s be the subclass of all groups in C p (D) which are ends of chains of length ≤ s. We prove by induction on s, starting from s = 0, that every centralizer of an element in a group in C p (D)≤s again belongs to C p (D). The induction base is trivial because the assertion of Proposition 10 holds by assumption for all the groups in D. Now suppose it holds for all groups in C p (D)≤s . Suppose then H is a group in C p (D)≤s+1 constructed out of C p (D)≤s according to the rules in Definition 8. We have the following possibilities: (1) The group H is a finite product H = H1 × · · · × HN of groups Hi in C p (D)≤s . Then the centralizer ZH (x) of an element x ∈ H is the product of the centralizers of the components xi of x. Each of the ZHi (xi ) belongs to C p (D) by the induction hypothesis, hence so does the product as C p (D) is closed under taking finite products by definition. (2) The group H is isoclinic to a group H in C p (D)≤s . By Remark 7, centralizers of elements in H are isoclinic to centralizers of elements in H . The latter however belong to C p (D) by induction. As C p (D) is closed under passage to isoclinic groups, H belongs to C p (D), too, in this case. (3) The group H is a wreath product H = H Z/p,

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where H belongs to C p (D)≤s . According to Lemma 9 above, we see, using the induction hypothesis and the definition of the class C p (D), that H also belongs to C p (D).

This concludes the proof. Theorem 11. for any group

The stable cohomology Hs∗ (G, Z/p) is detected by abelian subgroups G in C p (D).

Proof. Here we use the main Theorem 2.6 of [5]. Everything follows from Proposition 10, saying that C p (D) is closed under taking centralizers, and induction over the cohomological degree: note that all groups in C p (D) have trivial higher unramified cohomology and that the stable cohomology Hs1 (G, Z/p) of any finite group G is detected by abelian subgroups: this is so because any nontrivial character χ : G → Z/p is nontrivial on a cyclic subgroup in G. Corollary 12. The stable cohomology Hs∗ (Gn , Z/p) is detected by abelian subgroups. Proof. Take D = {Z/p} and apply Theorem 11.

Hs∗ (Gn ,

The following result allows us to determine Z/p) rather precisely. It follows from Corollary 12 but requires some additional work. Theorem 13. The stable cohomology Hs∗ (Gn , Z/p) is detected by elementary abelian subgroups. Proof. It will be sufficient to prove: p Every nontrivial class α ∈ Hsk (Gn , Z/p), k > 1, is nontrivial on the subgroup Gn−1 ⊂ Gn−1 Z/p = Gn .

Then the assertion of Theorem 13 will follow by induction and the fact that Hs1 (Gn , Z/p) is always detected by elementary abelian subgroups: in fact, every p nontrivial character χ : Gn → Z/p is nontrivial on Gn−1 or else nontrivial on the quotient Z/p. p By Theorem 12 a class α as above which is trivial on Gn−1 must be nontrivial on some abelian subgroup A which surjects onto Z/p under the composite map p A → Gn Z/p. Then A is contained in a subgroup BZ/p ⊂ Gn where B p ⊂ Gn−1 is p abelian: we can take for B the image of A∩Gn−1 in Gn−1 under any of the coordinate p projections Gn−1 → Gn−1 . Note that if x = ((a, 1, 1, . . . , 1), σ ) is an element in A p such that x surjects onto Z/p and if b = ((b1 , . . . , b p ), 1) ∈ A ∩ Gn−1 , then the equation xbx−1 = b together with the fact that a commutes with every bi implies that b1 = · · · = b p , so all coordinate projections are the same. Thus we get a nontrivial class in Hsk (B Z/p, Z/p) which with slight abuse of notation we denote again by α . We have to recall some results about the structure of the cohomology of B Z/p: by Nakaoka’s Theorem (see [13]) one has an isomorphism H ∗ (B Z/p, Z/p) H ∗ (Z/p, H ∗ (B, Z/p)⊗p ),

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where we consider the cohomology H ∗ (B, Z/p)⊗p as a Z/p[Z/p]-module (with nontrivial action). However, H ∗ (B, Z/p)⊗p is a direct sum of trivial Z/p[Z/p]modules and free Z/p[Z/p]-modules (see [1], p. 117). The trivial modules are generated by norm elements x ⊗ · · · ⊗ x ∈ H ∗ (B, Z/p)⊗p , x ∈ H ∗ (B, Z/p). The free modules do not contribute to the cohomology. Let b1 , b2 , . . . be a basis for H ∗ (B, Z/p). Hence there is a natural splitting H ∗ (B Z/p, Z/p) = H 0 (Z/p, H ∗ (B, Z/p)⊗p ) ⊕

H k (Z/p, Ti )

k>0, i

H ∗ (B p , Z/p)Z/p ⊕

H k (Z/p, Ti ) ,

k>0, i

the direct sum running over all the trivial modules Ti , generated by bi ⊗ · · · ⊗ bi , which occur. Consider now a faithful toric representation RB for B that stabilizes the cohomology of B with open part RoB where the B action is free. We construct the faithful B Z/p representation RBp ⊕ C where Z/p acts via a p-th root of unity in C and rotates the copies of RB . This has an open toric free part (RoB ) p × C∗ and the quotient Q := ((RoB ) p × C∗ )/(B Z/p) has the structure of a torus fibration with fiber (RoB ) p /B p over C∗ (C∗ )/(Z/p). The fundamental group π1 (Q) yields a partial stabilization for the cohomology of B Z/p and is of the form Bs Z where Bs = π1 (RoB /B) and stabilizes the cohomology of B (here partial stabilization means simply that the kernel of H ∗ (B Z/p, Z/p) → H ∗ (π1 (Q), Z/p) consists of unstable classes, so the stabilization map factors over the former map). We consider the image of the cohomology of B Z/p in the cohomology of Bs Z/p. That map can be factored f

f

H ∗ (B Z/p, Z/p) −−−1−→ H ∗ (Bs Z/p, Z/p) −−−2−→ H ∗ (Bs Z, Z/p), and the cohomologies of Bs Z/p and also of Bs Z can be described analogously to what was said above: First, clearly, H ∗ (Bs Z/p, Z/p) H ∗ (Z/p, H ∗ (Bs , Z/p)⊗p ), and the description is entirely the same as before. Now for Bs Z we also have (see [13], discussion on p. 19 and proof of Theorem 5.3.1,) that H ∗ (Bs Z, Z/p) H ∗ (Z, H ∗ (Bs , Z/p)⊗p ), where now we consider H ∗ (Bs , Z/p)⊗p as a Z/p[Z] module via the quotient map Z → Z/p. In particular, the free Z/p[Z/p]-submodules of H ∗ (Bs , Z/p)⊗p may contribute to the cohomology H ∗ (Z, H ∗ (Bs , Z/p)⊗p ) now (but those classes do not come from H ∗ (B Z/p, Z/p)). The maps f1 and f2 have a natural description using the previous isomorphisms: the map f1 is just induced by the map of coefficients H ∗ (B, Z/p)⊗p → H ∗ (Bs , Z/p)⊗p (which is a stabilization map for the cohomology of B p ), and f2 is induced by the natural surjection of groups Z Z/p.

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From this description we see that if a class α is in H ∗ (B p , Z/p)Z/p and stable, then it is detected already on B p . Moreover, the classes in H k (Z/p, Ti ) can only be stable if k = 1 and Ti is generated by β ⊗ · · · ⊗ β with β stable (and part of the chosen basis for H ∗ (B, Z/p)). Let deg(β ) =: b, and let us show that in fact all classes α = τ ∪ (β ⊗ · · · ⊗ β ) with b > 0 and τ some generator of H 1 (Z/p, Z/p) are unstable in H ∗ (B Z/p, Z/p), which will prove Theorem 13. The degree of α is 1 + pb. However, α is then induced from a class α in E Z/p, where E (Z/p)b is elementary abelian, via some surjection B E: in fact, we may assume β is a monomial in e1 , . . . , er , the latter being some basis of H 1 (B, Z/p) and then the surjection is just a coordinate projection followed by reduction to Z/p. If α were stable, then α would be stable. This is however clearly not so if b > 0 for in that case E Z/p has a faithful representation of dimension pb (let each standard copy of Z/p in E Z/p × · · · × Z/p act on C via a nontrivial character and let Z/p rotate those copies). Hence all classes in the cohomology of E Z/p of degrees > pb are killed under stabilization for dimension reasons. We have already proven a lot more, but let us record the easy. Corollary 14. The stable cohomology Hs∗ (Gn , Z/p) is detected by the two subp groups Gn−1 and Gn−1 × Z/p. Proof. It is known ([16], p. 349) that every maximal elementary abelian p-subgroup p of Gn is contained in Gn−1 or Gn−1 × Z/p. We can now also say very precisely how the cohomology ring Hs∗ (Gn Z/p) is structured. Theorem 15. The stable cohomology Hs∗ (Gn , Z/p) is determined inductively as follows: there is an isomorphism of groups (respecting the grading) p , Z/p)Z/p ⊕ Hs1 (Z/p, Z/p), i : Hs∗ (Gn , Z/p) Hs∗ (Gn−1 n where i = (i1 , i2 ) and i1 is the restriction map resG p G

to the subgroup Z/p in Gn = Gn−1 Z/p.

n−1

and i2 is the restriction map

Proof. This follows rather easily from the proof of Theorem 13, but let us give another proof using Steenrod’s operations and the Bloch-Kato conjecture to show how everything falls into place. We use triviality of the Steenrod operations in stable cohomology which is proved in [5], Lemma 2.7; for general information on the Bloch-Kato conjecture in relation to the birational anabelian geometry program, see [8]. By Steenrod’s description of the cohomology of wreath products (see, e.g., [1], IV.4, and IV.7) one has p H ∗ (Gn−1 Z/p) = H ∗ (Gn−1 , Z/p)Z/p + {Γ (α ) ∪ θi }, p p , Z/p)Z/p is the image of the restriction map to Gn−1 and α ∈ where H ∗ (Gn−1 k kp H (Gn−1 , Z/p), Γ (α ) ∈ H (Gn−1 Z/p, Z/p) is the total Steenrod power Γ

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applied to α , and θi are some classes in H i (Z/p, Z/p) with i ≥ 1. A little more conceptually, one can say that by Nakaoka’s Theorem, as above, one has an isomorphism H ∗ (Gn−1 Z/p, Z/p) H ∗ (Z/p, H ∗ (Gn−1 , Z/p)⊗p ) and there is a natural splitting H ∗ (Gn−1 Z/p, Z/p) = H 0 (Z/p, H) ⊕

H k (Z/p, Tx )

k>0, Tx p H ∗ (Gn−1 , Z/p)Z/p ⊕

H k (Z/p, Tx ) ,

k>0, Tx

the direct sum running over all the trivial modules Tx , generated by x ⊗ · · · ⊗ x, which occur. The point of the total Steenrod power Γ is then that it realizes the generator of H k (Z/p, Tx ) as an explicit class Γ (x) ∪ θk in the cohomology of Gn (here θk ∈ H k (Z/p, Z/p)) in a way that is compatible with cup products and functorial for group homomorphisms. We still need to remark that Γ (x) restricts to p H ∗ (Gn−1 , Z/p) as x ⊗ · · · ⊗ x and its restriction to Gn−1 × Z/p is of the form

∑ D j (x) ∪ g j , j

where, keeping in mind H ∗ (Gn−1 × Z/p, Z/p) = H ∗ (Gn−1 , Z/p) ⊗ H ∗ (Z/p, Z/p), the element g j is a generator of H j (Z/p, Z/p) and D j (x) is a certain class in H deg(x)p− j (Gn−1 , Z/p). Here D j (x) is, if nontrivial, equal—up to a sign—to a Steenrod power Ps (x) or β Ps (x), where β is the Bockstein operator. We do not need the exact formula [which is in [1], p. 184, (1.12)], but just that, due to the fact that the Steenrod operations Ps and the Bockstein are zero in stable cohomology, D j (x) ∪ g j can just be nontrivial in stable cohomology if D j is the identity up to a sign and j ≤ 1. This means deg(x)p− j = deg(x), and this can happen for odd primes p only when x is in degree 0. Hence we see that elements of the form Γ (α ) ∪ θi give nontrivial classes in stable cohomology (in view of Corollary 14, which of course we use all the time now) if and only if this is in fact just a class θi coming from H 1 (Z/p, Z/p). p Moreover the elements in H ∗ (Gn−1 , Z/p)Z/p we are getting are exactly of two types: norms and traces; norms being elements of the form x ⊗ · · · ⊗ x, x ∈ H ∗ (Gn−1 , Z/p) (these are stable if and only if x is stable) and traces being of the form p

∑ σ i (x1 ⊗ · · · ⊗ x p)

i=0

p x1 ⊗ · · · ⊗ x p ∈ H ∗ (Gn−1 , Z/p) and σ is a cyclic shift. These are stable if and only if all the xi are stable.

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5 Applications to Finite Groups of Lie Type We would like to give some simple applications of the foregoing material to the computation of stable cohomology of some finite groups of Lie type. Consider the group GLn (Fq ) of automorphisms of Fnq where Fq is the finite field with q = pρ elements. We assume that p and l are prime numbers with (p, l) = 1 and l = 2. Then by [1], Theorem VII.4, 4.1, and Corollary 4.3, the l-Sylow subgroup Syll (GLn (Fq ) is a product of groups of the form Z/l r Z/l · · · Z/l . Hence Theorem 11 shows Theorem 16. Let p and l be two primes with (p, l) = 1, l = 2, q = pρ , ρ , n ≥ 1 integers. Then the stable cohomology of Hs∗ (GLn (Fq ), Z/l) is detected by abelian ∗ (GL (F ), Z/l) is trivial. subgroups. In particular, the unramified cohomology Hnr n q The same statements hold for SLn (Fq ) (the l-Sylow is the same as that of GLn (Fq )). One should compare this to the results obtained in [6]. We hope to give a more exhaustive treatment of other classes of finite groups of Lie type elsewhere. Acknowledgements The first author was supported by NSF grant DMS-1001662 and by AG Laboratory GU- HSE grant RF government ag. 11 11.G34.31.0023. The second author was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft) through Heisenberg-Stipendium BO 3699/1-1.

References 1. ADEM, A. AND M ILGRAM, R. J., Cohomology of finite groups, 2nd ed., Grundlehren der Mathematischen Wissenschaften 309, Springer-Verlag, Berlin, 2004. 2. BENSON , D.J., Representations and Cohomology II. Cohomology of groups and modules, Cambridge studies in advanced mathematics 31, Cambridge University Press (1991) 3. BOGOMOLOV, F., Stable Cohomology of Groups and Algebraic Varieties, Russian Acad. Sci. Sb. Math. 76, no. 1, 1–21, (1993). 4. BOGOMOLOV, F., Stable cohomology of finite and profinite groups, in: Algebraic Groups (Yuri Tschinkel (Ed.)), Mathematisches Institut Georg-August-Universität Göttingen Summer School 27.06.-13.07.2005, Universitätsverlag Göttingen 2007 5. BOGOMOLOV, F. AND BÖHNING , CHR ., Stable cohomology of alternating groups, arXiv:1108.2814v2, (2011). 6. BOGOMOLOV, F., PETROV, T., AND TSCHINKEL, YU ., Unramified cohomology of finite groups of Lie type, in Cohomological and geometric approaches to rationality problems, Progr. Math., 282, Birkhäuser Boston, Inc., Boston, MA, 55–73, (2010). 7. BOGOMOLOV, F. AND PETROV, T., Unramified cohomology of alternating groups, Cent. Eur. J. Math. 9 (5), 936–948, (2011). 8. BOGOMOLOV, F. AND TSCHINKEL, Y., Introduction to birational anabelian geometry, in: Current Developments in Algebraic Geometry, MSRI Publications Volume 59, (2011)

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9. COLLIOT-THÉLÈNE, J.-L., Birational invariants, purity and the Gersten conjecture, in: K-theory and Algebraic Geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992) (B. Jacob and A. Rosenberg, eds.), Proceedings of Symposia in Pure Mathematics 58.1 (1995), 1–64 10. GARIBALDI , S., M ERKURJEV, A., AND SERRE, J.-P., Cohomological invariants in Galois cohomology, University Lecture Series 28, American Mathematical Society, Providence, RI, 2003. 11. HALL, P., The classification of prime power groups, Journal für die Reine und Angewandte Mathematik 182, 130–141, (1940). 12. HOSHI , A., KANG , M.-CH . AND KUNYAVSKII , B. E., Noether’s Problem and Unramified Brauer Groups, arXiv:1202.5812v1, (2012). 13. EVENS, L., The Cohomology of Groups, Oxford Mathematical Monographs, Oxford University Press (1991) 14. KARPILOVSKY, G., The Schur multiplier, London Mathematical Society Monographs New Series 2, Clarendon Press Oxford (1987) 15. M ORAVEC , P., Unramified Brauer groups and Isoclinism, arXiv:1203.2422v1, (2012). 16. M ÙI , H., Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 no. 3, 319–369, (1975). 17. QUILLEN , D., The Adams Conjecture, Topology 10, 67–80, (1971). 18. QUILLEN , D., The spectrum of an equivariant cohomology ring I, Annals of Math. 94, 549–572, (1971). 19. QUILLEN , D., The spectrum of an equivariant cohomology ring II, Annals of Math. 94, 573–602, (1971). 20. STEENROD , N.E., Cohomology Operations, Lectures by N.E. Steenrod, written and revised by D.B.A. Epstein, Annals of Mathematics Studies 50, Princeton University Press 1962

Unirationality and Existence of Infinitely Transitive Models Fedor Bogomolov, Ilya Karzhemanov, and Karine Kuyumzhiyan

Mathematics Subject Classification codes (2000): 14M20, 14M17, 14R20

1 Introduction This article aims to relate unirationality of a given algebraic variety with the property of being a homogeneous space with respect to unipotent algebraic group action. More precisely, let X be an algebraic variety defined over a field k, and Aut(X) be the group of regular automorphisms of X. Let also SAut(X) ⊆ Aut(X) be the subgroup generated by algebraic groups isomorphic to the additive group Ga . Definition 1.1 (cf. [1]). We call variety X infinitely transitive if for any k ∈ N and any two collections of points {P1 , . . . , Pk } and {Q1 , . . . , Qk } on X there exists an element g ∈ SAut(X) such that g(Pi ) = Qi for all i. Similarly, we call X stably infinitely transitive if X × km is infinitely transitive for some m. F. Bogomolov Courant Institute of Mathematical Sciences, New York University, 251 Mercer str., New York, NY 10012, USA National Research University Higher School of Economics, GU-HSE,7. Vavilova Str., Moscow 117 312, Russia e-mail: [email protected] I. Karzhemanov Courant Institute of Mathematical Sciences, New York University, 251 Mercer str., New York, NY 10012, USA e-mail: [email protected] K. Kuyumzhiyan () National Research University Higher School of Economics, GU-HSE,7. Vavilova Str., Moscow 117 312, Russia e-mail: [email protected] F. Bogomolov et al. (eds.), Birational Geometry, Rational Curves, and Arithmetic, DOI 10.1007/978-1-4614-6482-2__4, © Springer Science+Business Media New York 2013

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Recall that in birational geometry adding a number m of algebraically independent variables to the function field k(X) is referred to as stabilization. Geometrically this precisely corresponds to taking the product X × km with the affine space. Note also that if X is infinitely transitive, then it is unirational, i.e., k(X) ⊆ k(y1 , . . . .ym ) for some k-transcendental elements yi (see [1, Proposition 5.1]). This suggests to regard (stable) infinite transitivity as a birational property of X (in particular, we will usually assume the test variety X to be smooth and projective): Definition 1.2. We call variety X stably b-infinitely transitive if the field k(X) (y1 , . . . .ym ) admits an infinitely transitive model (not necessarily smooth or projective) for some m and k(X)-transcendental elements yi . If m = 0, we call X b-infinitely transitive. Example 1.3. The affine space X := kdim X is stably infinitely transitive (and infinitely transitive when dimX ≥ 2); see [8]. More generally, any rational variety is stably b-infinitely transitive, and it is b-infinitely transitive if the dimension ≥ 2. Example 1.3 suggests that being stably b-infinitely transitive does not give anything interesting for rational varieties. In the present article, we put forward the following: Conjecture 1.4. Any unirational variety X is stably b-infinitely transitive. Thus, Conjecture 1.4 together with the above-mentioned result from [1, Proposition 5.1] provides a (potential) characterization of unirational varieties among all those which are rationally connected. Note also that the class of rationally connected varieties contains all stably b-infinitely transitive varieties. We think that not every rationally connected variety is stably birationally infinitely transitive. In particular we expect that generic Fano hypersurfaces from the family considered by Kollar in [10] are not stably birationally infinitely transitive. These are generic smooth 2 hypersurfaces of degree d in Pn+1 , d > (n + 3). Our expectations are based on 3 the Kollar’s fundamental observation (see [10, Theorem (4.3)]) which yields strong restrictions on any surjective map of a uniruled variety of the same dimension on such a hypersurface. Remark 1.5. Originally, the study of infinitely transitive varieties was initiated in the paper [8]. We also remark one application of these varieties to the Lüroth problem in [1], where a nonrational infinitely transitive variety was constructed. See [4] for the properties of locally nilpotent derivations (LNDs for short), [15] for the Makar-Limanov invariant, and [2, 5, 6, 9, 12, 14] for other results, properties, and applications of infinitely transitive (and related) varieties. We are going to verify Conjecture 1.4 for some particular cases of X (see Theorems 2.1, and 2.2 and Propositions 3.4, 3.5, and 3.6 below). At this stage, one should note that it is not possible to lose the stabilization assumption in Conjecture 1.4: Example 1.6. Any three-dimensional algebraic variety X with an infinitely transitive model is rational. Indeed, let us take a one-dimensional algebraic subgroup

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G ⊂ SAut(X) acting on X with a free orbit. Then X is birationally isomorphic to G × Y (see Remark 2.16 below), where Y is a rational surface (since X is unirational). On the other hand, if X := X3 ⊂ P4 is a smooth cubic hypersurface, then it is unirational but not rational (see [3]). However, Conjecture 1.4 is true as stated for X3 , because X3 is stably b-infinitely transitive (see Proposition 3.4 below). In this context, it would be also interesting to settle down the case of the quartic hypersurface X4 in P4 (or, more generally, in Pn for arbitrary n), which relates our subject to the old problem of (non-)unirationality of (generic) X4 (cf. Remark 3.7 below). Notations 1.7. Throughout the paper we keep up with the following: • k is an algebraically closed field of characteristic zero and k× is the multiplicative group of k. • X1 ≈ X2 denotes birational equivalence between two algebraic varieties X1 and X2 . • We abbreviate infinite transitivity (transitive, transitively, etc.) to inf. trans.

2 Varieties with Many Cancellations 2.1 The Set-Up The goal of the present section is to prove the following: Theorem 2.1. Let K := k(X) for some (smooth projective) algebraic variety X of dimension n over k. We assume there are n presentations (we call them cancellations (of K or X)) K = K (xi ) for some K -transcendental elements xi , algebraically independent over k. Then there exists an inf. trans. model of K(y1 , . . . , yn ) for some K-transcendental elements yi . Let us put Theorem 2.1 into a geometric perspective. Namely, the presentation K = K (xi ) reads as there exists a model of K, say Xin , with a surjective regular map πi : Xin → Yin−1 and general fiber P1 such that πi admits a section over an open subset in Yin−1 . Moreover, by resolving indeterminacies, we may assume Xin := X fixed for all i. Then, since K admits n cancellations, n vectors, each tangent to a fiber of some πi , span the tangent space to X at the general point. Indeed, we have a map to Pn X Pn ,

x → (1 : x1 (x) : . . . : xn (x)).

It is dominant since elements x1 , . . . , xn are algebraically independent over k, and the tangent map is surjective at the general point. So we obtain the geometric counterpart of Theorem 2.1: Theorem 2.2. Let X be a smooth projective variety of dimension n. Assume that there exist n morphisms πi : X → Yi satisfying the following:

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(1) Yi is a (normal) projective variety such that πi admits a section over an open subset in Yi . (2) For the general point ζ ∈ X and the fiber Fi = P1i := πi−1 (πi (ζ )) P1 , vector fields TF1 , ζ , . . . , TFn , ζ span the tangent space TX, ζ . Then X is stably b-inf. trans. Note that existence of a section over an open subset on Yi means (almost by definition) birational triviality of the fibration πi . In Sects. 2.2 and 2.3 we illustrate our arguments by considering the cases when dim X = 1 and 2, respectively. In higher dimensions we additionally need the following: (3) For some ample line bundles Hi on Yi and their pullbacks πi∗ Hi to X, the n × n-matrix (1) (πi∗ Hi · P1i ) is of maximal rank (in particular, the classes of π1∗ H1 , . . . , πn∗ Hn in Pic(X) are linearly independent). In particular, this means that the fibers P11 , P12 , . . ., P1n are linearly independent in H2 (X). In Sects. 2.4, 2.6, and 2.7 we prove Theorem 2.2, assuming that the condition (3) is satisfied. Furthermore, adding new variables (i.e., forming the product of X and an affine space) and passing to a (good) birational model, we may assume that (3) holds; see Sects. 2.8 and 2.9.

2.2 One-Dimensional Case Variety O(m)× (or, equivalently, O(−m)× ) is just an affine cone minus the origin P1 P1 over a rational normal curve of degree m. Thus O(m)× is a quasiaffine toric variety, P1 so it is infinitely transitive by [2, Theorem 0.2(3)]. Indeed, we can use only those automorphisms which preserve the origin, i.e., for m-transitivity on k2 \ {0}, we use (m + 1)-transitivity on k2 .

2.3 Two-Dimensional Case Let us study now the next simplest case when X = P1 × P1 . Choose H2 := O(1) on the first factor P1 and, similarly, H1 := O(1) on the second factor P1 . Now take the pullbacks π1∗ H1 and π2∗ H2 to X and throw away their zero sections. We obtain a toric bundle over X isomorphic to (k2 \ {0}) × (k2 \ {0}). The latter is inf. trans. since k2 \ {0} is (cf. the one-dimensional case above). More generally, if one starts with H2 = O(m2 ) and H1 = O(m1 ) for some mi 1, then the resulting variety will be ((k2 \ {0})/(Z/m1Z)) × ((k2 \ {0})/(Z/m2Z)). It is again inf-transitive being the product of two inf-transitive varieties. Indeed, for


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mi > 1, the corresponding variety is just the smooth locus on the corresponding toric variety, and its inf-transitivity is shown in [2, Theorem 0.2(3)]. Remark 2.3. The product of two (quasiaffine or affine) inf-transitive varieties is inftransitive. Indeed, we call variety X flexible if the tangent space at every smooth point on X is generated by the tangent vectors to the orbits of one-parameter unipotent subgroups in Aut(X). It was shown in [1] that for affine X, being flexible is equivalent to inf-transitivity. But clearly the product of two flexible varieties is again flexible.

2.4 Construction of an Infinitely Transitive Model in the Simplest Case Recall the setting. In the notation of Theorem 2.2, we choose very ample line bundles Hi on each Yi , i = 1, . . . , n, take their pullbacks πi∗ Hi to X, put mi j := (πi∗ Hi )|P1 , and form the intersection matrix j

Mn = Mn (X) = (mi j )1i, jn ,

mi j = (πi∗ Hi )|P1 .

(1)

j

Clearly, for all i, we have (πi∗ Hi )|P1 = 0; however, for i = j, (πi∗ Hi )|P1 > 0, being i

j

equal the restriction of Hi to an image of a generic P1j via pi (i.e., the restriction of a bundle on the variety Yi ). The matrix Mn defines a linear map from a subgroup of the Pickard group Pic X to Zn . In this section we suppose that the classes of P11 , . . ., P1n in H2 (X) are linearly independent and also that det Mn = 0. Our goal is to construct a quasiaffine variety TX , TX ≈ X × kN for some N, equipped with a collection of projections to quasiaffine varieties Y¯i with generic fibers being equal to (k2 \ {0})/(Z/mZ) and such that an open subset of TX is inf-transitive, cf. Sect. 2.3. The existence of a good open subset will be shown in Sect. 2.6. To start with, let us set Y¯i := the affine cone OYi (Hi )× minus the origin over Yi embedded via Hi , 1 i n. It is a quasiaffine variety.

2.4.1 Technical Step: Adding One More Coordinate We already embedded Yi into affine varieties; now we also need to embed X. For this purpose, we take a very ample line bundle H0 on X and replace X with X = X × P1 , and Yi with Yi = Yi × P1 . Let also Y0 = X; clearly we have X → Y0 = X, which makes the situation absolutely symmetric with respect to indices 0, 1, . . . , n. We modify the set of Hi s in the following way: for every i > 0, we construct Hi on Yi being the sum

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of the trivial lift of Hi from Yi and O(1) on the new P1 (in fact here we can take any O(ni )). Now the intersection matrix Mn+1 (X ) takes the form ⎛ ⎞ 0 k1 . . . kn ⎜1 ⎟ ⎜ ⎟ Mn+1 (X ) = ⎜ . ⎟. ⎝ .. Mn ⎠ 1 Here ki := H0 · P1i . We further denote X just by X and n + 1 = dim X just by n, keeping in mind that one of our projections is just a trivial projection. We also assume that one column of our matrix contains only 1s (and one 0 on the diagonal). 2.4.2 The Construction of TX We construct a vector bundle H1 ×X π2∗ H2 ×X π3∗ H3 ×X . . . ×X πn∗ Hn and furthermore a toric bundle TX = (H1 )× ×X (π2∗ H2 )× ×X . . . ×X (πn∗ Hn )× .

(2)

We denote by δ the canonical projection TX → X. Line operations in the intersection matrix (1) correspond to base changes in this toric bundle (Neron–Severi torus). For our convenience, we fix below the following set of line bundles L1 , . . . , Ln ∈ H1 , π2∗ H2 , . . . , πn∗ Hn : (i) Each of them should be primitive in the lattice Z(H1 , π2∗ H2 , . . . , πn∗ Hn ). (ii) All in total, they should be linearly independent in the lattice Z(H1 , π2∗ H2 , . . . , πn∗ Hn ). They can be chosen in the following way. There is a map Mn ith coordinate → Zn −−−−−−−−−→ Z. Z(H1 , . . . , Hn ) −−−−

Its kernel has dimension n − 1, and Hi itself belongs to the kernel. So in fact there is a map Z(H1 , . . . , Hˇ i , . . . , Hn ) → Z, and Li is any covector defining this map. All in total, they can be chosen linearly independent.

2.4.3 Construction of Local Two-Dimensional Coordinates Recall that we denote by Y¯i the total space of Hi× → Yi .


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Lemma 2.4. For each i, there is a fibration ϕi : TX → Y¯i such that its general fiber equals ((k2 \ {0})/(Z/miZ)) × Tin−2 , where Tin−2 (k∗ )n−2 . and a linear map Zn → Z2 which is just Proof. Choose a basis Hi , Li , H1 , . . . , Hn−2 taking the two first coordinates in the new basis. Its kernel will correspond precisely to a (n − 2)-dimensional torus, the bundle Hi will provide us with the affine cone Y¯i over Yi , and the bundle Li restricted to P1i will form a quasiaffine fiber of form (k2 \ {0})/(Z/mZ) over a general point of Y¯i .

We have a commutative diagram: T n−2

i × −→ L× TX −−− i × Hi ⏐ ⏐ ⏐ 2 ⏐ (k \{0})/(Z/niZ)×...

X −−−−→ n−2

×

(3)

Yi

Ti ×L1 ×P where T − → Y¯i −−−−→ Yi . X −−−−−−− 1

(4)

This realization will be intensively used below. Note that the fibration is trivial over any open subset U in Yi such that all the fibers of πi are P1 s over U and the restriction of all Hs are generic on these fibers and respectively over Y¯i . So if one fixes a finite number of points P1 , . . . , Ps in Y¯i , we can choose an open subset U in Y¯i containing the fibers passing through all these points (since it is quasiaffine). Lemma 2.5. At the general point x on TX , local coordinates on ((k2 \ {0})/(Z/ mi Z)-fibers from Lemma 2.4, i = 1, . . . , n, form a system of local coordinates on TX at x. Proof. In the notation of Lemma 2.4, tangent space to each fiber of ϕi is spanned by a pair of the tangent vectors to (k2 \ {0})/(Z/mi Z) and by tangent vectors to Tin−2 . By the condition (2) of Theorem 2.2 and by nondegeneracy of matrix Mn , the tangent vectors to (k2 \ {0})/(Z/miZ) are linearly independent, which proves the assertion. 2.4.4 Quasiaffinneness of TX Here we exploit the projections (4) and the technique from the proof of Lemma 2.4. Lemma 2.6. The variety TX is quasiaffine. Proof. The bundle H1 gives an embedding of X to a projective space PN1 , and every Hi , i = 2, . . . , n, embeds Yi to a PNi . The variety TX is now {(x, l2 , . . . , ln )} such that x ∈ X, li ∈ cone(πi (x)) in AN1 +N2 +...+Nn . Note that TX → X is a principal toric bundle which has a section (the diagonal), and all the fibers are isomorphic to (k× )n [see formula (2)]. In particular, we have TX ≈ X × kn .

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2.4.5 Idea of Further Proof Proposition 2.7. The variety TX is stably b-inf. trans. Its proof will be given in Sect. 2.7. We use the ideas from [1, 2, 8] to move an m-tuple of general (in the sense of Sect. 2.6) points to another such m-tuple.

2.5 Stratification on X Let q ∈ X be an arbitrary point. We denote by X(q) the locus of all points on X connected to q by a sequence of smooth fibers P1i of the projections πi , 1 i n. Lemma 2.8. Let Z be an irreducible subvariety of X. Consider all smooth fibers P1i passing through the points of Z and the union Z of all such fibers. Then either dim Z > dim Z or all smooth fibers P1i which contain points in Z are actually ¯ contained in the closure Z. Proof. If the curve P1i intersects Z but is not contained in Z then the curves in the same family intersect an open subvariety in Z since the subvariety X˜i consisting of curves P1i is an open subvariety of X. Hence in the latter case dim Z > Z. Otherwise all the smooth fibers P1i which contain points in Z are actually contained in the closure of Z. Note that the same holds even if a line P1i intersects the closure Z¯ but ¯ is not contained in Z. Corollary 2.9. Every point in X(q) is connected to q by a chain of P1i of length at most n2 . Proof. Indeed, let X p (q) be a subvariety obtained after adding the points connected by the chains of curves of length at most p. It is a union of algebraic subvarieties of X of dimension p. Then by adding the curves from all n families of P1i we either increase the dimension of every component of maximal dimension, or one of them X p0 (q) is invariant, i.e., all smooth fibers P1i which contain points in X p0 (q) are actually contained in the closure of X p0 (q). Note that in the latter case since q ∈ X p0 (q), all other components are contained in X p0 (q), and hence X p0 (q) = X(q). Thus after adding lines from different families we obtain either X(q) or a variety X p+n(q) with maximal component of greater dimension. Thus we will need at most n2 lines to get X(q). Remark 2.10. If we started with a generic point q ∈ X, then it follows from Lemma 2.8 that dim X(q) = n. Indeed, the condition (2) of Theorem 2.2 implies that the tangent vectors to the smooth P1i -fibers in q generate the full tangent space in q, and if X(q) was of lower dimension then the tangent space would also be of lower dimension.


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Remark 2.11. The bound in Corollary 2.9 is not effective. By a more thorough examination one can show that the sequence X0 (q) ⊆ X1 (q) ⊆ . . . stabilizes earlier than at the n2 th step. Corollary 2.12. We can apply the same in reverse. Consider x ∈ X(q). Then all points in X which are connected to x ∈ X(q) can be connected by a chain of length at most n2 + n. Proof. Indeed, for any such point x , we have Xn2 (x ) of dimension n and hence contains an open subvariety in X. It may take at most n P1i s to connect to x. Thus the variety obtained from general points in X in n2 steps coincides with the subvariety of all points in X connected to general point by a chain of smooth lines.

2.6 Construction of a Big Open Subset in TX Now we pass from stratification on X to stratification on TX . We stratify TX in the following way: we take toric preimages for every strata in X. For our needs we take the toric preimage of X(q) for a general point q ∈ X. Note that for every fiber P1i of πi its preimage is (((k2 \ {0})/(Z/mZ)) × T n−2 , and for every chain P1 − P2 − . . . − Pk connecting two points in X there is a chain P¯1 − P¯2 − . . . − P¯k in TX such that every two adjacent points belong to the same ((k2 \ {0})/(Z/mZ)) for one of the projections.

2.7 Proof of Proposition 2.7 Let a variety TX be as in (2). For each i we have a fibration with the quasiaffine base and fiber being ((k2 \ {0})/(Z/miZ) × T n−2 ; see (4). Definition 2.13. For the points C1 , . . . ,Cr in the base of projection (4), let StabC1 ,...,Cr be the subgroup in SAut(TX ) preserving all the fibers of the projection and fixing pointwise the fibers above C1 , . . . ,Cr . Now we need some technique concerning LNDs. To lift automorphisms, we need to extend an LND on (k2 \ {0})/(Z/mZ) to a LND on TX (i.e., to a LND of the algebra k[TX ]). More precisely, suppose that we chose a fiber of form (k2 \ {0})/(Z/mZ) of the projection (4) and some other fibers that we want to fix. We can project all these subvarieties to Y¯i and then take a regular function on Y¯i which equals 1 at the projection of the first fiber and 0 in the projections of other fibers. If we multiply the LND by this function (obviously belonging to the kernel

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of the derivation) and trivially extend it to the toric factor, we will obtain a rational derivation well-defined on an open subset U of TX corresponding to the smooth locus of the corresponding πi . Now we can take a regular function on TX (lifted from a regular function on Y¯i ) such that its zero locus contains the singular locus of the projection, multiply the derivation by some power of this function, and obtain a regular LND on TX .1 For a given m-tuple of points P1 , P2 , . . . , Pm , we need the following lemma: Lemma 2.14. In the notation as above (4), let C0 be a point on a base such that the fiber over this point is general, and P1 , . . . , Ps be some points from this fiber with different projections to Y¯i . Let also C1 , . . . ,Cr be some other points of the base. Then the subgroup StabC1,...,Cr acts infinitely transitively on the fiber over C0 , i.e., can map P1 , . . . , Ps to any other subset in the same fiber. Proof. By [2, Theorem 0.2(3)], the fiber is infinitely transitive. For every oneparameter unipotent subgroup of automorphisms on this fiber, we can lift it to TX , fixing pointwise a given finite collection of fibers; see above. Now it remains to prove infinite transitivity for TX . There are two ways to show it.

2.7.1 Way 1 Lemma 2.15. For m + 1 points P1 , P1 , P2 , P3 , . . . , Pm projecting to the chosen above open subset in X, there exists an automorphism mapping P1 to P1 and preserving all the other points. Proof. There always exists a small automorphism which moves the initial set to a set where for all i all the (k2 \ {0})/(Z/mi Z)-coordinates of the given points are different. Let us connect the projections of P1 and P1 by a chain of smooth P1i curves in X. We denote by Q1 , . . . , Qs the intersection points of these curves, Qi ∈ X, Q1 = δ (P1 ), and Qs = δ (P1 ). For i = 2, . . . , (s − 1) we take some lifts Ri ∈ TX of these points in such a way that all their (k2 \ {0})/(Z/mi Z)-coordinates do not coincide with the corresponding coordinates of the previous points. Let R1 := P1 and Rs = P1 . For every i, 1 i (s − 1), we want to map Ri to Ri+1 by an automorphism of TX preserving all the other points in the given set. We may assume that Ri and Ri+1 belong to one two-dimensional fiber of form (k2 \ {0})/(Z/mZ) of one of the projections to Y¯ j × T jn−2 (the toric fibration is generated by Li s, and we can densify the sequence of Ri s if needed to change only one Li -direction at every step to fulfill this condition). Every such two-dimensional fiber is inf-transitive. Now we need to lift the corresponding automorphism to TX . We need two following observations. First, if we are lifting a curve with respect to the projection πi , then the resulting automorphism is well defined over the singular fibers and is trivial there. Second,

the terminology of [1], if ∂ is a LND, g = exp(t ∂ )|t=1 the corresponding automorphism, and f a function from the kernel of ∂ , then the automorphism g1 = exp(t f ∂ )|t=1 is called a replica of g.

1 In


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all the two-dimensional fibers belonging to the same fiber of ϕi move together, and if several Pj belong to the same fiber as the Ri which we are moving, then we use that their projections to the two-dimensional fiber are different and also different from the projection of Ri+1 , and we use inf-transitivity (not only 1-transitivity) of the corresponding fiber. Now for every i we lift the corresponding automorphism of the two-dimensional fiber to an automorphism of TX from the corresponding subgroup Stab fixing the points from the other fibers of δ and multiply all these automorphisms. It does not change P2 , . . . , Pm and maps P1 to P1 . This ends the proof. Now infinite transitivity follows easily: to map P1 , P2 , . . . , Pm to Q1 , Q2 , . . ., Qm , we map P1 to Q1 fixing P2 , . . . , Pm , Q2 , . . . , Qm , etc.

2.7.2 Way 2 The other way to finish the proof is as follows. It is enough to show 1-transitivity while fixing some other points of a given finite set. Let us consider automorphisms of bounded degree fixing P2 , . . . , Pm and the orbits of P1 and P1 under this group. Clearly, by flexibility, every orbit is an open subset in TX , and every two dominant subsets should have a nonempty intersection. So there is a common point, which means that P1 can be mapped to P1 by a subgroup in SAut(TX ) fixing P2 , . . . , Pm . Remark 2.16. Conversely, in view of Theorem 2.2, given a b-inf. trans. variety X, there exist dimX cancellations of X. Indeed, for general point ζ ∈ X, we can find dim X tangent vectors spanning TX,ζ , such that each vector generates a copy of Ga =: Gi ⊆ SAut(X), 1 i n. Let G ⊆ SAut(X) be the subgroup generated by the groups G2 , . . . , Gn . Then we have X ≈ G1 × G · ζ .

2.8 Increasing the Rank of the Corresponding Subgroup in H2 (X) We want to treat here the case when rkP11 , . . . , P1n is t, t < n, as of a subgroup in H2 (X). Remark 2.17. Here we precise the ancient construction of TX . Indeed, if the rank is not maximal, then the toric bundle contains a trivial part, and we need to get rid of it. One way is to change it with the trivial vector bundle part. However here we give another construction which uses stabilization. Lemma 2.18. There is a stabilization X of X such that dim X − t(X ) < (n − t). Proof. We assume that the cycles P1i are dependent and in particular that an integer multiple of P1n is contained in the envelope of P1i , i < n, on X. There is a natural projection pn,n+1 : X × P1 → Yn with a generic fiber P1 × P1 . Let us take P1 × P1 and

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blow it up at three points. Thus we will have P2 with 5 blown up points. For every four points there is a pencil of conics passing through four points. Indeed, if we fix two smooth conics, there is a pencil of conics passing through the intersection. So on P1 × P1 blown up at three points we can choose two different 4-tuples of points on P2 and define two projections π¯i : BlQ1 ,Q2 ,Q3 (P1 × P1 ) → P1 . Now we can extend them to P1 × P1 × B by blowing up three constant sections and similarly extend projections. The projections π¯1 , π¯2 provide cancellations with new P¯ 1i , i = 1, 2, independent with generic P1j , j = 1, 2 on the blown-up X. We denote the resulting ˜ it is a smooth model of X × P1 . Here the rank t 1 = t + 2. variety by X,

2.9 Increasing the Rank of the Matrix M For a variety X with a given set of cancellations and corresponding bundles, we constructed (1) a matrix Mn of restrictions. To prove birational stable infinite transitivity, we need the rank of this matrix to be full. The aim of this section is to prove the following lemma. Lemma 2.19. Let the rank of the subgroup generated by P1i in H2 (X, Z) be n, and let matrix Mn = M(X) = (mi, j ) be as in (1) and its rank be s < n = dim X. Then there exists a birational model X˜ for X × P1 with n + 1 projections corresponding to cancellations and a family Hi1 , i = 1, 2, . . . , (n + 1), such that s1 ≥ n + 2 for the new matrix M(X˜ ). Proof. By Lemma 2.18, we may assume that all the classes [P11 ], [P12 ], . . ., [P1n ] are independent in H2 (X). If s < n = dim X, then due to Hodge duality, there is a divisor with nonzero positive pairings with all the fibers P1i , i.e., there is an ample divisor Hn+1 on X such that it defines an element in the lattice Zn which is not contained in M(Z(H1 , H2 , . . . , Hn )) (here we identify Hi with the elements of the standard basis in Zn ). Let us define in this case πi1 : X × P1 → (Yi × P1); take Hi1 = Hi + OP1 (ni ) for some positive numbers ni ; πn+1 : X × P1 → X the trivial projection and Hn+1 ˜ chosen above. Then if the restriction of Hn+1 on P1i is O(ti ), the new matrix M(X) is as follows: ⎞ ⎛ n1 ⎜ .. ⎟ ⎜ .⎟ M(X˜ ) = ⎜ M(X) ⎟. ⎝ nn ⎠ t1

. . . tn 0

˜ in this case has rank Note that all the diagonal elements mi,i = 0. The matrix M(X) ˜ s + 2 for some choice of ni . Indeed, the last row of M(X) is independent with other rows by the assumption on Hn+1 . Now we can add ni in such a way that the rank of ˜ will be (rk M + 1) + 1 (if rk M < n). Hence rk M1 = s + 2 in this case. M(X) Corollary 2.20. In finite number of steps (not more than 2n), using Lemmas 2.18 ˜ = dim(X × Pr ) = dim(X). ˜ and 2.19, we obtain a model X˜ of X × Pr with rk M(X)


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3 Examples Here we collect several examples and properties of (stably) b-inf. trans. varieties.

3.1 Quotients Let us start with the projective space Pn , n ≥ 2, and a finite group G ⊂ PGLn+1 (k). Notice that the quotient Pn /G is stably b-inf. trans. Indeed, let us replace G by its finite central extension G˜ acting linearly on V := kn+1 , so that V /G˜ ≈ Pn /G × P1. ˜ Further, form the product V × V with the diagonal G-action, and take the quotient ˜ Then, projecting on the first factor, we get V ≈ V × V /G˜ and V := (V × V )/G. similarly for the second factor. This implies that V admits 2n + 2 cancellations (cf. Theorem 2.1). Hence V is stably b-inf. trans. by Theorem 2.2. The argument just used can be summarized as follows: Lemma 3.1. Let X → S be a Pm -fibration for some m ∈ N. Then the product X ×S X ≈ X × km admits 2m algebraically independent cancellations over S. Proof. Note that X ×S X has two projections (left and right) onto X, both having a section (the diagonal ΔX ⊂ X ×S X); hence the corresponding Pm -fibrations are birational (over S) to X × km . This gives 2m algebraically independent cancellations over S. Corollary 3.2. Assume that X carries a collection of distinct birational structures of Pmi -bundles, πi : X → Si , with the condition that the tangent spaces of generic fibers of πi span the tangent space of X at the generic point. Then X is stably b-inf. trans. Proof. Indeed, after multiplying by the maximum of mi , we may assume that all Pmi -bundles provide with at least 2mi different cancellations (see Lemma 3.1). We can now apply Theorem 2.2. Remark 3.3. It seems plausible that given an inf. trans. variety X and a finite group G ⊂ Aut(X), variety X/G is stably b-inf. trans. (though the proof of this fact requires a finer understanding of the group SAut(X)). At this stage, note also that if G is cyclic, then there exists a G-fixed point on X. Indeed, since X is alg unirational (cf. Sect. 1), it has trivial algebraic fundamental group π1 (X) (see [11]). alg Then, if the G-action is free on X, we get G ⊂ π1 (X/G) = {1} for X/G smooth unirational, a contradiction. This fixed-point-non-freeness property of X relates X to homogeneous spaces, and it would be interesting to investigate whether this is indeed the fact, i.e., in particular, does X, after stabilization and passing to birational

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model, admit a uniformization which is a genuine (finite dimensional) algebraic group?2

3.2 Cubic Hypersurfaces Let X3 ⊂ Pn+1 , n ≥ 2, be a smooth cubic. Then Proposition 3.4. X3 is stably b-inf. trans. Proof. Smooth cubic X3 contains a two-dimensional family of lines which span P4 . Let L ⊂ X3 be a line and π : X3 Pn−1 the linear projection from L. Let us resolve the indeterminacies of π by blowing up X3 at L. We arrive at a smooth variety XL together with a morphism πL : XL → Pn−1 whose general fiber is P1 ( a conic in P2 ). Varying L ⊂ X3 , we then apply Lemma 3.1 and Corollary 3.2 to get that X3 is stably b-inf. trans.

3.3 Quartic Hypersurfaces Let X4 ⊂ Pn , n ≥ 4, be a quartic hypersurface with a line L ⊂ X4 of double singularities. Then Proposition 3.5. X4 is stably b-inf. trans. Proof. Consider the cone X4 ⊂ Pn+1 over X4 . Then X4 contains a plane Π of double singularities. Pick up a (generic) line L ⊂ Π and consider the linear projection X4 Pn−1 from L . This induces a conic bundle structure on X4 , similarly as in the proof of Proposition 3.4, and varying L in Π as above we obtain that X4 is stably b-inf. trans. Then, since X4 ≈ X4 × k, Proposition 3.5 follows.

3.4 Complete Intersections Let X2·2·2 ⊂ P6 be the smooth complete intersection of three quadrics. Then

2 This question was suggested

by J.-L. Colliot-Thélène in connection with Conjecture 1.4. However, there are reasons to doubt the positive answer, since, for example, it would imply that X is (stably) birationally isomorphic to G/H, where both G, H are (finite dimensional) reductive algebraic groups. Even more, up to stable birational equivalence, we may assume that X = G /H , where H is a finite group and G is the product of a general linear group, spinor groups, and exceptional Lie groups. The latter implies, among other things, that there are only countably many stable birational equivalence classes of unirational varieties, but we could not develop a rigorous argument to bring this to contradiction.


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Proposition 3.6. X2·2·2 is stably b-inf. trans. Proof. The threefold X2·2·2 contains at least a one-dimensional family of lines. Let L ⊂ X2·2·2 be a line and XL → X2·2·2 the blowup of L. Then the threefold XL carries the structure of a conic bundle (see [7, Chap. 10, Example 10.1.2, (ii)]). Now, varying L and applying the same arguments as in the proof of Proposition 3.4, we obtain that X2·2·2 is stably b-inf. trans. Remark 3.7. Fix n, r ∈ N, n r, and a sequence of integers 0 < d1 . . . dm , m di + r m ≥ 2. Let us assume that (n − r)(r + 1) ≥ ∑ . Consider the complete r i=1 intersection X := H1 ∩ . . . ∩ Hm of hypersurfaces Hi ⊂ Pn of degree di . Then it follows from the arguments in [13] that X contains a positive dimensional family of linear subspaces Pr . Moreover, X is unirational, provided X is generic. It would be interesting to adopt the arguments from the proofs of Propositions 3.4, 3.5, and 3.6 to this more general setting in order to prove that X is stably b-inf. trans. Remark 3.8. Propositions 3.4, 3.5, and 3.6 (cf. Remark 3.7) provide an alternative method of proving unirationality of complete intersections (see [7, Chap. 10] for recollection of classical arguments). Note also that (generic) X2·2·2 is nonrational (see, e.g., [16]), and (non)rationality of the most of other complete intersections considered above is not known. At the same time, verifying stable b-inf. trans. property of other (nonrational) Fano manifolds (cf. [7, Chap. 10, Examples 10.1.3, (ii), (iii), (iv)]) is out of reach for our techniques at the moment. Acknowledgements The first author was supported by NSF grant DMS-1001662 and by AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023. The third author was supported by AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023, RFBR grants 12-01-31342 and 1201-33101, by the “EADS Foundation Chair in Mathematics”, Russian-French Poncelet Laboratory (UMI 2615 of CNRS MK-983.2013.1), and Dmitry Zimin fund “Dynasty.” The first author wants to thank Sh. Kaliman and M. Zaidenberg for useful discussions. The second author would like to thank Courant Institute for hospitality. The second author has also benefited from discussions with I. Cheltsov, Yu. Prokhorov, V. Shokurov, and K. Shramov. The third author would like to thank I. Arzhantsev for fruitful discussions. The authors are grateful to the referee for valuable comments. Finally, the authors thank the organizers of the summer school and the conference in Yekaterinburg (2011), where the work on the article originated.

References 1. I. V. ARZHANTSEV, H. FLENNER , S. KALIMAN , F. KUTZSCHEBAUCH , AND M. ZAIDEN BERG , Flexible varieties and automorphism groups, Duke Math. J. 162, No 4 (2013), 60p. (to appear), arXiv:1011.5375, (2010), 41p. 2. I. V. ARZHANTSEV, K. KUYUMZHIYAN , AND M. ZAIDENBERG , Flag varieties, toric varieties, and suspensions: three instances of infinite transitivity, Sbornik:Mathematics, russian version: 203 7, 3–30, (2012), arXiv:1003.3164. 3. C. H. CLEMENS AND P. A. GRIFFITHS, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95, 281–356, (1972).

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4. G. FREUDENBURG , Algebraic theory of locally nilpotent derivations, Encyclopaedia of Mathematical Sciences, 136, Springer, Berlin (2006). 5. M. H. GIZATULLIN AND V. I. DANILOV , Examples of nonhomogeneous quasihomogeneous surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 38, 42–58, (1974). 6. J. HUISMAN AND F. M ANGOLTE, The group of automorphisms of a real rational surface is n-transitive, Bull. Lond. Math. Soc. 41, no. 3, 563–568, (2009). 7. V. A. ISKOVSKIKH AND YU . G. PROKHOROV , Fano varieties, Algebraic geometry, V, 1–247, Encyclopaedia Math. Sci., 47 Springer, Berlin (2000). 8. SH . KALIMAN AND M. ZAIDENBERG , Affine modifications and affine hypersurfaces with a very transitive automorphism group, Transform. Groups 4, no. 1, 53–95, (1999). 9. T. KISHIMOTO , Y. PROKHOROV, AND M. ZAIDENBERG , Group actions on affine cones, in Affine algebraic geometry, 123–163, CRM Proc. Lecture Notes, 54 Amer. Math. Soc., Providence, RI (2011). 10. J. KOLLÁR , Nonrational hypersurfaces, J. Amer. Math. Soc. 8, no. 1, 241–249, (1995). 11. J. KOLLÁR , Y. M IYAOKA , AND S. M ORI , Rational connectedness and boundedness of Fano manifolds, J. Differential Geom. 36, no. 3, 765–779, (1992). 12. K. KUYUMZHIYAN AND F. M ANGOLTE, Infinitely transitive actions on real affine suspensions, Journal of Pure and Applied Algebra 216, 2106–2112, (2012). 13. K. PARANJAPE, V. SRINIVAS, Unirationality of the general Complete Intersection of sufficiently small multidegree, in Flips and abundance for algebraic threefolds, (ed. J. Kollár), Proc. of Summer Seminar at University of Utah, Salt Lake City, U.S.A. 1991, Astérisque 211 (1992), 241–248. 14. A. PEREPECHKO , Flexibility of affine cones over del Pezzo surfaces of degree 4 and 5, arXiv:1108.5841 (2011), to appear in: Func. Analysis and Its Appl. 15. V. L. POPOV , On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, CRM Proceedings and Lecture Notes 54 , 289–311, (2011), arXiv:1001.1311. 16. A. N. TJURIN , The intersection of quadrics, Uspehi Mat. Nauk 30, no. 6 (186), 51–99, (1975).

Birational Geometry via Moduli Spaces Ivan Cheltsov, Ludmil Katzarkov, and Victor Przyjalkowski

Mathematics Subject Classification codes (2000): 14M25, 14H10, 14Q15, 14D07

1 Introduction 1.1 Moduli Approach to Birational Geometry In recent years, we have witnessed many significant developments in the Minimal Model Program (MMP) (see, e.g., [10]), based on major advances in the study of singularities of pairs. An alternative approach to MMP, relying on the correspondence between Mori fibrations (MF) and semiorthogonal decompositions (SOD), was proposed by Kawamata [67]; it does not involve the study of discrepancies or the effective cone. In the meantime, a new epoch, the epoch of wall-crossing, has emerged. At present, after papers of Seiberg–Witten [91], Gaiotto–Moore–Neitzke [40], and Cecotti–Vafa [20] and seminal works by Donaldson–Thomas [33], Joyce–Song [62],

I. Cheltsov University of Edinburgh, Edinburgh, EH1 2QL, UK e-mail: [email protected] L. Katzarkov University of Miami, Coral Gables, FL, USA University of Vienna, Vienna, Austria e-mail: [email protected] V. Przyjalkowski () Steklov Mathematical Institute, Moscow, Russia e-mail: [email protected] F. Bogomolov et al. (eds.), Birational Geometry, Rational Curves, and Arithmetic, DOI 10.1007/978-1-4614-6482-2__5, © Springer Science+Business Media New York 2013

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Maulik–Nekrasov–Okounkov–Pandharipande [79, 80], Douglas [35], Bridgeland [13], and Kontsevich–Soibelman [72, 73], the situation with the wall-crossing phenomenon is quite similar to the situation with the theory of Higgs Bundles after Higgs and Hitchin: it was clear that a general “Hodge type” theory existed and that it needed to be developed. This led to major mathematical applications, for example, in uniformization and in the Langlands program, to mention a few. Similarly, in the framework of wall-crossing, it is also apparent that a “Hodge type” theory needs to be developed in order to reap mathematical benefits, e.g., solve some long-standing problems in algebraic geometry. Already, several attempts have been made to connect homological mirror symmetry (HMS) to birational geometry. For instance, the paper [1] was motivated by the idea that a proof of HMS is likely to involve a detailed study of birational transformations (including, noncommutative) on the A and B sides of HMS. Later, new ideas were introduced in [4, 27, 28, 45, 68, 70]. These can be summarized as follows: 1. The moduli spaces of stability conditions of Fukaya–Seidel categories can be included in a one-parametric family with the moduli space of Landau–Ginzburg (LG) models as a central fiber. 2. The moduli space of LG models determines the birational geometry; this was proved in the toric case in [27]. In this paper, we simultaneously consider all Mori fiber spaces, all Sarkisov links and all relations between Sarkisov links, and explore these in terms of the geometry of moduli space of LG models and the moving schemes involved. We introduce the following main ideas: 1. All Fano varieties are connected, via their degenerations, by simple basic links— projections of a special kind. We show this for Fano threefolds of Picard rank 1 (see Table 2). This agrees with their toric Landau–Ginzburg models (see Theorem 3.12). 2. All Fano manifolds can be considered together, i.e., there exists a big moduli space of LG models, which includes mirrors of all Fano’s. We demonstrate this partially for two- and three-dimensional Fano’s. 3. We introduce an analog of the canonical divisor measure for a minimal model. For us, this is the local geometry of singularities and a fiber at infinity of the LG model. The last one affects the geometry of moduli spaces of LG models, the stability conditions. We propose local models for these moduli spaces (stacks). The correspondence between the usual and the categorical approach to birational geometry is displayed in Table 1. 4. Following [28] and the pioneering work [12], we develop the notion of phantom category, emphasizing its connection with the notion of a moving scheme, defined in this paper. Moving schemes determine the geometry of the moduli space of LG models and thus the geometry of the initial manifold. We conjecture:

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Table 1 Extended Kawamata program Classical Mori fibrations (KX + Δ ) log differentials

Sarkisov links Relations between Sarkisov links

Derived SOD or circuits Moving schemes of the fiber at ∞ in Landau–Ginzburg model described by differentials Two-dimensional faces Three-dimensional faces

A. For classical surfaces of general type (Campedelli, Godeaux, Burniat, Dolgachev surfaces, quotients of products of higher-genus curves and fake P2 s) there should exist quasi-phantoms in their SOD, i.e., nontrivial categories with trivial Hochschild homology. These surfaces are clearly not rational as they have nontrivial fundamental groups, but also since, conjecturally, they have quasi-phantom subcategories in their SOD. On the Landau– Ginzburg side, these quasi-phantoms are described by the moving scheme. The deformation of the Landau–Ginzburg models is determined by the moving scheme so the quasi-phantoms factor in the geometry of the Landau– Ginzburg models. On the mirror side, this translates to the fact that the Ext groups between the quasi-phantom and the rest of the SOD determine the moduli space. B. For surfaces of general type with trivial fundamental groups (e.g., the Barlow surface, see [28]), we conjecture the existence of a nontrivial phantom category, a category with a trivial K 0 group. The deformation of the Landau– Ginzburg models is determined by the moving scheme, so the phantom factors in the geometry of the Landau–Ginzburg models. Similarly, on the mirror side this translates to the fact that the Ext groups between the phantom and the rest of the SOD determine the moduli space. C. We relate the existence of such phantom categories with nonrationality questions. For surfaces, it is clear that phantoms lead to nonrationality. For threefolds, we exhibit examples (of Sarkisov type) of nonrational threefolds where phantoms conjecturally imply nonrationality. We introduce “higher” categorical invariants detecting nonrationality: gaps of spectra. Conjecturally, these are not present in the Sarkisov examples. These ideas are a natural continuation of [50]. 5. We introduce conjectural invariants associated to our moduli spaces—gaps and local differentials. We suggest that these numbers (changed drastically via “wallcrossing”) produce strong birational invariants. We relate these invariants to the “Hassett–Kuznetsov–Tschinkel” program (see [46,77])—a program for studying rationality of the four-dimensional cubic and its “relatives.” This paper is organized as follows. In Sects. 2 and 3 we relate degenerations of Fano manifolds via projections. Using mirror symmetry in Sect. 3 we transfer

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these connections to the side of Landau–Ginzburg model. Based on that, in Sect. 4 we suggest a generalization of Kawamata’s categorical approach to birational geometry, enhancing it via the geometry of moduli spaces of Landau–Ginzburg models. We give several applications, most notably a conjectural application to the Hassett–Kuznetsov–Tschinkel program. Our approach is based on two categorical nonrationality invariants—phantoms and gaps. Full details will appear in a future paper. Notations. Smooth del Pezzo threefolds (smooth Fano threefolds of index 2) are denoted by Vn , where n is its degree with respect to a Picard group generator, except for the quadric denoted by Q. Fano threefolds of Picard rank 1, index 1, and degree n are denoted by Xn . The remaining Fano threefolds are denoted by Xk.m , where k is a Picard rank of a variety and m is its number according to [57]. The Laurent polynomials from the k-th line of Table 2 are denoted by fk . A toric variety whose fan polytope is a Newton polytope of fk is denoted by Fk or just a variety number k. The Landau–Ginzburg model for a variety X is denoted by LG(X).

2 “Classical” Birational Geometry In this section we recall some facts from classical birational geometry of threedimensional Fano varieties. Our presentation of this geometry is adopted to HMS.

2.1 The Importance of Being Gorenstein Among singular Fano varieties, those with canonical Gorenstein singularities are of special importance. They arise in many different geometrical problems: degeneration of smooth Fano varieties with a special regard to mirror symmetry (see [6,7]), classification of reflexive polytopes (see [74,75]), midpoints of Sarkisov links and bad Sarkisov links (see [22,24]), compactification of certain moduli spaces (see [81]), etc. Historically, Fano varieties with canonical Gorenstein singularities are the original Fano varieties. Indeed, the name Fano varieties originated in the works of Iskovskikh (see [54, 55]) that filled the gaps in old results by Fano who studied in [36, 37] anticanonically embedded Fano threefolds with canonical Gorenstein singularities without naming them so (cf. [15]). In dimension two, canonical singularities are always Gorenstein, so being Gorenstein is a vacuous condition. Surprisingly, the classification of del Pezzo surfaces with canonical singularities is simpler than the classification of smooth del Pezzo surfaces (see [26, 84]). Fano threefolds with canonical Gorenstein singularities are not yet classified, but first steps in this directions are already have been made by Mukai, Jahnke, Radloff, Cheltsov, Shramov, Przyjalkowski, Prokhorov, and Karzhemanov (see [17, 60, 65, 66, 81, 85]).


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2.2 Birational Maps Between Fano Varieties and their Classification V. Iskovskikh used birational maps between Fano threefolds to classify them. Indeed, he discovered smooth Fano varieties of degree 22 and Picard group Z by constructing the following commutative diagram: ρ

U

W

α

(1) β

ψ

V5

X22 ,

where V5 is a smooth section of the Grassmannian Gr(2, 5) ⊂ P9 by a linear subspace of codimension 3 (they are all isomorphic), X22 is a smooth Fano threefold of index 1 and degree 22 mentioned above, i.e., Pic(X22 ) = Z[−KX22 ] and −KX322 = 22, and α is a blowup of a quintic rational normal curve C, ρ is a flop of the proper transforms of the secant lines to C, β contracts a surface to a curve L ⊂ X22 with −KX22 · L = 1, and ψ is a double projection from the curve L (see [54, 55]). While very powerful, this approach does not always work (see Example 2.1). Iskovskikh gave many other examples of birational maps between smooth Fano threefolds (see [57]); similar examples were found by Takeuchi [94]. Recently, P. Jahnke, I. Radloff, and I. Karzhemanov produced many new examples of Fano threefolds with canonical Gorenstein singularities by using elementary birational transformation between them.

2.3 Birational Maps Between Fano Varieties and Sarkisov Program Results of V. Iskovskikh, Yu. Manin, V. Shokurov, and K. Takeuchi were used by V. Sarkisov and A. Corti to create what is now known as the three-dimensional Sarkisov program (see [22]). In particular, this program decomposes birational maps between terminal Q-factorial Fano threefolds with Picard group Z into a sequence of so-called elementary links (often called Sarkisov links). Recently, the threedimensional Sarkisov program has been generalized to higher dimensions by Hacon and McKernan (see [44]). Unfortunately, the Sarkisov program is not applicable to Fano varieties with non-Q-factorial singularities, it is not applicable to Fano varieties with nonterminal singularities, and it is not applicable to Fano varieties whose Picard group is not Z. Moreover, in a dimension bigger than two, the Sarkisov program is not explicit, except in the toric case. In dimension three, the description of Sarkisov links is

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closely related to the classification of terminal non-Q-factorial Fano threefolds whose class group is Z2 . In general this problem is very far from being solved. But in the Gorenstein case we know a lot (see [11, 25, 58, 59, 63]).

2.4 Basic Links Between del Pezzo Surfaces with Canonical Singularities The anticanonical linear system | − KP2 | gives an embedding P2 → P9 . Its image is a surface of degree 9, which we denote by S9 . Let π : S9 S8 be a birational map induced by the linear projection P9 P8 from a point in S9 (the center of the projection), where S8 is surface of degree 8 in P8 obtained as the image of S9 under this projection. For simplicity, we say that π9 is a projection of the surface S9 from a point. We get a commutative diagram S˜9 β9

α9

S9

S8 ,

π9

where α9 is a blowup of a smooth point of the surface S9 and β9 is a birational morphism that is induced by | − KS˜9 |. Note that S8 is a del Pezzo surface and (−KS8 )2 = 8. Iterating this process and taking smooth points of the obtained surfaces Si as centers of projections, we get the following sequence of projections: P2 = S 9

π9

π8

π6

π7

S8

S7

S6

π5

S5

π4

S4

S3 , (2)

where every Si is a del Pezzo surface with canonical singularities, e.g., S3 is a cubic surface in P3 with isolated singularities that is not a cone. Note that we have to stop our iteration at i = 3, since the projection of S3 from its smooth point gives a rational map of degree 2. For every constructed projection πi : Si Si−1 , we get a commutative diagram S˜i αi

Si

(3) βi

πi

Si−1 ,


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where αi is a blowup of a smooth point of the surface Si and βi is a birational morphism that is induced by | − KS˜i |. We say that diagram (3) is a basic link between del Pezzo surfaces. Instead of P2 , we can use an irreducible quadric as a root of our sequence of projections. In this way, we obtain all del Pezzo surfaces with canonical singularities except for P2 , quadric cone and quartic hypersurfaces in P(1, 1, 1, 2) and sextic hypersurfaces in P(1, 1, 2, 3). Note that S3 is not an intersection of quadrics (trigonal case), anticanonical linear system of every quartic hypersurface in P(1, 1, 1, 2), with canonical singularities is a morphism that is not an embedding (hyperelliptic case), and anticanonical linear system of every sextic hypersurface in P(1, 1, 2, 3) has a unique base point. Let us fix an action of a torus (C∗ )2 on P2 . So, if instead of taking smooth points as projection centers, we take toric smooth points (fixing the torus action), then the constructed sequence of projections (2) and the commutative diagram (3) are going to be toric as well. In this case we say that diagram (3) is a toric basic link between toric del Pezzo surfaces. Recall that there are exactly 16 toric del Pezzo surfaces with canonical singularities. In fact, we can explicitly describe all possible toric projections of toric del Pezzo surfaces from their smooth toric points (this is purely combinatorial problem), which also gives the complete description of all toric basic link between toric del Pezzo surfaces. The easiest way of doing this is to use reflexive lattice polytopes that correspond del Pezzo surfaces with canonical singularities.1 The answer is given by Fig. 1.

2.5 Basic Links Between Gorenstein Fano Threefolds with Canonical Singularities Similar to the two-dimensional case, it is tempting to fix few very explicit basic links between Fano threefolds with canonical Gorenstein singularities (explicit here means that these basic links should have a geometric description like projections from points or curves of small degrees) and describe all such threefolds using these links. However, this is impossible in general due to the following: Example 2.1 (Iskovskikh–Manin). Any Fano threefold with canonical singularities that is birational to a smooth quartic threefold is itself a smooth quartic threefold (see [16, 56]). However, if we are only interested in classification up to a deformation, then we can try to fix few very explicit basic links between Fano threefolds with canonical Gorenstein singularities and describe all deformation types of such threefolds using

1 Recall

that toric n-dimensional Fano varieties with canonical Gorenstein singularities up to isomorphism are in one-two-one correspondence with reflexive n-dimensional lattice polytopes in Rn up to SLn (Z) action.

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Fig. 1 del Pezzo tree

these links. Moreover, it seems reasonable to expect that this approach allows us to obtain all smooth Fano threefolds in a unified way. We can define three-dimensional basic links similar to the two-dimensional case. Namely, let X be a Fano threefold with canonical Gorenstein singularities. Put g = KX3 /2 + 1. Then, g is a positive integer and h0 (OX (−KX )) = g + 1. Let ϕ|−KX | : X → Pg+1 be a map given by | − KX |. Then, / and all such X are found in [60]. 1. Either Bs| − KX | = 0, 2. Or ϕ|−KX | is not a morphism, the threefold X is called hyperelliptic, and all such X are found in [17]. 3. Or ϕ|−KX | is a morphism and ϕ|−KX | (X) is not an intersection of quadrics, the threefold X is called trigonal, and all such X are found in [17].


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4. Or ϕ|−KX | (X) is an intersection of quadrics. Thus, we always can assume that ϕ|−KX | is an embedding and ϕ|−KX | (X) is an intersection of quadrics. Let us identify X with its anticanonical image ϕ|−KX | (X). Let Z be either a smooth point of the threefold X, terminal cDV point (see [88]) of the threefold X, line in X ⊂ Pg+1 that does not pass through a non-cDV point, or a smooth irreducible conic in X ⊂ Pg+1 that does not pass through a non-cDV point. Let α : X˜ → X be a blowup of the ideal sheaf of the subvariety Z ⊂ X. Lemma 2.2. Suppose that Z is either a cDV point or a line. Then | − KX˜ | is free from base points. Proof. This follows from an assumption that ϕ|−KX | (X) is an embedding.

If Z is a smooth point, let β : X → X be a morphism given by | − KX˜ |. Lemma 2.3. Suppose that Z is either a cDV point or a line. Then the morphism β is birational and X is a Fano variety with canonical Gorenstein singularities such that −KX is very ample. Proof. The required assertion follows from the fact that X is an intersection of quadrics. If Z is a conic, then we need to impose a few additional assumptions on X and Z (cf. [94, Theorem 1.8]) to be sure that the morphism β is birational, and X is a Fano variety with canonical Gorenstein singularities such that −KX is very ample. In the toric case, these conditions can be easily verified. Let π : X X be a projection from Z. If Z is not a smooth point, then the diagram X˜

X

(4) β

α

π

X

commutes. Unfortunately, if Z is a smooth point, then diagram (4) does not commute. In this case, we should define the basic link between Fano threefolds in a slightly different way. Namely, if Z is a smooth point, we still can consider the commutative diagram (4), but we have to assume that π is a projection from the projective tangent space to X at the point Z (instead of projection from Z like in other cases). Moreover, if Z is a smooth point, similar to the case when Z is a conic, we must impose a few additional assumptions on X and Z to be sure that the morphism β is birational, and X is a Fano variety with canonical Gorenstein singularities such that −KX is very ample. These conditions can be easily verified in many cases—in the toric case or in the case of index bigger than 1, see Remark 2.4. We call diagram (4) a basic link between Fano threefolds of type: • IIp if Z is a smooth point

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• IIdp (or IIo or IIcDV , respectively) if Z is a double point (ordinary double point or non-ordinary double point, respectively) • IIl if Z is a line • IIc if Z is a conic Moreover, in all possible cases, we are going to call Z a center of the basic link (4) or projection center (of π ). Remark 2.4. Suppose that Z is a smooth point, and −KX ∼ 2H for some ample Cartier divisor H. Put d = H 3 . Then the linear system |H| induces a rational map ϕ|H| : X Pd+1 (this follows from the Riemann–Roch Theorem and basic vanishing theorems). If ϕ|H| is not an embedding, i.e., H is not very ample, then X can be easily described exactly as in the smooth case (see [57]). Namely, one can show that X is either a hypersurface of degree 6 in P(1, 1, 1, 2, 3) or a hypersurface of degree 4 in P(1, 1, 1, 1, 2). Similarly, if H is very ample and ϕ|H| (X) is not an intersection of quadrics in Pd+1 , then X is just a cubic hypersurface in P4 . Assuming that ϕ|H| (X) is an intersection of quadrics in Pd+1 (this is equivalent to (−KX )3 > 24) and identifying X with its image ϕ|H| (X) in Pd+1 , we see that there exists a commutative diagram X˜

X

(5) β

α

π

X ,

where π : X X is a projection of the threefold X ⊂ Pd+1 from the point Z. Then X is a Fano threefold with canonical Gorenstein singularities whose Fano index is divisible by 2 as well. Similar to the two-dimensional case, we can take P3 or an irreducible quadric in P3 and start applying basic links iteratively. Hypothetically, this should give us all (or almost all) deformation types of Fano threefolds with canonical Gorenstein singularities whose anticanonical degree is at most 64 (the anticanonical degree decreases after the basic link).

2.6 Toric Basic Links Between Toric Fano Threefolds with Canonical Gorenstein Singularities Let X be a toric Fano threefold with canonical Gorenstein singularities. Fix the action of the torus (C∗ )3 on X. Suppose that −KX is very ample and X is not trigonal. Then we can identify X with its anticanonical image in Pg+1 , where


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g = (−KX )3 /2 + 1 (usually called the genus of the Fano threefold X). If Z is not a smooth point of the threefold X, then the commutative diagram (4) is torus invariant as well, and we call the basic link 4 a toric basic link. This gives us three types of toric basic links: IIdp if Z is a double point (IIo if Z is an ordinary double point and IIcDV if Z is non-ordinary double point), IIl if Z is a line, and IIc if Z is a conic. When Z is a smooth torus invariant point we proceed as in Remark 2.4 and obtain the toric basic link of type IIp , assuming that the Fano index of the threefold X is divisible by 2 or 3 and (−KX )3 > 24. We can take X = P3 and start applying toric basic links until we get a toric Fano threefold with canonical Gorenstein singularities to which we can not apply any toric basic link (e.g., when we get a toric quartic hypersurface in P4 ). Hypothetically, this would give us birational maps between almost all toric Fano threefolds with canonical Gorenstein singularities whose anticanonical degree is at most 64. Similarly, we can take into account irreducible quadrics in P4 to make our picture look more complicated and, perhaps, refined. Moreover, we can start with X = P(1, 1, 1, 3) or X = P(1, 1, 4, 6), which are the highest anticanonical degree Fano threefolds with canonical Gorenstein singularities (see [85]) to get possibly all toric Fano threefolds with canonical Gorenstein singularities. Keeping in mind that there are 4319 such toric Fano threefolds, we see that this problem requires some computational effort and use of databases of toric Fano threefolds (see [14]). Let us restrict our attention to toric Fano threefolds with canonical Gorenstein singularities that are known to be smoothable to smooth Fano threefolds with Picard group Z. Starting with P3 and with a singular quadric in P4 with one ordinary double point and taking into account some toric basic links, we obtain Fig. 2, where we use bold fonts to denote Fano threefolds with Picard group Z. (See the text preceding Table 2 for more explanation of the notation.) Recent progress in mirror symmetry for smooth Fano threefolds (see [21, 23, 42, 43,86,87]) shed new light on and attracted a lot of attention to toric degenerations of smooth Fano threefolds (see [6–8, 19, 38, 52]). It would be interesting to understand the relation between toric basic links between smoothable toric Fano threefolds with canonical Gorenstein singularities, basic links between smoothable Fano threefolds with canonical Gorenstein singularities, their toric degenerations, and the geometry of their Landau–Ginzburg models (cf. [86]). Proposition 2.5 ([53, Theorem 2.8]). Consider a Laurent polynomial p1 = xg1 g2 + g3 + g4 /x, where gi are Laurent polynomials that do not depend on x. Let p2 = xg1 + g3 + g2g4 /x. Let Ti be a toric variety whose fan polytope is a Newton polytope of pi . Then T2 deforms to T1 . Remark 2.6. In [61, Example 2.3], Jahnke and Radloff considered an anticanonical cone over the del Pezzo surface S6 (the rightmost on the fourth line of Fig. 1) of degree 6 and showed that it has two smoothings, to X2.32 and X3.27 . Notice that S6 has 4 canonical toric degenerations; all of them are projections from P2 and two of them, S6 and S6 (the third on the fourth line of Fig. 1), are projections of a smooth quadric surface. Cones over these varieties have fan polytope numbers 155

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P3

X2.30

II p

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V2.35

X3.23 II odp

II p

V2.32

II c

X3.24

II odp

X4.9

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X22

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X6

II cDV

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II cDV

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II odp

X12

II odp

X14

II odp

X16

II p

V3 II p

V2

Fig. 2 Fano snake

and 121 (according to [21]) correspondingly. Two of these polytopes are exactly the ones having two Minkowski decompositions, each of which gives constant terms series for X2.32 and X3.27 . So we have two smoothings corresponding to two pairs of Minkowski decompositions. The question is why the existence of two deformations to two different varieties corresponds to the fact that the toric varieties are projections from a quadric surface. Example 2.7. Consider a Laurent polynomial p1 = xy + xz + xyz + x/y + x/z + x + 1/x. One can prove that it is a toric Landau–Ginzburg model for X2.35 . Indeed, one can directly check the period and Calabi–Yau conditions. To prove the toric condition one can observe that p1 = x(z + z/y + 1)(y + 1/z) + 1/x. So, by Proposition 2.5, the toric variety Tp1 associated with p1 can be deformed to the toric variety associated with p2 = x(z + z/y + 1) + (y + 1/z)/x,


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which after toric change of variables coincides with f2 . We get F2 which can be smoothed to X2.35 by Theorem 3.12. A variety Tp1 is nothing but a cone over a toric del Pezzo surface S7 (the rightmost in the third column of Fig. 1). Consider a basic link—projection from a smooth point on Tp1 . One gets a toric variety—a cone over the del Pezzo surface S6 . It has two smoothings (see Remark 2.6). Moreover, there are two toric Landau–Ginzburg models, p3 = xy + xz + xyz + x/y + x/z + x/y/z + 2x + 1/x and p4 = xy + xz + xyz + x/y + x/z + x/y/z + 3x + 1/x, one for X2.32 and another one for X3.27 . Indeed, as before the period and Calabi–Yau conditions can be checked directly. Notice that p3 = x ((yz + 1)/z/y)(y + 1)(z + 1) + 1/x and p4 = x ((yz + z + 1)/y/z)(yz + y + 1) + 1/x. After a cluster change of variables by Proposition 2.5 one gets two polynomials associated with F3 and F4 . These varieties by Theorem 3.12 can be smoothed to X2.32 and X3.27 . Nevertheless, the correct target for the basic link between the last two varieties is X2.32 , since projecting a general X2.32 from a point, we always obtain X3.27 . Remark 2.8. The same can be done with another variety, the cone over S6 (the third in the fourth column of Fig. 1). Indeed, by Proposition 2.5 a variety S6 is a degeneration of S6 so cones over them degenerate as well.

3 Classical Theory of Landau–Ginzburg Models From now on we concentrate on the theory of Landau–Ginzburg models and their moduli. First we recall the classical definition of the Landau–Ginzburg model of a single Fano variety (see, e.g., [86] and the references therein). Let X be a smooth Fano variety of dimension n and QH ∗ (X) = H ∗ (X, Q) ⊗Z Λ its quantum cohomology ring, where Λ is the Novikov ring for X. The multiplication in QH ∗ (X), i.e., the quantum multiplication, is given by (genus zero) Gromov–Witten invariants—numbers counting rational curves on X. Let QX be the associated regularized quantum differential operator (the second Dubrovin connection)—the regularization of an operator associated with the connection in the trivial vector bundle given by quantum multiplication by the canonical class KX . Solutions of an equation given by this operator are given by I-series for X—generating series for its one-pointed Gromov–Witten invariants. In particular, one “distinguished” solution

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is a constant term (with respect to cohomology) of I-series. Let us denote it by I = 1 + a1t + a2t 2 + . . .. Definition 3.1. A toric Landau–Ginzburg model is a Laurent polynomial f ∈ ±1 C[x±1 1 , . . . , xn ] such that: ±1 Period condition: The constant term of f i ∈ C[x±1 1 , . . . , xn ] is ai for any i (this ∗ n means that I is a period of a family f : (C ) → C; see [86]). Calabi–Yau condition: There exists a fiberwise compactification (the Calabi–Yau compactification) whose total space is a smooth (open) Calabi–Yau variety. Toric condition: There is an embedded degeneration X T to a toric variety T whose fan polytope (the convex hull of generators of its rays) coincides with the Newton polytope (the convex hull of nonzero coefficients) of f .

Remark 3.2. This notion can be extended to some non-smooth cases; see, for instance, [23]. Theorem 3.3 ([87, Theorem 18] and [52, Theorem 3.1]). Smooth Fano threefolds of Picard rank 1 have toric Landau–Ginzburg models. Remark 3.4. Toric Landau–Ginzburg models for Picard rank 1 Fano threefolds are found in [87]. However they are not unique. Some of them coincide with ones from Table 2. Anyway Theorem 3.3 holds for all threefolds from Table 2; see Theorem 3.12. Theorem 3.5 ([34]). Let X be a Fano threefold of index i and (−KX )3 = i3 k. Then fibers of toric weak Landau–Ginzburg model from [87] can be compactified to Shioda–Inose surfaces with Picard lattice H ⊕ E8 (−1) ⊕ E8(−1) + −ik. Remark 3.6. This theorem holds for toric Landau–Ginzburg models for Fano threefolds of Picard rank 1 from Table 2. This theorem means that fibers of compactified toric Landau–Ginzburg models are mirrors of anticanonical sections of corresponding Fano varieties, and this property determines compactified toric Landau–Ginzburg models uniquely as the moduli spaces of possible mirror K3’s are just P1 ’s. The discussion above can be summarized to the following mirror symmetry conjecture. Conjecture 3.7. Every smooth Fano variety has a toric Landau–Ginzburg model. Theorem 3.3 shows that this conjecture holds for Fano threefolds of Picard rank 1. Theorem 3.12 shows that the conjecture holds for Fano varieties from Table 2.

3.1 The Table Now we study toric Landau–Ginzburg models for Fano threefolds of Picard rank 1 given by toric basic links from P3 and the quadric. We give a table of such toric


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Landau–Ginzburg models and prove in Theorem 3.12 that Laurent polynomials listed in the table are toric Landau–Ginzburg models of Fano threefolds. Table 2 is organized as follows. N is the number of a variety in the table. “Var.” is a Fano smoothing numerated following [57]. “Deg.” is a degree of a variety. “Par.” is the number of varieties giving our variety by a projection. “BL” is a type of toric basic link(s). “Desc.” stands for descendants—varieties that can be obtained by projection from given variety. The last column is a toric Landau–Ginzburg model for the variety. Remark 3.8. F2 is a blowup of F1 = P3 at one point with an exceptional divisor E. F3 = X2.35 is a projection from a point lying far from E. If we project from a point lying on E we get another (singular) variety, F3 , with corresponding weak Landau– Ginzburg model 2 yz 1 + + . x+y+z+ xyz x x Remark 3.9. Variety 24, the toric quartic, has no cDV points or smooth toric lines. So we cannot proceed to make basic links. However, it has 4 singular canonical (triple) points and we can project from any of them. In other words, we can project the quartic {x1 x2 x3 x4 = x40 } ⊂ P[x0 , x1 , x2 , x3 , x4 ] from the point, say, (0 : 0 : 0 : 0 : 1). Obviously, we get variety 1, that is, P3 again. Proposition 3.10. Families of hypersurfaces in (C∗ )3 given by Laurent polynomials from Table 2 can be fiberwise compactified to (open) Calabi–Yau varieties. Proof. Let f be a Laurent polynomial from the table. Compactify the corresponding family { f = λ } ⊂ Spec C[x±1 , y±1 , z±1 ] × SpecC[λ ] fiberwise using the standard embedding Spec C[x±1 , y±1 , z±1 ] ⊂ Proj C[x, y, z,t]. In other words, multiply it by a denominator (xyz) and add an extra homogenous variable t. For varieties 11, 12, 13 do toric change of variables xy → x, yz → z. We get a family of singular quartics. Thus it has trivial canonical class. The threefold singularities we get are du Val along lines and ordinary double points; the same type of singularities arises after crepant blow-ups of singular lines and small resolutions of ordinary double points. Thus the threefold admits a crepant resolution; this resolution is the Calabi–Yau compactification we need. Proposition 3.11. Toric varieties from Table 2 are degenerations of corresponding Fano varieties. Proof. Varieties 1–5, 9–15, 25–27 are terminal Gorenstein toric Fano threefolds. So, by [82] they can be smoothed. By [38, Corollary 3.27], the smoothings are Fano’s with the same numerical invariants as the initial toric varieties. The only smooth Fano threefolds with given invariants are listed at the second column. In other words, the statement of the proposition for varieties 1–5, 9–15, 26–28 follows from the proof of [38, Theorem 2.7].

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Table 2 Weak Landau–Ginzburg models for Fano threefolds N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Var. 1.17 2.35 2.32 3.27 1.15 1.14 1.13 1.12 3.24 4.9 4.6 3.12 3.10 4.1 1.10

Deg. 23 · 8 23 · 7 23 · 6 23 · 6 23 · 5 23 · 4 23 · 3 23 · 2 42 40 34 28 26 24 22

Par. 0/ 1 2 0/ 3, 4 5 6 7 3 9, 27 10 11 12 13 14

BL 0/ IIp IIp 0/ IIp , IIp IIp IIp IIp IIc IIo , IIo IIc IIc IIo IIo 2 IIo

Desc. 2 3 5, 9 5 6 7 8 0/ 10 11 12 13 14 15 16

16

2.13

20

15

IIo

17

17

1.9

18

16

IIo

18

18

1.8

16

17

IIcDV

19

Weak LG model 1 x + y + z + xyz 1 x + y + z + xyz + 1x 1 x + y + z + xyz + 1x + 1y x + y + z + 1x + 1y + 1z 1 x + y + z + xyz + 1x + 1y + 1z 1 x + y + z + xyz + 2x + 1y + 1z + yzx 1 x + y + 2z + xyz + 2x + 2y + 1z + yzx + xzy 1 2x + 2y + 2z + xyz + 2x + 2y + 2z + yzx + xzy + xyz 1 1 x + y + z + xyz + x + 1y + yz1 1 1 x + y + z + xyz + 1x + 1y + yz1 + xy 1 1 1 1 1 x + y + z + xyz + x + y + yz + xy + yz 1 1 x + y + z + xyz + 1x + 1y + yz1 + xy + yz + xy 1 1 x + y + z + xyz + 1x + 1y + yz1 + xy + yz + xy + 1z x + y + z + 1x + 1y + 1z + xy + xz + xy + yz + xz + yz x + y + z + 1x + 1y + 1z + xy + xz + yx + yz + xz + z x y + yz x + y + z + 1x + 1y + 2z + xy + xz + yx + yz + xz + y z x y + yz + xz 2 x + y + z + x + 2y + 2z + xy + xz + yx + yz + xz + y z x z y + yz + xz + xy 3 3 x + y + z + x + y + 3z + xy + xz + yx + yz + xz +

20

1 2 + yzx + xzy + xyz + xyz + xy + yz2 + xz2 3 3 4 x 2x x + y + z + x + y + z + y + z + yx + 2yz + xz +

21

1 2 + yzx + xzy + xyz + xyz + xy + yz2 + xz2 + xyz 3 4 4 2x 2x + y + z + x + y + z + y + 2xz + yx + 2yz +

z y

19

1.7

14

18

IIo

z y

20

1.6

12

19

IIo

z 2z x x + y + yz xy xz z + y

21

1.5

10

20

IIo

22

2y 2y 2x 2x + 2y+ 2z+ 4x + 4y + 4z + 2x y + z + x + z + y 2z x z 1 2 2 y + yz + xz + xy + xyz + xy + yz + xy yz + z + xzy + x 2y 2y 2x 4x + 4y+ 4z+ 4x + 4y + 4z + 2x y + z + x + z + y 2z 2z x z 1 2 2 x + y + yz + xz + xy + xyz + xy + yz + xy yz 2 xz xz + z + y + x +xyz + 2xy + 2xz + 2yz (y+z+1)2 ((y+z+1)2 +2x(y+z+1)+x2 ) − 12 xyz (x+y+z+1)4 xyz 1 x + y + z + xy + yz1 1 1 x + y + z + xy + yz1 + xyz 1 1 1 x + y + z + xy + yz + xyz + 1x 2z x 2 xz

22

1.4

8

21

IIcDV

23

23

1.3

6

22

IIcDV 3

24

24 25 26 27

1.2 1.16 2.30 3.23

4 54 46 42

23 0/ 27 28

IIcDV 0/ IIp IIl

0/ 26 27 10

1 2 + xzy + xyz + xyz + xy + yz2 + xz2 +

+


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Varieties 6–8 are complete intersections in (weighted) projective spaces. We can write down the dual polytopes to their fan polytopes. The equations of the toric varieties correspond to homogenous relations on integral points of the dual polytopes. One can see that the relations are binomials defining corresponding complete intersections and the equations of a Veronese map v2 . So the toric varieties can be smoothed to the corresponding complete intersections. For more details see [52, Theorem 2.2]. Variety X2.13 can be described as a section of P2 × P4 by divisors of type (1, 1), (1, 1), and (0, 2) (see, say, [21]). Equations of P2 × P4 in Segre embedding can be described as all (2 × 2)-minors of a matrix ⎛

⎞ x00 x01 x02 x03 x04 ⎝ x10 x11 x12 x13 x14 ⎠ . x20 x21 x22 x23 x24 Consider its section T given by equations x00 = x11 , x11 = x22 , x01 x02 = x03 x04 . These equations give divisors of types (1, 1), (1, 1), and (0, 2), respectively. They are binomial, which means that T is a toric variety. The equations giving variety 16 are homogenous integral relations on integral points of a polytope dual to a Newton polytope of f16 . It is easy to see that these relations are exactly ones defining T . Thus T = F16 and F16 can be smoothed to X2.13 . Varieties 17 and 18 correspond to ones from [87]. Thus, by [52, Theorem 3.1] they can be smoothed to corresponding Fano threefolds. The dual polytope to the fan polytope for variety 19 is drawn on Fig. 3. It obviously has a triangulation on 14 triangles satisfying conditions of [18, Corollary 3.4]. By this corollary variety 19 can be smoothed to the variety we need.

Fig. 3 Polytope defining variety 19

Finally the existence of smoothings of varieties 20–24 to corresponding Fano varieties follows from [19].

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Theorem 3.12. Laurent polynomials from Table 2 are toric Landau–Ginzburg models for corresponding Fano threefolds. Proof. The period condition follows from direct computations; see [21]. The Calabi–Yau condition holds by Proposition 3.10. The toric condition holds by Proposition 3.11.

4 Landau–Ginzburg Considerations 4.1 Categorical Background Examples in the previous sections suggest a new approach to birational geometry of Fano manifolds. In this section we summarize this approach and develop the relevant technical tools. We proceed by extending Kawamata’s approach described in Table 1 by adding further data to the categorical approach recorded in the geometry of the moduli space of Landau–Ginzburg models. The main points are: 1. There exists a moduli space of Landau–Ginzburg models for many (possibly all) three-dimensional Fano manifolds. 2. The topology of this compactified moduli space of Landau–Ginzburg models determines Sarkisov links among these Fano manifolds. We conjecture that the geometry of the moduli space of Landau–Ginzburg models gives answers to many questions related to rationality and birational equivalence—we suggest some invariants and give examples. The geometry of the moduli space of Landau–Ginzburg models was introduced in [27,28,68] as an analogy with nonabelian Hodge theory. We describe this analogy. We build the “twistor” family so that the fiber over zero is the “moduli space” of Landau–Ginzburg models and the generic fiber is the Stability Hodge Structure (SHS) (see below). Noncommutative Hodge theory endows the cohomology groups of a dg-category with additional linear data—the noncommutative Hodge structure—which records important information about the geometry of the category. However, due to their linear nature, noncommutative Hodge structures are not sophisticated enough to codify the full geometric information hidden in a dg-category. In view of the homological complexity of such categories it is clear that only a subtler nonlinear Hodge theoretic entity can adequately capture the salient features of such categorical or noncommutative geometries. In this section by analogy with “classical nonabelian Hodge theory” we construct and study from such prospective a new type of entity of exactly such type—the SHS associated with a dg-category. As the name suggests, the SHS of a category is related to the Bridgeland stabilities on this category. The moduli space StabC of stability conditions of a triangulated dg-category C is, in general, a complicated curved space, possibly with fractal boundary. In the special case when C is the Fukaya category of a Calabi– Yau threefold, the space StabC admits a natural one-parameter specialization to a


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much simpler space S0 . Indeed, HMS predicts that the moduli space of complex structures on the mirror Calabi–Yau threefold maps to a Lagrangian subvariety geom StabC ⊂ StabC . The space S0 is the fiber at 0 of this completed family and conjecturally S → C is one chart of a twistor-like family S → P1 which is by definition the Stability Hodge Structure associated with C. SHSs are expected to exist for more general dg-categories, in particular for Fukaya–Seidel categories associated with a superpotential on a Calabi–Yau space or with categories of representations of quivers. Moreover, for special non-compact Calabi–Yau threefolds, the zero fiber S0 of a SHS can be identified with the Dolbeault realization of a nonabelian Hodge structure of an algebraic curve. This is an unexpected and direct connection with Simpson’s nonabelian Hodge theory (see [92]) which we exploit further, suggesting some geometric applications. We briefly recall nonabelian Hodge theory settings. According to Simpson (see [92]), we have a one-parametric twistor family such that the fiber over zero is the moduli space of Higgs bundles and the generic fiber is the moduli space— MBetti —of representations of the fundamental group of the space over which the Higgs bundle lives. By analogy with the nonabelian Hodge structure we have: Conjecture 4.1 ([68]). The moduli space of stability conditions of Fukaya–Seidel category can be included in one-parametric “twistor” family. In other words SHS exists for Fukaya–Seidel categories. Parts of this conjecture are checked in [45, 68]. We give a brief example of SHS. Example 4.2. We will give a brief explanation of the calculation of the “twistor” family for the SHS for the category An recorded in the picture above. We start with the moduli space of stability conditions for the category An , which can be identified with differentials e p(z) dz, where p(z) is a polynomial of degree n + 1. Classical work of Nevanlinna identifies these integrals with graphs (see Fig. 4)— graphs connecting the singularities of the function given by an integral against the exponential differentials. Now we consider the limit e p(z)/u dz. Geometrically, the limit differential can be identified with polynomials, e.g., with Landau–Ginzburg models (see Fig. 4)—for more see [45].

4.2 The Fiber Over Zero The fiber over zero (described below) plays an analogous role to the moduli space of Higgs Bundles in Simpson’s twistor family in the theory of nonabelian Hodge

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∞

∞

∞ ∞

∞

∞ ∞

u →0 p(z)

∞

∞

∞

Fig. 4 Taking limit

structures. As alluded to earlier, an important class of examples of categories and their stability conditions arises from HMS—Fukaya–Seidel categories. Indeed, such categories are the origin of the modern definition of such stability conditions. The prescription given by Batyrev–Borisov and Hori–Vafa in [9, 47] to obtain homological mirrors for toric Fano varieties is perfectly explicit and provides a reasonably large set of examples to examine. We recall that if Σ is a fan in Rn for a toric Fano variety XΣ , then the homological mirror to the B model of XΣ is a Landau–Ginzburg model w : (C∗ )n → R, where the Newton polytope Q of w is the fan polytope of Σ . In fact, we may consider the domain (C∗ )n to occur as the dense orbit of a toric variety XA where A is Q ∩ Zn and XA indicates the polytope toric construction. In this setting, the function w occurs as a pencil Vw ⊂ H 0 (XA , LA ) with fiber at infinity equal to the toric boundary of XA . A similar construction works for generic nontoric Fano’s. In this paper we work with the directed Fukaya category associated to the superpotential w—Fukaya–Seidel categories. To build on the discussion above, we discuss here these two categories in the context of stability conditions. The fiber over zero corresponds to the moduli of complex structures. If XA is toric, the space of complex structures on it is trivial, so the complex moduli appearing here are a result of the choice of fiber H ⊂ XA and the choice of pencil w, respectively. The appropriate stack parameterizing the choice of fiber contains the quotient [U/(C∗ )n ] as an open dense subset where U is the open subset of H 0 (XA , LA ) consisting of those sections whose hypersurfaces are nondegenerate (i.e., smooth and transversely intersecting the toric boundary) and (C∗ )n acts by its action on XA . To produce a reasonably well-behaved compactification of this stack, we borrow from the work of Alexeev (see [2]), Gelfand–Kapranov–Zelevinsky (see [41]), and Lafforgue (see [78]) to construct the stack XSec(A) with universal


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hypersurface stack XLa f (A) . We quote the following theorem which describes much of the qualitative behavior of these stacks: Theorem 4.3 ([29]). (i) The stack XSec(A) is a toric stack with moment polytope equal to the secondary polytope Sec(A) of A. (ii) The stack XLa f (A) is a toric stack with moment polytope equal to the Minkowski sum Sec(A) + ΔA where ΔA is the standard simplex in RA . (iii) Given any toric degeneration F : Y → C of the pair (XA , H), there exists a unique map f : C → XSec(A) such that F is the pullback of XLa f (A) . We note that in the theorem above, the stacks XLa f (A) and XSec(A) carry additional equivariant line bundles that have not been examined extensively in existing literature but are of great geometric significance. The stack XSec(A) is a moduli stack for toric degenerations of toric hypersurfaces H ⊂ XA . There is a hypersurface EA ⊂ XSec(A) which parameterizes all degenerate hypersurfaces. For the Fukaya category of hypersurfaces in XA , the compliment XSec(A) \ EA plays the role of the classical stability conditions, while including EA incorporates the compactified version where MHS come into effect. To find the stability conditions associated with the directed Fukaya category of (XA , w), one needs to identify the complex structures associated with this model. In fact, these are described as the coefficients of the superpotential, or in our setup, the pencil Vw ⊂ H 0 (XA , w). Noticing that the toric boundary is also a toric degeneration of the hypersurface, we have that the pencil Vw is nothing other than a map from P1 to XSec(A) with prescribed point at infinity. If we decorate P1 with markings at the critical values of w and ∞, then we can observe such a map as an element of M0,Vol(Q)+1(XSec(A) , [w]) which evaluates to EA at all points except one and ∂ XA at the remaining point. We define the cycle of all stable maps with such an evaluation to be WA and regard it as the appropriate compactification of complex structures on Landau–Ginzburg A models. Applying techniques from fiber polytopes we obtain the following description of WA : Theorem 4.4 ([29]). The stack WA is a toric stack with moment polytope equal to the monotone path polytope of Sec(A). The polytope occurring here is not as widely known as the secondary polytope but occurs in a broad framework of “iterated fiber polytopes” introduced by Billera and Sturmfels. In addition to applications of these moduli spaces to stability conditions, we also obtain important information on the directed Fukaya categories and their mirrors from this approach. In particular, the theorem above may be applied to computationally find a finite set of special Landau–Ginzburg models {w1 , . . . , ws } corresponding to the fixed points of WA (or the vertices of the monotone path polytope of Sec(A)). Each such point is a stable map to XSec(A) whose image in moment space lies on the 1-skeleton of the secondary polytope. This gives a natural semiorthogonal decomposition of the directed Fukaya category into pieces

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corresponding to the components in the stable curve which is the domain of wi . After ordering these components, we see that the image of any one of them is a multicover of the equivariant cycle corresponding to an edge of Sec(A). These edges are known as circuits in combinatorics and we study the categories defined by each such component in [29]. Now we put this moduli space as a “zero fiber” of the “twistor” family of moduli family of stability conditions. Theorem 4.5 (see [68]). The fiber over zero is a formal scheme F over WA determined by the solutions of the Maurer–Cartan equations for a dg-complex 0 ←−−−− Λ 3 TY ←−−−− Λ 2 TY ←−−−− TY ←−−−− OY ←−−−− 0. −3 −2 −1 0 A Sketch of the Proof. The above complex describes deformations with fixed fiber at infinity. We can associate with this complex a Batalin–Vilkovisky algebra. Following [68] we associate with it a smooth stack. In the case of Fukaya–Seidel category of a Landau–Ginzburg mirror of a Fano manifold X the argument above implies that the dimension of the smooth stack of Landau–Ginzburg models is equal to h1,1 (X) + 1. We also have a C∗ action on F with fixed points corresponding to limiting stability conditions. Conjecture 4.6 (see [69]). The local completion of fixed points over X has a mixed Hodge structure. In the same way as the fixed point set under the C∗ action plays an important role in describing the rational homotopy types of smooth projective varieties we study the fixed points of the C∗ action on F and derive information about the homotopy type of a category. Similarly we can modify the above complex by fixing only a part of the fiber at infinity and deforming the rest. Similar Batalin–Vilkovisky algebra technique allows us to prove Theorem 4.7 ([68]). We obtain a smooth moduli stack of Landau–Ginzburg models if we fix only a part of the fiber at infinity. This means that we can allow different parts of the fiber at infinity to move—we call this part a moving scheme. The geometrical properties of the moving scheme contain deep birational, categorical, and algebraic cycle information. We record this information in new invariants, mainly emphasizing the birational content.

4.3 Birational Applications In this subsection we look at the data collected from Sects. 2 and 3 from a new categorical prospective. We apply the theorems above to the case of Landau–Ginzburg


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Table 3 Moving points del Pezzo

Moving scheme

P2

I9

∞ Pˆpt2

I8

∞

models for del Pezzo surfaces—this gives a new read of Sect. 2.4. The basic links among del Pezzo surfaces can be interpreted as follows. Theorem 4.8 ([29]). There exists an 11-dimensional moduli stack of all Landau– Ginzburg models of all del Pezzo surfaces. This moduli stack has a cell structure with the biggest cells corresponding to the del Pezzo surfaces of big Picard rank. The basic links correspond to moving to the boundary of this stack. Proof. The proof of this statement amounts to allowing all points at the fiber at infinity to move (the case of rational elliptic fibration) and then fixing them one by one (for the first step see Table 3). Theorem 4.8 suggests that we can extend the construction to rational blowdowns. We associate a moduli space of Landau–Ginzburg models to a rational blowdown of a rational surface by fixing corresponding subschemes of the fiber at infinity. This is a new construction in category theory, where the compactified moduli spaces of Landau–Ginzburg models play the role of the moduli space of vector bundles in the Donaldson theory of polynomial invariants. As a result we get a tool for studying the SODs by putting a topological structure on them based on the compactification of moduli spaces of Landau–Ginzburg models. We conjecture the following (see also [28]).

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Table 4 Moduli of Landau–Ginzburg models for surfaces Configuration at ∞ and a moving scheme Wheel of 9 lines, all points are fixed—no moving scheme

Surface P2

E(1)—rational elliptic surface with 12 singular fibers Rational blow-down of 2 E(1)#7CP

CP2 #8CP

2

Wheel of 9 lines, all points can move—moving scheme is all intersection points Wheel of 9 lines s.t. six points with multiplicities configure a moving scheme

Wheel of 9 curves, the moving scheme consist of eight points

MLG as an invariant Two-dimensional moduli space, FS(LG(P2 )) has a semiorthogonal decomposition Ten-dimensional moduli space The moduli space is parameterized by Ext’s from Φ to Ei in the category E1 , . . ., E9 , Φ , K 0 (Φ ) = 0, π1 (MLG(E(1)#7CP2 ) \ D) as an invariant The moduli space is parameterized by Ext’s among Ei ’s in the category E1 , . . ., E11 , π1 (MLG(CP2 #8CP2 ) \ D) as an invariant

Conjecture 4.9 (see [69]). The derived categories of the Barlow surface and of the rational blowdown described above contain as a semiorthogonal piece a phantom category, i.e., a nontrivial category with trivial K 0 group. This conjecture is rather bold and will make the study of algebraic cycles and rationality questions rather difficult. Some evidence for it has appeared in [50]. We summarize our findings in Table 4. In the leftmost part of this table we consider different surfaces. In the second part we describe the fiber at infinity with the corresponding moving scheme. In the last part we comment what is the moduli of Landau–Ginzburg models and what are some of its invariants. In most cases this is the fundamental group of the non-compactified moduli space. In case of rational blowdown this fundamental group suggests the appearance of new phenomenon a nontrivial category with trivial K 0 group—a phantom category, which we will discuss later. This also appears in the Barlow surface. The connection with the Godeaux surface (see [12]) suggests that the fundamental group of the non-compactified moduli space differs from the fundamental group of the noncompactified moduli space of LG models for del Pezzo surfaces of degree 1. Remark 4.10. Figure 2 suggests that different Fano’s are connected in the big moduli of Landau–Ginzburg models either by wall-crossings or by going to the boundary of such a moduli space.


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4.4 High-Dimensional Fano’s We concentrate on the case of high-dimensional Fano manifolds. We give the findings in Sect. 2.6 in the following categorical read. Conjecture 4.11. There exists a moduli stack of Landau–Ginzburg models of all three-dimensional Fano’s. It has a cell structure parallel to the basic links from Sect. 2.6. This conjecture is based on the following implementation of the theory of Landau–Ginzburg models. In the same way as in the case of del Pezzo surfaces we can allow moving different subschemes at the fiber at infinity. The first threedimensional examples were worked out in [1, 64]. In these cases the moving scheme at infinity corresponds to a Riemann surface so modified Landau–Ginzburg models correspond to Landau–Ginzburg mirrors of blown up toric varieties. In higher dimensions of course the cell structure is more elaborate. By fixing different parts at the divisor at infinity we can change the Picard rank of the generic fiber. Modifications, gluing, and conifold transitions are needed in dimension three and four. These lead to the need for Landau–Ginzburg moduli spaces with many components. The next case to consider is the case of the three-dimensional cubic. In this case the moving scheme is described in [50]. A similar moving scheme is associated with the threefold X14 . The next theorem follows from [29]. Theorem 4.12. The moduli space of the Landau–Ginzburg mirrors for the smooth three dimensional cubic and X14 are deformations of one another. As it follows from [29] this would imply their birationality since it means that some Mori fibrations associated with the three-dimensional cubic and X14 are connected via Sarkisov links. The A side interpretation of this result is given in [5]. It implies that the SODs of the derived categories of three-dimensional cubic and X14 have a common semiorthogonal piece and differ only by several exceptional objects—a result obtained by Kuznetsov in [77]. Similar observations can be made for other three-dimensional Fano manifolds, whose Landau–Ginzburg models can be included in one big moduli space. So studying and comparing these Landau–Ginzburg models at the same time brings a new approach to birational geometry. The material described in Sect. 2.6 suggests that there are many other three-dimensional Fano manifolds related as the threedimensional cubic and X14 , that is, related by one only non-commutative cobordism. Moving from one Landau–Ginzburg model associated to one Fano threefold to another can be considered as a certain “wall-crossing.” As the material of Sect. 2.6 suggests we can include singular Fano threefolds as boundary of the moduli space of Landau–Ginzburg models — i.e.,“limiting stability conditions” on which even more dramatic “wall-crossing” occurs. The experimental material from Sect. 2.6

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and Conjecture 3.7 also brings the idea that studying birational geometry of Fano threefolds and proving HMS for them might be closely related problems. Similar picture exists in higher dimension. We give examples and invariants connected with moduli spaces of Landau–Ginzburg models associated with very special four-dimensional Fano manifolds—four dimensional cubics and their “relatives.” For these fourfolds we look at the Hassett–Kuznetsov–Tschinkel program from the Landau–Ginzburg perspective. It is expected that there are many analogs in dimension four to the behavior of three-dimensional cubics and X14 , namely, they have a common semiorthogonal piece and differ only by several exceptional objects. We indicate several of them. Four-dimensional cubics have been studied by many people: Voisin, Beauville, Donagi, Hassett, and Tschinkel. On the level of derived categories, a lot of fundamental work was done by Kuznetsov and then Addington and Thomas. On the Landau–Ginzburg side, calculations were done by [51, 70]. We extend our approach in the case of three-dimensional Fano threefolds to some fourfolds. Recall the following theorem by Kuznetsov. Theorem 4.13 ([77]). Let X be a smooth four-dimensional cubic. Then Db (X) = Db (K3), E1 , E2 , E3 . Here Db (K3) is the derived category of a noncommutative K3 surface. This noncommutative K3 surface is very non-generic. Moving the subscheme at infinity corresponding to a generic K3 surface we obtain the moduli spaces of Landau– Ginzburg models associated to four-dimensional X10 . This suggests Conjecture 4.14. Let X be a smooth four-dimensional variety X10 . Then Db (X) = Db (K3), E1 , E2 , E3 , E4 . Here Db (K3) is the derived category of a generic noncommutative K3 surface. We expect that this conjecture will follow from some version of homological projective duality. Similarly to the three-dimensional case, there is an overlap between the Landau– Ginzburg moduli spaces of the four-dimensional cubic and four-dimensional X10 . Conjecture 4.15. There is an infinite series of moduli of Landau–Ginzburg models associated with special (from the Noether–Lefschetz loci) four-dimensional cubics and four-dimensional X10 ’s, which can be deformed one to another and therefore they are birational (see Table 5). This series corresponds to cases when the moving scheme at infinity is associated with commutative K3 surface. According to [29] such a deformation between moduli of Landau–Ginzburg models implies the existence of a Sarkisov links between such cubics and X10 . We will return to rationality questions in the next subsection. We would like to mention here that a generalization of homological projective duality of Kuznetsov’s arrives at


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Table 5 Noncommutative Sarkisov program Dim 3

Landau–Ginzburg moduli behavior

∞

X14 V3

link

4

LG models fibers over ∞

M1 moving schemes

M2

generic V3 generic X10 special X10

M3

link special V3

Conjecture 4.16. Let X be a smooth four-dimensional Kuechle manifold (see [76]) Then Db (X) = Db (K3), E1 , E2 , . . . , En . Here Db (K3) is the derived category of a generic noncommutative K3 surface. As a consequence we have Conjecture 4.17. There is an infinite series of moduli of Landau–Ginzburg models associated with special four-dimensional cubics, four-dimensional X10 s, and Kuechle manifolds, which can be deformed one to another. This conjecture suggests that the rationality question for Kuechle manifolds can be treated similarly as the questions for four-dimensional cubic and X10 —see the next subsection.

4.5 Invariants In this subsection we introduce two types of invariants, which are connected with SHS and moduli spaces of Landau–Ginzburg models. The first type is a global

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invariant—the Orlov spectra of a category. It was conjectured in [4] that it is an invariant measuring nonrationality. Its relation to Landau–Ginzburg models was emphasized in [50,51,70]. In this subsection we relate it to the Hassett–Kuznetsov– Tschinkel program—a program relating the Noether–Lefschetz components to rationality of four-dimensional cubic. The second type of invariant is of local nature—the local singularity of the Landau–Ginzburg models. We relate these invariants to stability conditions. We suggest that they play the role of discrepancies and thresholds in the Kawamata’s correspondence described in Table 1. In other words, these invariants measure if two Landau–Ginzburg moduli spaces can be deformed one to another and according to [29] if there are Sarkisov links connecting the Fano manifolds from the A side. Noncommutative Hodge structures were introduced by Kontsevich, Katzarkov, and Pantev in [68], as means of bringing techniques and tools of Hodge theory into the categorical and noncommutative realm. In the classical setting, much of the information about an isolated singularity is recorded by means of the Hodge spectrum, a set of rational eigenvalues of the monodromy operator. The Orlov spectrum (defined below) is a categorical analog of this Hodge spectrum appearing in the works of Orlov [83] and Rouquier [89]. The missing numbers in the spectra are called gaps. Let T be a triangulated category. For any G ∈ T denote by G0 the smallest full subcategory containing G which is closed under isomorphisms, shifting, and taking finite direct sums and summands. Now inductively define Gn as the full subcategory of objects, B, such that there is a distinguished triangle, X → B → Y → X[1], with X ∈ Gn−1 and Y ∈ G0 . Definition 4.18. Let G be an object of a triangulated category T . If there is some number n with Gn = T , we set t(G) := min{n ≥ 0 | Gn = T }. Otherwise, we set t(G) := ∞. We call t(G) the generation time of G. If t(G) is finite, we say that G is a strong generator. The Orlov spectrum of T is the union of all possible generation times for strong generators of T . The Rouquier dimension is the smallest number in the Orlov spectrum. We say that a triangulated category, T , has a gap of length s if a and a + s + 1 are in the Orlov spectrum but r is not in the Orlov spectrum for a < r < a + s + 1. The first connection to Hodge theory is given by the following theorem. Theorem 4.19 (see [4]). Let X be an algebraic variety with an isolated hypersurface singularity. The Orlov spectrum of the category of singularities of X is bounded by twice the embedding dimension times the Tjurina number of the singularity. The following conjecture plays an important role in our considerations. Conjecture 4.20 (see [4]). Let X be a rational Fano manifold of dimension n > 2. Then a gap of spectra of Db (X) is less or equal to n − 3.


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Table 6 Summary Dim

Examples

2

Rational blowdown X1 2 of CP2 #6CP

2

Barlow surface X2 = 2 CP2 #8CP

Hodge diamond 1 7 1 1 9 1

Categories

Invariants

Db (X1 ) = E1 , . . ., E9 , A, K 0 (A) = 0

π1 (MLG \ D)

Db (X2 ) = E1 , . . ., E11 , A, K 0 (A) = 0

π1 (MLG \ D)

Db (X1 ) = E1 , E2 , A

Monodromy of Landau– Ginzburg models, gap in spectra is at most 1

Db (X2 ) = E1 , . . ., E10 , A, K 0 (A) = Z2

Monodromy of Landau– Ginzburg models, gap in spectra is 0

Db (X) = E1 , E2 , E3 , A

Monodromy of Landau– Ginzburg models, gap in spectra is at most 2

1 1 3

3

4

Cubic threefold X1

Artin– Mumford example X2

Cubic fourfold X

5

5 1 1

1 1 1 1

1 1 1 21 1 1 1

After this brief review of theory of spectra and gaps we connect them with the SHS and moduli of Landau–Ginzburg models. Conjecture 4.21 (see [68]). The monodromy of the Landau–Ginzburg models for Fano manifold X determines the gap of spectra of Db (X). This conjecture was partially verified in [50, 51, 70]. We record our findings in Table 6. It is clear from our construction that the monodromy of Landau–Ginzburg models depends on the choice of moving scheme. This suggests that the classical Hodge theory cannot distinguish rationality. We employ the geometry of the moduli spaces of Landau–Ginzburg models in order to do so. These moduli spaces measure the way the pieces in the SODs are put together—this information computes the spectra of a category.

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This was first observed in [50, 51, 70]. Applying the theory of Orlov’s spectra to the case of four-dimensional cubic, four-dimensional X10 , and Kuechle manifolds we arrive at the following conjecture suggested by the Hassett–Kuznetsov– Tschinkel program. Conjecture 4.22. The four-dimensional cubic, four-dimensional X10 , and Kuechle manifolds are not rational if they do not contain derived categories of commutative K3 surfaces in their SODs (see [77] for cubic fourfold). In other words this conjecture implies that the generic fourfold as above is not rational since the gap of their categories of coherent sheaves is equal to two. In case the SODs contain the derived category of a commutative K3 surface—the issue is more delicate and requires the use of a Noether–Lefschetz spectra—see [50]. Remark 4.23. We are very grateful to A. Iliev who has informed us that some checks of Conjecture 4.22 were done by him, Debarre, and Manivel. We proceed with a topic which we have started in [50]—how to detect rationality when gaps of spectra cannot be used. The example we have considered there was the Artin–Mumford example. Initially it was shown that the Artin–Mumford example is not rational since it has torsion in its third cohomology group. Our conjectural interpretation in [50] is that the Artin–Mumford example is not rational since it is a conic bundle which contains the derived category of an Enriques surface in the SOD of its derived category. The derived category of an Enriques surface has 10 exceptional objects and a category A, which does not look like a category of a curve, in its SOD. We conjecture in [50] that derived category of the Artin–Mumford example has no gap in its spectra, but it is the moving scheme which determines its nonrationality. We also exhibit the connection between category A and its moving scheme—see also [49]. We bring a totally new prospective to rationality questions—the parallel of spectra and gaps of categories with topological superconductors. Indeed, if we consider generators as Hamiltonians and generation times as states of matter we get a far-reaching parallel. The first application of this parallel was a prediction of existence of phantom categories which we have defined above. The phantoms are the equivalent of topological superconductors in the above parallel, which, we conjecture, allows us to compute spectra in the same way as Turaev–Viro procedure allows us to compute topological states in the Kitaev–Kong models. In fact, the parallel produces a new spectra code which can be used in quantum computing, opening new horizons for research. We outline this parallel in Table 7. The discovery of phantoms came contrary to the expectations of the founding fathers of derived categories. Today it is known that phantoms are everywhere, in the same way as topological insulators and topological superconductors—a truly groundbreaking unconventional parallel. We anticipate striking applications of phantoms in the study of rationality of algebraic varieties. Let us briefly outline one of these applications. We consider another conic bundle—Sarkisov’s example; see [90] and Table 8. This example can be described as follows—we start with


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Table 7 Gaps, spectra, and topological superconductors Topological states of matter Hamiltonians Topological states Real three-dimensional manifolds in Kitaev–Kong model Topological superconductors

Gaps and phantoms Generators of the category Generation time Singularities of Landau–Ginzburg model Phantoms as limits of gaps

an irreducible singular plane curve Csing in P2 of degree d 3 that has exactly (d − 1)(d − 2)/2 − 1 ordinary double points (such curves exists for every d 3). Then we blow up P2 at the singular points of Csing . Denote by S the resulting surface and by C the proper transform of the curve Csing . Then C is a smooth elliptic curve (easy genus count). Let τ : C˜ → C be some unramified double cover. Then it follows from [90, Theorem 5.9] there exists a smooth threefold X of Picard rank rk(Pic(S)) + 1 with a morphism π : X → S whose general fiber is P1 , i.e., π is a conic bundle, such that C is the discriminant curve of π , and τ is induced by interchanging components of the fibers over the points of C. Moreover, it follows from [90, Theorem 4.1] that X is not rational if d 12. On the other hand, we always have H 3 (X, Z) = 0, since C is an elliptic curve. Note that birationally, X can be obtained as a degeneration of a standard conic bundle over P2 whose discriminant curve is a smooth curve of degree d. On the side of Landau–Ginzburg models we can observe the following. The mirrors of conic bundles are partially understood—see [1]. The degeneration procedure on the B side amounts to conifold transitions on the A side. These conifold transitions define a moving scheme for the Landau–Ginzburg model which suggests the following conjecture. Conjecture 4.24. The SOD of Db (X) contains a phantom category—a nontrivial category with trivial K 0 group. This phantom category is the reason for nonrationality of Sarkisov’s conic bundle, which we conjecture has no gaps in the spectra of its derived category— see Table 8. So conjecturally we have Db (X) = A, E1 , . . . , E112 , K 0 (A) = 0, A = 0. The degeneration construction above suggests an ample opportunity of constructing phantom categories. Conjecture 4.25. The SOD of derived category of degeneration of a generic quadric bundles over a surface contains a phantom category. As a result, we conjecture nonrationality of such quadric bundles. An interesting question is where this phantom categories come from. The analysis of Sarkisov’s example suggests the following. We start with a conic bundle over P2 with a curve

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Table 8 Sarkisov example Conic bundle Conic bundle X with degeneration curve C, pa (C) = 1

Hodge diamond

Phantom category

1 0 0 0

0 55

0 0

0 0

55 0

0 0

0 1

of degeneration C. Such a conic bundle has a nontrivial gap in the spectra of its derived category. Via degeneration we reduce this gap in the same way as via degeneration we get rid of the intermediate Jacobian. The degeneration of the two sheeted covering of P2 produces a surface with a phantom category in its SOD. This observation provides us with many possibilities to construct geometric examples of phantom categories. After all, many of the classical examples of surfaces of general type are obtained from rational surfaces by taking double coverings, quotienting by group actions, degenerations, and smoothings. So the existence of nontrivial categories in SOD with trivial Hochschild homology can be conjecturally seen in the case of classical surfaces of general type, Campedelli, Godeaux (see [12]), Burniat (see [3]), Dolgachev surfaces, product of curves ([39]), and also in the categories of quotients of product of curves and fake P2 . These surfaces are not rational since they have nontrivial fundamental groups but also since they have conjecturally a quasi-phantom subcategory in their SOD. We call a category a quasi-phantom if it is a nontrivial category with a trivial Hochschild homology. On the Landau–Ginzburg side these quasi-phantoms are described by the moving scheme. The deformation of Landau–Ginzburg models is determined by the moving scheme, thus the (quasi-)phantoms factor in the geometry of the Landau–Ginzburg models. On the mirror side this translates to the fact Ext’s between the (quasi-)phantom category and the rest of the SOD determine the moduli space. Finding phantoms—nontrivial categories with trivial K 0 groups—is a quantum leap more difficult than finding quasi-phantoms. We conjecture that derived categories of the Barlow surface (see [28]) and rational blow-downs contain phantoms in their SOD.


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Applying the quadric bundles construction described above one conjecturally can produce many examples of phantom categories. There are two main parallels we build our quadric bundles construction on: 1. Degenerations of Hodge structures applied to intermediate Jacobians. This construction goes back to Clemens and Griffiths and later to Alexeev. They degenerate intermediate Jacobians to Prym varieties or completely to algebraic tori. The important information to remember are the data of degeneration. For us the algebraic tori is analogous to the phantom and degeneration data to the gap in the spectrum. The data to analyze is how the phantom fits in SOD. This determines the gap and the geometry of the Landau–Ginzburg moduli space. It is directly connected with the geometry of the moving scheme and the monodromy of LG models. 2. The Candelas idea to study rigid Calabi–Yau manifolds by including them in Fano varieties, e.g., four-, seven-, and ten-dimensional cubics. This gives him the freedom to deform. Similarly, by including the phantom in the quadric bundles we get the opportunity to deform and degenerate. The rich SOD of the quadric bundle allows us to study the phantom. This of course is a manifestation of the geometry of the moving scheme and the monodromy at infinity of the moduli space of LG models. We turn to the A side and pose the following question. Question 4.26. Do A side phantoms provide examples of nonsymplectomorphic symplectic manifolds with the same Gromov–Witten invariants? Remark 4.27. An initial application of the conic bundles above to classical Horikawa surfaces (see [48]) seems to suggest that after deformation we get a phantom in the Fukaya category for one of them and not for the other. It would be interesting to see if a Hodge type argument would lead to the fact that these two types of Horikawa surfaces have different gaps of spectra of their Fukaya categories and as a result are not symplectomorphic. It will be analogous to degenerating Hodge structures to nonisomorphic ones for the benefit of geometric consequences. In what follows we move to finding quantitative statements for gaps and phantoms. We have already emphasized the importance of quadric bundles. In what follows we concentrate on moduli space of a stability conditions of local CY obtained as quadric bundles. We move to a second type of invariants we have mentioned. We take the point of view from [45] that for special type of Fukaya and Fukaya wrapped categories locally stability conditions are described by differentials with coefficients irrational or exponential functions. The main idea in [45] is that for such categories we can tilt the t-structure in a way that the heart of it becomes an Artinian category. Such a simple t-structure allows description of stability condition in terms of geometry of Lefschetz theory and as a result in terms of the moduli space of Landau–Ginzburg models.

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As it is suggested in [4] there is a connection between monodromy of Landau– Ginzburg models and the gaps of a spectrum. We record our observations in Tables 9 and 10. 1. In the case of An category the stability conditions are just exponential differentials as we have demonstrated in Example 4.2. In this case the simple objects for the t-structures are given by the intervals connecting singular points of the function given by the central charge. 2. Similarly for one-dimensional Fukaya wrapped categories the simple objects for the t-structures are given by the intervals connecting zero sets of the differentials. This procedure allows us to take categories with quivers. 3. For more complicated Fukaya–Seidel categories obtained as a superposition of one-dimensional Fukaya wrapped—see [71]—we describe the stability conditions by intertwining the stability conditions for one-dimensional Fukaya wrapped—look at the last line of Table 10. At the end we obtain a number d/k, where d is the degree of some of the polynomials p(z) involved in the formula and k is the root we take out of it. Such a number can be associated with Fukaya–Seidel category associated with a local Calabi–Yau manifold obtained as a quadric bundle. The following conjecture suggests local invariants: Conjecture 4.28. The geometry of the moving set determines the number d/k and the gap of the spectra of the corresponding category. This conjecture suggests the following question. Question 4.29. Is the number d/k a birational invariant? In Table 10 we give examples of categories and their stability conditions. These categories serve as building blocks for more involved categories of Fano manifolds. In case the question above has a positive answer we get a way of comparing the moving schemes of the divisors at infinity. We will also get a way of deciding if the corresponding moduli spaces of Landau–Ginzburg models can be deformed to each other which according to [29] is a way of deciding if we can build a Sarkisov link between them. It is clear that the numbers d/k fit well in the landscape of quadric bundles. We expect particularly interesting behavior from the numbers d/k coming from the phantom categories the existence of which was conjectured earlier. Question 4.30. Can we read off the existence of phantoms and gaps in terms of the numbers d/k? It is also clear that the geometry of the moving scheme in general has a deep connection with the geometry of the Fano manifold. In fact, following [93], we associate a complex of singularities with this moving scheme. So it is natural to expect that we can read many geometrical properties of Fano manifolds from this complex of singularities. For example, it has been conjectured by S.-T. Yau, G. Tian, and S. Donaldson that some kind of stability of Fano manifolds is a necessary


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Table 9 Landau–Ginzburg models and stability conditions Category An

Imaginary part of a central charge Im e p(z) dz

Landau–Ginzburg models and hearts of a t-structure

simple objects in the heart C

R

One-dimensional Fukaya wrapped category or Fukaya category of ddimensional Calabi–Yau category

Im q(z)dz, where q(z) is a quadratic differential with all zeros of multiplicities d −2

Db (Pn )

Imz : K 0 (Db (Pn )) →C Moduli space of Landau– Ginzburg models

Db (X), X is a Fano variety

a1

a2

a3

R ai ’s are simple objects The Pn quiver Intertwining Landau–Ginzburg models and hearts via monodromy—creating gaps in spectra

and sufficient condition for the existence of Kähler–Einstein metrics on them. This conjecture has been verified in the two-dimensional case (see [95]) and in the toric case (see [97]). Moreover, one direction of this conjecture is now almost proved by Donaldson, who showed that the existence of the Kähler–Einstein metric implies the so-called K-semistability (see [31,32]). Recall that Tian (see [96]) defined the notion of K-stability, arising from certain degenerations of the manifold or, as he called them, test configurations. Proving Yau–Tian–Donaldson conjecture is currently a major research program in Differential Geometry (see [30]). We finish with the following question: Question 4.31. Can we read off the existence of a Kähler–Einstein metric on the Fano manifold from this complex of singularities?

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Table 10 Conjectural duality Fukaya categories Fuk(F), where F is a local Calabi–Yau variety 2 }, where p is a {y2 + r(x) = α12 + . . . + αd−2 polynomial of degree n + 1 Fukwrapped (C)(d), where C is a Riemann surface with punctures (Fuk(F1 ) × . . . × Fuk(Fm ) × Fukwrapped (C1 ) × . . . ×Fukwrapped (Cn ) /Zk

Stability conditions q(z), where q is a quadratic differential with roots of multiplicities d q(z)e p(z) dz, where q is a quadratic differential with roots of multiplicities d and p is a polynomial of degree 2g(C) + 1 k q1 (z) · . . . · qm+n (z) ·e p1 (z)+...+pn (z) dz

Acknowledgements We are very grateful to A. Bondal, M. Ballard, C. Diemer, and D. Favero, F. Haiden, A. Iliev, A. Kasprzyck, G. Kerr, M. Kontsevich, A. Kuznetsov, T. Pantev, Y. Soibelman, D. Stepanov for the useful discussions. I. C. is grateful to Hausdorff Research Institute of Mathematics (Bonn) and was funded by RFFI grants 11-01-00336-a and AG Laboratory GU-HSE and RF government grant ag. 11 11.G34.31.0023. L. K was funded by grants NSF DMS0600800, NSF FRG DMS-0652633, NSF FRG DMS-0854977, NSF DMS-0854977, NSF DMS-0901330, grants FWF P 24572-N25 and FWF P20778, and an ERC grant—GEMIS, V.P. was funded by Dynasty Foundation, grants NSF FRG DMS-0854977, NSF DMS-0854977, and NSF DMS-0901330; grants FWF P 24572-N25 and FWF P20778; RFFI grants 11-01-00336-a, 11-01-00185-a, 1201-33024, and 12-01-31012; grants MK−1192.2012.1, NSh−5139.2012.1, and AG Laboratory GU-HSE and RF government grant ag. 11 11.G34.31.0023.

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Curves of Low Degrees on Fano Varieties Olivier Debarre

Mathematics Subject Classification codes (2010): 14C05, 14C30, 14C34, 14D20, 14E05, 14E08, 14E20, 14H10, 14J10, 14J30, 14J35, 14J45, 14J60, 14J70, 14M20, 14M22, 14N25

1 Introduction We work over the field of complex numbers. If X ⊂ PN is a smooth projective variety, we let Cdg (X) be the (quasi-projective) subscheme of the Hilbert scheme of X corresponding to smooth, irreducible, genus-g, degree-d curves on X. g These schemes, and their compactifications C d (X), particularly for low d and g, have proved very useful in the study of the geometry and the period maps of some Fano varieties of low degrees. In this informal survey article, we explain, mainly without proofs, what is known (or conjectured) in the following cases: cubic threefolds, Fano threefolds of degree 14 and index 1, cubic fourfolds, Fano varieties of degree 10, coindex 3, and dimensions 3, 4, or 5. In the case of a smooth cubic threefold X ⊂ P4 (Sect. 2), Hilbert schemes of curves have long been used to parametrize useful subvarieties of the intermediate Jacobian J(X). For example, the (smooth projective) surface C10 (X) parametrizing lines contained in X was an essential ingredient in the proofs by Clemens and Griffiths of the non-rationality of X and of the Torelli theorem (that J(X) determines g X). The Abel–Jacobi maps aj : C d (X) → J(X) have been since proved in some cases to be mrc fibrations (see Sect. 2 for definitions and results). O. Debarre () Département de Mathématiques et Applications – CNRS UMR 8553, École Normale Supérieure, 45 rue d’Ulm, 75230 Paris cedex 05, France e-mail: [email protected] F. Bogomolov et al. (eds.), Birational Geometry, Rational Curves, and Arithmetic, DOI 10.1007/978-1-4614-6482-2__6, © Springer Science+Business Media New York 2013

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Curves on Fano threefolds can also be used to construct interesting birational correspondences. This is illustrated in Sect. 2.9, where it is explained that Fano threefolds of degree 14 and index 1 are birational to cubic threefolds. The situation is very rich and gives rise to a beautiful description of the period maps for these Fano threefolds. In Sect. 3, we move on to smooth cubic fourfolds Y ⊂ P5 , for which there are g no intermediate Jacobians. Nevertheless, the mrc fibrations of C d (Y ) should be substitutes for the Abel–Jacobi maps and should provide interesting information. In another direction, ideas of Mukai show how to construct symplectic forms on some moduli spaces of sheaves (Sect. 3.1). This can be used to prove that the space of lines C10 (Y ) is a symplectic fourfold (Beauville and Donagi), whose Hodge structure is closely related to that of Y . This was the basis of Voisin’s proof of the injectivity of the period map for smooth cubic fourfolds. This period map is now completely described by work of Looijenga and Laza and identifies the moduli space of smooth cubic fourfolds with an explicit open subset of the Baily–Borel compactification of the period domain (Sect. 3.2). Analogous results are available for the moduli space of smooth cubic threefolds (Sect. 3.3). 4 of degree Mukai’s construction was also shown to apply to Fano fourfolds X10 0

4 ) leads to the 10 and index 2 (Example 3.5). In this case, the scheme C 2 (X10 construction of a symplectic fourfold called a double EPW sextic (Iliev and Manivel, O’Grady). In Sect. 4, we study in some details the period maps of these Fano varieties of degree 10, in dimensions 3, 4, and 5. There is beautiful geometry at work here, with intriguing interplay with that of EPW sextics (work in progress with Iliev and Manivel).

2 Cubic Threefolds The maximal rationally connected fibration (mrc fibration for short) of a smooth proper (complex) variety Y is a rational dominant map ρ : Y R(Y ) such that for z ∈ R(Y ) very general, the fiber ρ −1 (z) is proper and rationally connected (two general points can be joined by a rational curve), and any rational curve in Y that meets ρ −1 (z) is contained in ρ −1 (z). The mrc fibration exists and is unique up to birational equivalence. The variety R(Y ) is called the reduction of Y . Let X ⊂ P4 be a general (although some results are known for any smooth X) cubic hypersurface. The intermediate Jacobian J(X) := H 2,1 (X)∨ /H3 (X, Z) is a 5-dimensional principally polarized abelian variety. We let X3 be the (irreducible, 10-dimensional) moduli space for smooth cubic threefolds, and we let An be the moduli space for principally polarized abelian varieties of dimension n. Taking X to J(X) defines the period map X3 → A5 . We have:

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• For g = 0 or d ≤ 5, Cdg (X) is integral of dimension 2d ([7], Theorem 1.1); g • For d ≤ 5, the Abel–Jacobi map aj : C d (X) → J(X) induces the mrc fibration ([6], Theorem 1.1); 0 • For d ≥ 4, aj : C d (X) → J(X) is dominant with irreducible general fibers. 0

It is natural to ask whether the Abel–Jacobi map aj : C d (X) → J(X) is the mrc fibration for all d ≥ 4 (i.e., are general fibers rationally connected?).

2.1 Lines • C10 (X) (which parametrizes lines on X) is a smooth projective irreducible surface of general type; • The image of aj : C10 (X) → J(X) is a surface S with minimal class [Θ ]3 /3! and S − S is a theta divisor Θ ; • aj induces an isomorphism between the Albanese variety of C10 (X) and J(X) ([2]). The second item yields a proof of Torelli: the period map X3 → A5 is injective.

2.2 Conics 0

• C 2 (X) is a smooth projective irreducible fourfold; 0 • aj : C 2 (X) → J(X) is a P2 -bundle over the surface S of Sect. 2.1 (a conic is uniquely determined by the plane that it spans and the residual line in X). The mrc 0 fibration is therefore C 2 (X) → S.

2.3 Plane Cubics 1

• C 3 (X) is isomorphic to the Grassmannian G(3, 5); • The Abel–Jacobi map is therefore constant.

2.4 Twisted Cubics • aj : C30 (X) → J(X) is birational to a P2 -bundle over a theta divisor Θ ([6], Sect. 4). 0

The mrc fibration is therefore C 3 (X) → Θ .

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2.5 Elliptic Quartics • aj : C41 (X) → J(X) is birational to a P6 -bundle over the surface S of Sect. 2.1 ([6], 1

Sect. 4.1). The mrc fibration is therefore C 4 (X) → S.

2.6 Normal Rational Quartics 0

• aj : C 4 (X) → J(X) is dominant and the general fiber is birational to X ([9], Theorem 5.2), hence unirational. It is therefore the mrc fibration.

2.7 Normal Elliptic Quintics • C51 (X) is an irreducible tenfold; • There is a factorization ([14], Theorem 5.6; [9], Theorem 3.2) 1

α

β

aj : C 5 (X) − → MX (2; 0, 2) − → J(X), where – MX (2; 0, 2) is some component of the moduli space of rank-2 stable vector bundles on X with Chern classes c1 = 0 and c2 = 2; – α is a P5 -bundle over a dense open subset of MX (2; 0, 2); – β is birational (this is proved in [9] via ingenious geometrical constructions). 1

In particular, C 5 (X) → J(X) is therefore the mrc fibration. This is seen as follows. The map α is obtained via the Serre construction: to C ∈ C51 (X), one associates a stable rank-2 vector bundle EC on X with Chern classes c1 = 0 and c2 = 2 such that C is the zero-locus of a section of EC (1). The fibers of α are P(H 0 (X, EC (1))) P5 ; hence the Abel–Jacobi map factors through α . According to Murre, the Chow group of algebraic 1-cycles of fixed degree on X modulo rational equivalence is canonically isomorphic to J(X). The map β can then be defined directly as E → c2 (E ).

2.8 Normal Elliptic Sextics • C61 (X) is an irreducible 12-fold; 1

• aj : C 6 (X) → J(X) is the mrc fibration ([21], Theorem 2.1).


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2.9 Fano Threefolds of Degree 14 and Index 1 There is a very interesting relationship between cubic threefolds and Fano threefolds of degree 14 and index 1 [9, 10, 14]. The latter are obtained as linear sections of G(2, 6) ⊂ P14 by a P9 . Let X ⊂ P4 be a smooth cubic threefold, let C be general in C51 (X), and let π : X˜ → X be its blow-up, with exceptional divisor E. We have −KX˜ ≡ −π ∗ KX − E ≡ 2π ∗ H − E. lin

lin

This linear system induces a morphism X˜ → P4 which induces a small contraction ϕ onto the normalization X¯ of its image. Its non-trivial fibers are the strict transforms of the 25 lines bisecant to C: the divisor E is ϕ -ample hence there is a flop ϕ

ϕ

χ : X˜ −→ X¯ ←− X˜ , where X˜ is smooth projective and χ (E) is ϕ -antiample. We have ρ (X˜ ) = 2. Since the extremal ray generated by the classes of curves contracted by ϕ has KX˜ -degree 0 and KX˜ is not nef (−KX˜ ≡ ϕ ∗ H¯ ), the other lin

extremal ray is KX˜ -negative and defines a contraction π : X˜ → X . One checks that: • X is a smooth Fano threefold of degree 14 and index 1, with Picard group generated by H := −KX ; • π is the blowup of a smooth elliptic quintic curve C ⊂ X , with exceptional divisor E ≡ 5ϕ ∗ H¯ − 3χ∗(E) and χ ∗ π ∗ H ≡ 7H − 4E. lin

lin

Conversely, given a general elliptic quintic curve C in a smooth Fano threefold X of degree 14 and index 1, one can reverse the construction above and obtain a cubic threefold X with a quintic curve C ⊂ X. In other words, if C51 (X3 ) denotes the moduli space of all pairs (X,C) as above, and C51 (X14 ) the moduli space of all pairs (X ,C ), we have a birational isomorphism C51 (X3 ) C51 (X14 ) between (irreducible, 20-dimensional) varieties. We have the following two properties. • The intermediate Jacobians of X and X are isomorphic (this holds because J(X) × J(C) J(X ) × J(C ), J(X ) is not a product, and J(X) and J(X ) have the same dimension). Since we have Torelli for cubic threefolds (Sect. 2.1), the cubic obtained from a pair (X ,C ) only depends on X . • The variety X only depends on the vector bundle EC defined in Sect. 2.7 (Kuznetsov proves in [10] that if S is the rank-2 tautological vector bundle on G(2, 6), the variety P(S |X ) is obtained by a flop of P(EC )).

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So, if X14 is the moduli space for smooth Fano threefolds of degree 14 and index 1, and if MX3 (2; 0, 2) is the moduli space of pairs (X, E ), with [E ] ∈ MX (2; 0, 2) (see Sect. 2.7), we get a commutative diagram: C51 (X3 )

dim. 20

C51 (X14 )

dim. 20

X14

dim. 15

P5 −bundle

MX3 (2; 0, 2)

dim. 15

γ isom.

δ

X3

dim. 10

period map

period map

A5 . The map γ is actually an isomorphism of stacks and the fiber of δ (between stacks) at [X] is the (quasi-projective) moduli space of stable rank-2 vector bundles E such that c1 (E ) = 0 and H 1 (X, E (−1)) = 0 (instanton bundles), an open subset of J(X) ([10], Theorem 2.9).

3 Cubic Fourfolds Let Y ⊂ P5 be a general cubic hypersurface. 0 The variety C d (Y ) is integral of dimension 3d + 1 ([5], Proposition 2.4). To study these varieties, one could use, instead of the Abel–Jacobi map, their mrc fibration [5]. Here is what is known, or conjectured: • C10 (Y ) is a symplectic fourfold [1] hence is its own reduction; 0

• The map C 2 (Y ) C10 (Y ) given by residuation is a P3 -bundle so this is the mrc fibration; 0 • C 3 (Y ) is uniruled: a general cubic curve lies on a unique cubic surface and moves in a 2-dimensional linear system on it; so its reduction has dimension ≤ 8; 0 • There is a map C 4 (Y ) J (Y ) (where J (Y ) → (P5 )∨ is the relative intermediate Jacobian of smooth hyperplane sections of Y ) whose general fibers are these cubic threefolds (Sect. 2.6) hence are unirational; moreover, J (Y ) should be non-uniruled (see also Example 3.4 below), so this should be the mrc fibration; • For d ≥ 5 odd, there is ([5], Theorem 1.2; [11]) a holomorphic 2-form on (a smooth non-singular model of) Cd0 (Y ) which is non-degenerate at a general 0

point. In particular, C d (Y ) is not uniruled.


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3.1 Constructing Symplectic Forms on Moduli Spaces Mukai proved in 1984 that the moduli space M of simple sheaves on a K3 or abelian surface carries a closed non-degenerate holomorphic 2-form: the tangent space to M at a point [F ] representing a simple sheaf F on a smooth projective variety Z is isomorphic to Ext1 (F , F ). The Yoneda coupling Ext1 (F , F ) × Ext1 (F , F ) −→ Ext2 (F , F ) is skew-symmetric whenever [F ] is a smooth point of M . When Z is a symplectic surface S with a symplectic holomorphic form ω ∈ H 0 (S, ΩS2 ), Mukai composes the Yoneda coupling with the map ∪ω

Tr

→ H 2 (S, OS ) −−→ H 2 (S, ΩS2 ) = C, Ext2 (F , F ) − and this defines the symplectic structure on the smooth locus of M . Over an n-dimensional variety Z such that hq,q+2(Z) = 0 for some integer q, we: • Pick a non-zero element ω ∈ H n−q−2(Z, ΩZ ); q • Use the exterior power At(F )∧q ∈ Extq (F , F ⊗ ΩZ ) of the Atiyah class1 1 1 At(F ) ∈ Ext (F , F ⊗ ΩZ ). n−q

We then define At(F )∧q ◦•

Tr

→ Ext2 (F , F ) −−−−−−→ Extq+2 (F , F ⊗ ΩZq ) − ∪ω

H q+2 (Z, ΩZ ) −−→ H n (Z, ΩZn ) C. q

Composing the Yoneda coupling with this map provides a closed (possibly degenerate) 2-form on the smooth locus of the moduli space. Example 3.3. Let Y ⊂ P5 be a smooth cubic fourfold. Since h1,3 (Y ) = 1, the construction provides a (unique) 2-form on the smooth loci of moduli spaces of sheaves on Y . Kuznetsov and Markushevich use this construction in a roundabout

Δ : Z → Z × Z be the diagonal embedding and let Δ (Z)(2) ⊂ Z × Z be the closed subscheme defined by the sheaf of ideals IΔ2(Z) . Since IΔ (Z) /IΔ2(Z) ΩZ ; we have an exact sequence

1 Let

0 → Δ∗ ΩZ → OΔ (Z)(2) → Δ∗ OZ → 0. If F is a locally free sheaf on Z, we obtain an exact sequence 0 → F ⊗ ΩZ → p1∗ (p∗2 (F ⊗ OΔ (Z)(2) )) → F → 0, hence an extension class AtF ∈ Ext1 (F , F ⊗ ΩZ ). The same construction can be extended to any coherent sheaf on Z by working in the derived category (Illusie).

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way to produce a symplectic structure on the (smooth) fourfold C10 (Y ) of lines L ⊂ Y (originally constructed by Beauville and Donagi by a deformation argument; note that the simple-minded idea to look at sheaves of the form OL does not work). Example 3.4. Let Y ⊂ P5 be a smooth cubic fourfold. Let NY be the (quasiprojective) moduli space of sheaves on Y of the form i∗ E , where i : X → Y is a non-singular hyperplane section of Y and [E ] ∈ MX (2; 0, 2). By Sect. 2.7, NY is a torsor under the (symplectic) relative intermediate Jacobian J (Y ) of smooth hyperplane sections of Y . The Donagi–Markman symplectic structure ([4], 8.5.2) on J (Y ) induces a symplectic structure on NY which should be the same as the Kuznetsov–Markushevich structure ([11, 15], Theorem 7.3 and Remark 7.5). Note that since we do not know whether the Donagi–Markman symplectic form on J (Y ) extends to a smooth compactification, we cannot deduce that J (Y ) is not uniruled. 4 be a smooth Fano fourfold obtained by intersecting the Example 3.5. Let X10 Grassmannian G(2, 5) in its Plücker embedding with a general hyperplane and a 4 ) = 1. The Hilbert scheme C 0 (X 4 ) general quadric (see Sect. 4). We have h1,3 (X10 2 10 4 is smooth [8]; hence it is endowed, by the of (possibly degenerate) conics in X10 0

4 ) construction above, with a canonical global holomorphic 2-form. Since C 2 (X10 has dimension 5, this form must be degenerate. There is a naturally defined morphism 0

4 C 2 (X10 ) → P(IX 4 (2)) P5 10

whose image is an Eisenbud–Popescu–Walter (EPW for short) sextic hypersurface ZX 4 ⊂ P5 (see [16] for the definition of these sextics). In the Stein factorization 10

0

4 C 2 (X10 ) → YX 4 → ZX 4 , 10

10

0

4 ) is (essentially) the projective variety YX 4 is a smooth fourfold over which C 2 (X10 10

0

4 ) thus descends to Y a smooth fibration in projective lines. The 2-form on C 2 (X10 4 X10 and makes YX 4 into a holomorphic symplectic fourfold [8, 16] called a double EPW 10 sextic.

3.2 Periods for Cubic Fourfolds Let Y ⊂ P5 be a smooth cubic fourfold. Since Y has even dimension, it has no intermediate Jacobian but still an interesting Hodge structure “of K3-type”: H 4 (Y, C)prim = H 1,3 (Y ) ⊕ H 2,2 (Y )prim ⊕ H 3,1 (Y ) dimensions: 1 20 1


141

with period domain D 20 = {[ω ] ∈ P21 | Q(ω , ω ) = 0, Q(ω , ω¯ ) > 0}, a 20-dimensional bounded symmetric domain of type IV (where Q is the (non-degenerate) intersection form on H 4 (Y, C)). Let Y3 be the (irreducible, 20dimensional) moduli space for smooth cubic fourfolds. We get a period map Y3 → D 20 /Γ , where Γ is an explicit discrete arithmetic group. Voisin proved that it is injective and its (open) image was determined in [12].

3.3 Periods for Cubic Threefolds One can construct another period map for cubic threefolds (compare with Sect. 2): to such a cubic X ⊂ P4 , we associate the cyclic triple cover YX → P4 branched along X. It is a cubic fourfold; hence this construction defines a map X3 → Y3 → D 20 /Γ . Because of the presence of an automorphism of YX of order 3, we can restrict the image and define a period map (Allcock–Carlson–Toledo) X3 → D 10 /Γ , where

D 10 := {ω ∈ P10 | Q(ω , ω¯ ) < 0} B10

and Γ is a discrete arithmetic group. Again, it is an isomorphism onto an explicitly described open subset of D 10 /Γ .

4 Fano Varieties of Degree 10 Let V5 be a 5-dimensional complex vector space. We define, for k ∈ {3, 4, 5}, a smooth, degree-10, coindex-3, Fano k-fold by k X10 := G(2,V5 ) ∩ Pk+4 ∩ Ω ⊂ P(∧2V5 ),

where Pk+4 is a general (k + 4)-plane and Ω a general quadric. Let X10k be the moduli stack for smooth varieties of this type. 3 4 5 Enriques proved that all (smooth) X10 , X10 , and X10 are unirational. A general 3 5 4 X10 is not rational, whereas all (smooth) X10 are rational [19]. Some smooth X10 are 4 rational ([17, 18]), but the rationality of a general X10 is an open question.

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We have IX k (2) CΩ ⊕ V5 , 10

where V5 corresponds to the rank-6 Plücker quadrics ω → ω ∧ ω ∧ v (v ∈ V5 ). In the 5-plane P(IX k (2)), the degree-(k + 5) hypersurface corresponding to singular 10 quadrics decomposes as (k − 1)P(V5) + ZX∨k , 10

where ZX∨k ⊂ P(IX k (2)) is an EPW sextic ([8], Sect. 2.2; this is indeed the projective 10

10

dual of the sextic defined in Example 3.5 when k = 4). If E PW is the (irreducible, 20-dimensional) moduli space of EPW sextics, we get morphisms epwk : X10k → E PW , which are dominant ([8], Corollary 4.17). Note that dim(X105 ) = 25, dim(X104 ) = 24, dim(X103 ) = 22. Proposition 4.1 (Debarre–Iliev–Manivel). For a general EPW sextic Z, with projective dual Z ∨ , the fiber (epw3 )−1 ([Z]) is isomorphic to the smooth surface Sing(Z ∨ ).

4.1 Gushel Degenerations The link between these Fano varieties of various dimensions can be made through a construction of Gushel analogous to what we did with cubics in Sect. 3.3. Let CG ⊂ P(C⊕ ∧2V5 ) be the cone, with vertex v = P(C), over the Grassmannian G(2,V5 ). Intersect CG with a general quadric Ω ⊂ P(C ⊕ ∧2V5 ) and a linear space Pk+4 to get a Fano variety T k of dimension k. There are two cases: k of G(2,V ) • Either v ∈ / Pk+4 , in which case T k is isomorphic to the intersection X10 5 k+4 2 with the projection of P to P(∧ V5 ) and a quadric • Or v ∈ Pk+4 , in which case Pk+4 is a cone over a Pk+3 ⊂ P(∧2V5 ) and T k is a k−1 double cover XGk of G(2,V5 ) ∩ Pk+3 branched along its intersection X10 with a quadric

The second case is a specialization of the first, and the EPW sextics ZX k from 10 the first case degenerate to the sextics ZX k from the second case. Moreover, in the G second case, the sextics ZX k and ZX k−1 are the same. G

10


143

k and their Gushel Let XGk be the moduli stack for smooth varieties of type X10 degenerations. The Gushel constructions therefore yield morphisms X10k → XGk+1 such that the diagrams

X10k

XGk+1 epwk+1 G

epwk

E PW commute. We can perform a “double Gushel construction” as follows. Let CCG ⊂ P(C2 ⊕ 2 ∧ V5 ) be the cone, with vertex L = P(C2 ), over the Grassmannian G(2,V5 ). Intersect CCG with a general quadric Ω ⊂ P(C2 ⊕ ∧2V5 ) and a codimension-2 linear space P9 to get a Fano variety T of dimension 5. 5. • If L ∩ P9 = ∅, the variety T is smooth of type X10 9 9 7 • If L ⊂ P , in which case P is a cone over a P ⊂ P(∧2V5 ), the corresponding variety T0 meets L at two points p and q, and the projection from L induces a rational conic bundle T0 W54 := G(2,V5 ) ∩ P7 , undefined at p and q, whose 3 . Blowing up these two points, we obtain discriminant locus is a threefold X10 4 a conic bundle Tˆ0 → W5 with two disjoint sections corresponding to the two exceptional divisors. These two sections trivialize the canonical double étale 3 → X 3 of the discriminant. cover X˜10 10 5

We call the singular variety T0 a double-Gushel degeneration. If we let X GG be the 5 and their Gushel and double-Gushel moduli stack for smooth varieties of type X10 degenerations, we obtain a commutative diagram: XG4

X103 epw3

5 XGG

epw4G

E PW .

epw5GG

4.2 Period Maps 3 , Q) and H 5 (X 5 , Q) have dimension 20 and carry Hodge structures of Both H 3 (X10 10 weight 1. This gives rise to period maps

℘3 : X103 → A10

and ℘5 : X105 → A10 .

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By [3], the general fibers of ℘3 are unions of smooth proper surfaces that come in pairs: 0

3 ) by a fixed-point-free involution, and • FX 3 , isomorphic to the quotient of C 2 (X10 10 also to Sing(ZX∨3 ). 10

3 • FX3 , the analogous surface for any “line-transform” of X10 . 10

In particular, by Proposition 4.1, there is a commutative diagram: X3

epw3

E PW

E PW /duality

℘ ℘¯

℘3

A10 , where the map ℘¯ is generically finite (presumably birational) onto its image. Since 3 ) is isomorphic to the Albanese variety of the surface C 0 (X 3 ) [13], the map ℘ J(X10 2 10 is defined by sending the class of a general EPW sextic Z to the Albanese variety of the canonical double cover of its singular locus (the resulting principally polarized abelian varieties are isomorphic for Z and Z ∨ ). Theorem 4.2. The image of ℘3 and the image of ℘5 have same closures. Proof. This is proved using a double Gushel degeneration (see Sect. 4.1): with the notation above, one proves that intermediate Jacobian J(Tˆ0 ) is still a 10-dimensional principally polarized abelian variety which is a limit of intermediate Jacobians of 5 Fano fivefolds of type X10 , hence belongs to the closure of Im(℘5 ). Since, by Lefschetz theorem, we have H 5 (W54 , Q) = H 3 (W54 , Q) = 0, the follow3 ing lemma implies that the intermediate Jacobians J(Tˆ0 ) and J(X10 ) are isomorphic. 3 This proves already that the image of ℘ is contained in the closure of the image of ℘5 . Lemma 4.3. Let T and W be smooth projective varieties. Assume H k (W, Q) = H k−2 (W, Q) = 0. Let π : T → W be a conic bundle with smooth irreducible discriminant divisor X ⊂ W . Assume further that π −1 (X) is reducible. There is an isomorphism of polarized Hodge structures H k (T, Z)/tors H k−2 (X, Z)/tors. To finish the proof of the theorem, we prove, by computing the kernel of its differential, that the fibers of ℘5 have dimension at least 5. Question 4.4. Is ℘5 equal to ℘◦ epw5 ? Question 4.5. Can one extend ℘ to the GIT compactification of E PW studied in [20], with values in a suitable compactification of A10 ?


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Acknowledgements O. Debarre is part of the project VSHMOD-2009 ANR-09-BLAN-0104-01. These are notes from a talk given at the conference “Geometry Over Non-Closed Fields” funded by the Simons Foundation, whose support is gratefully acknowledged.

References 1. BEAUVILLE, A. AND DONAGI , R., La variété des droites d’une hypersurface cubique de dimension 4, C. R. Acad. Sci. Paris Sér. I Math. 301, 703–706, (1985). 2. CLEMENS, C. H. AND GRIFFITHS, P. A., The intermediate Jacobian of the cubic threefold, Ann. of Math. 95, 281–356, (1972). 3. DEBARRE, O., ILIEV, A., AND M ANIVEL, L., On the period map for prime Fano threefolds of degree 10, J. Algebraic Geom. 21, 21–59, (2012). 4. DONAGI , R. AND M ARKMAN , E., Spectral covers, algebraically completely integrable Hamiltonian systems, and moduli of bundles, In: Francaviglia, M. (ed.) et al., Integrable systems and quantum groups, CIME Lectures, Italy, June 14–22, 1993. Lect. Notes Math. 1620, 1–119, Springer-Verlag, Berlin, 1996. 5. DE JONG , A. J. AND STARR , J., Cubic fourfolds and spaces of rational curves, Illinois J. Math. 48, 415–450, (2004). 6. HARRIS, J., ROTH , M., AND STARR , J., Abel-Jacobi maps associated to smooth cubic threefolds, arXiv:math/0202080, (2002). 7. HARRIS, J., ROTH , M., AND STARR , J., Rational curves on hypersurfaces of low degree, J. Reine Angew. Math. 571, 73–106, (2004). 8. ILIEV, A. AND M ANIVEL, L., Fano manifolds of degree 10 and EPW sextics, Ann. Sci. École Norm. Sup. 44 393–426, (2011). 9. ILIEV, A. AND M ARKUSHEVICH , D., The Abel-Jacobi map for a cubic threefold and periods of Fano threefolds of degree 14, Doc. Math. 5 23–47, (2000). 10. KUZNETSOV, A., Derived category of a cubic threefold and the variety V14 , (in Russian) Tr. Mat. Inst. Steklova 246 (2004), Algebr. Geom. Metody, Svyazi i Prilozh., 183–207; English translation in Proc. Steklov Inst. Math. 246, 171–194, (2004). 11. KUZNETSOV, A. AND M ARKUSHEVICH , D., Symplectic structures on moduli spaces of sheaves via the Atiyah class, J. Geom. Phys. 59, 843–860, (2009). 12. LAZA , R., The moduli space of cubic fourfolds via the period map, Ann. of Math. 172, 673– 711, (2010). 13. LOGACHEV, D., Fano threefolds of genus 6, Asian J. Math. 16, 515–560, (2012). 14. M ARKUSHEVICH, D. AND TIKHOMIROV, A., The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold, J. Algebraic Geom. 10, 37–62, (2001). 15. M ARKUSHEVICH, D. AND TIKHOMIROV, A., Symplectic structure on a moduli space of sheaves on the cubic fourfold, Izv. Ross. Akad. Nauk Ser. Mat. 67, 131–158, (2003). 16. O’GRADY, K., Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics, Duke Math. J. 134, 99–137, (2006). 17. PROKHOROV , YU ., Rationality constructions of some Fano fourfolds of index 2, Moscow University Mathematics Bulletin 48, 32–35, (1993). 18. ROTH , L., Algebraic varieties with canonical curve sections, Ann. Mat. Pura Appl. (4) 29, 91–97, (1949). 19. SEMPLE, J. On quadric representations of the lines of four-dimensional space, Proc. London Math. Soc. 30, 500–512, (1930). 20. O’GRADY, K., Moduli of double EPW-sextics, arXiv:1111.1395, (2011). 21. VOISIN , C., Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal, J. Algebraic Geom. 22, 141–174, (2013).

Uniruledness Criteria and Applications Stefan Kebekus

Mathematics Subject Classification codes (2010): 14F10, 14C20

1 Introduction These are notes taken from an overview talk given at the 2012 Simons Symposium Geometry Over Nonclosed Fields. The talk focused on uniruledness criteria and their applications. One of the points made in the talk is that uniruledness criteria can be useful even in the study of varieties that are known a priori not to contain any rational curves. Two examples were given to illustrate this point: • Assume we are given two normal complex projective varieties X and Y , where Y is not uniruled. Perhaps somewhat surprisingly, uniruledness criteria apply to show that those components of Hom(X,Y ) whose points correspond to surjective morphisms are Abelian varieties. If Y is smooth, their dimension can be bounded in terms of the Kodaira dimension κ (Y ). • Given a smooth family f : X → Y of canonically polarised manifolds over a smooth quasi-projective base manifold Y , uniruledness criteria help to bound the variation of f in terms of the (logarithmic) Kodaira dimension of Y . The first item is discussed in detail in Sect. 2 below. We have chosen not to include any discussion of the moduli problems in this text because there are several surveys available, including [17, 21, Sect. 5].

S. Kebekus () Mathematisches Institut, Albert-Ludwigs-Universität Freiburg Eckerstraße 1, 79104 Freiburg im Breisgau, Germany e-mail: [email protected] F. Bogomolov et al. (eds.), Birational Geometry, Rational Curves, and Arithmetic, DOI 10.1007/978-1-4614-6482-2__7, © Springer Science+Business Media New York 2013

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Next, it was shown how uniruledness criteria help to study the geometry of varieties that are uniruled or even rationally connected. In essence, we aim to decompose a given variety into parts depending on “density” of rational curves. More precisely, given a polarised projective manifold X, we show that the Harder– Narasimhan filtration of the tangent sheaf TX induces a sequence of increasingly fine “partial rational quotients”. This construction will be discussed in Sect. 3. We list a number of relatively new results pertaining to the dependence of the partial rational quotients on the choice of the polarisation and to the relation between the partial rational quotients and the minimal model program. In spite of the progress made, a full understanding of the geometric meaning of the partial rational quotients is currently still missing.

1.1 Miyaoka’s Uniruledness Criterion and Other Criteria The prototypical uniruledness criterion that we will be using most in this survey is Miyaoka’s generic semipositivity. Theorem 1.1 (Miyaoka’s uniruledness criterion, [24, Cor. 8.6]). Let X be a normal complex projective variety of dimension dim X ≥ 2 and C ⊂ X a general complete intersection curve. Then X is smooth along C, and either X is uniruled or ΩX1 |C is a nef vector bundle. Miyaoka’s criterion was shown using the Harder–Narasimhan filtration of TX to exhibit a foliation along which the curve C can be deformed, in order to construct rational curves via bend-and-break. Over C, an elementary introduction to “deformation along a subsheaf” is found in the expository papers [15, 18]. The key ingredient in the construction of the partial rational quotients is a criterion used to guarantee that leaves of a foliation are algebraic and rationally connected, Theorem 3.3. This can be seen as a generalisation and improvement of Miyaoka’s Theorem 1.1.

1.2 Other Uniruledness Criteria For completeness’ sake, we mention a few other important criteria which were not discussed in the talk for lack of time. The most relevant is probably the result of Boucksom–Demailly–P˘aun–Peternell, [2], which asserts that the canonical bundle of any projective manifold is pseudo-effective, unless the manifold is uniruled. This result has recently been generalised by Campana–Peternell to higher tensor powers of sheaves of differentials.

Uniruledness Criteria and Applications

149

Theorem 1.2 (Pseudo-effectivity of quotients of pluri-forms, [5, Theorem 0.1]). Let X be a complex, projective manifold, m ∈ N and (ΩX1 )⊗m → F a torsion free coherent quotient. If X is not uniruled, then detF is pseudo-effective. In particular, if X is not uniruled, then ωX is pseudo-effective. Theorem 1.2 has been generalised to reflexive differentials on singular varieties; see [8, Prop. 5.6]. It plays a crucial role in recent generalisations of the Beauville– Bogomolov decomposition to varieties with trivial Chern class and singularities as they appear in minimal model theory. We refer to the paper [8] for more details and for an overview of this set of problems.

1.3 Disclaimer The talk given at the Simons Symposium aimed to survey how uniruledness criteria are used in algebraic geometry today. It did not contain any new results. There exists some overlap between this overview and other survey papers; see for instance [16, 21]. The results discussed at the end of Sect. 3 have, however, not all been presented in public yet. Throughout this survey, we work over the complex number field.

2 Applications to Non-uniruled Spaces: Description of the Hom-Scheme Let f : X → Y be a surjective morphism between normal complex projective varieties. A classical problem of complex geometry asks for a criterion to guarantee the (non-)existence of deformations of the morphism f , with X and Y fixed. More generally, one would like to understand the geometry of the connected component Hom f (X,Y ) ⊂ Hom(X,Y ) of the space of morphisms. Somewhat surprisingly, using Miyaoka’s uniruledness criterion, Theorem 1.1, we obtain a very precise description of Hom f (X,Y ) if the target manifold Y is not uniruled. Theorem 2.1 (Description of the Hom-scheme, [9, Thm 1.2]). Let f : X → Y be a surjective morphism between normal complex projective varieties, and assume that Y is not uniruled. Then Hom f (X,Y ) is an Abelian variety. Remark 2.2. If dim Hom f (X,Y ) = 0, then Theorem 2.1 implies that Hom f (X,Y ) is a reduced point. Formulated in different terms, Theorem 2.1 implies that deformation-rigid morphisms are in fact infinitesimally rigid.

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In fact, more is true. Up to a finite covering, the scheme Hom f (X,Y ) is the maximal connected subgroup of the automorphism group of a variety Z that admits a quasi-étale map1 to Y . Since the existence of an Abelian variety in the automorphism group has strong implications for many other invariants, one obtains the following results as nearly immediate corollaries. Corollary 2.3 (Deformations and Kodaira dimension, [9, Corollary. 1.3]). In the setup of Theorem 2.1, if Y is smooth and has non-negative Kodaira dimension κ (Y ) ≥ 0, then dim Hom f (X,Y ) ≤ dimY − κ (Y ). Corollary 2.4 (Deformations and fundamental group, [9, Corollary. 1.5]). In the setup of Theorem 2.1, let Y be a projective manifold which is not uniruled. If π1 (Y ) is finite, then Hom f (X,Y ) is a reduced point. Corollary 2.5 (Deformations and top Chern class, [9, Corollary. 1.6]). In the setup of Theorem 2.1, let Y be a projective n-dimensional manifold which is not uniruled. If cn (Y ) = 0, then Hom f (X,Y ) is a reduced point.

2.1 Idea of Proof We show only the much simpler assertion that Hom f (X,Y ) is smooth, that is, that every infinitesimal deformation of f is induced by a holomorphic one-parameter family of morphisms. To this end, assume we are given an infinitesimal deformation σ ∈ H 0 (X, f ∗ TY ). To avoid technical difficulties and quickly come to the core of the argument, we make the following extra assumptions: Additional Assumption 2.6. The varieties X and Y are smooth and the morphism f is finite. In particular, f∗ OX is a locally free sheaf on Y . 2.1.1 Step 1: The Composition Morphism Recall that the automorphism group of the complex variety Y is a complex Lie group. Composing the morphism f with elements of its maximal connected subgroup Aut0 (Y ), we obtain an injective composition morphism f ◦ : Aut0 (Y ) → Hom f (X,Y ). g → g◦ f

(1)

The tangent spaces to Aut0 (Y ) and Hom f (X,Y ) are well understood. The derivative of f ◦ at the identity e ∈ Aut0 (Y ) thus yields a diagram

1 Quasi-étale

= finite and étale in codimension one.


151 d f ◦ |e

TAut0 (Y ) |e

THom | f

∼ =

(2)

∼ =

H 0 (Y, TY )

pull-back

H 0 (X, f ∗ TY ).

The horizontal arrows in Diagram (2) are clearly injective. Since every infinitesimal deformation which is in the image of d f ◦ |e is clearly induced by a one-parameter group in Aut0 (Y ), we need to show that the infinitesimal deformation σ ∈ H 0 (X, f ∗ TY ) is obtained as the pull-back of a vector field on Y . We argue by contradiction and assume this is not the case. Additional Assumption 2.7. The infinitesimal deformation σ ∈ THom | f is not the pull-back of a vector field on Y .

2.1.2 Step 2: The Splitting of f* OX and Étale Covers Fix an ample divisor H ∈ Pic(Y ), and let C ⊂ Y be an associated general complete intersection curve. We recall a few facts about the push-forward sheaf f∗ OX that are relevant in our context. Fact 2.8 (Description of f* OX , [23] or [27, Theorem A]). The trace map tr : f∗ OX → OY yields a natural splitting f∗ OX ∼ = OY ⊕ E ∗ , where E is a locally free sheaf whose restriction E |C is nef. The following conditions are equivalent: 1. The morphism f is branched. 2. The morphism f is branched and its branch locus intersects the curve C. 3. The degree of the restricted sheaf is positive, deg(E |C ) > 0. Proposition 2.9 (Ampleness of E |C , [9]). The restricted sheaf E |C is either ample or there exists a non-trivial factorisation f

X

α , finite

Z

β , étale

Y .

Proof (Idea of proof). If the restricted sheaf E |C is ample, we are done. If not, then there exists a term A ⊂ E in the Harder–Narasimhan filtration of E such that the following holds:

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• The restriction A |C is an ample sub-vectorbundle of E |C . E | • The quotient C A | has degree zero. C

The second point follows from the fact that E |C is nef, so that quotients will always have semi-positive degrees. Dualising, we find a sheaf B ⊂ E ∗ of slope zero whose quotient has negative slope. Using that there is no map from a semistable sheaf of high slope to one of smaller slope, one observes that the subsheaf B ⊕OY ⊆ f∗ OX is closed under multiplication, hence forms a sheaf of OY -algebras. The variety Z is obtained as Spec of that sheaf. To continue the proof of Theorem 2.1, assume we are in a situation where E |C is not ample. Proposition 2.9 will then give a non-trivial decomposition of f , and it follows from the étaleness of β that the infinitesimal deformation σ can be interpreted as an infinitesimal deformation of the morphism α , that is,

σ ∈ H 0 (X, f ∗ TY ) = H 0 (X, α ∗ TZ ). To prove that σ comes from a one-parameter family of deformations of f , it clearly suffices to show that σ comes from a one-parameter family of deformations of α — composing with β will then give the deformation of f . Replacing Y by Z, iterating the argument, and using that f is of finite degree, we can therefore assume without loss of generality that the following holds: Additional Assumption 2.10. The restricted sheaf E |C is ample.

2.1.3 Step 3: End of Proof To end the proof, use the projection formula to obtain a decomposition H 0 X, f ∗ TY = H 0 Y, f∗ ( f ∗ TY ) = H 0 Y, TY ⊕ H 0 Y, E ∗ ⊗ TY = H 0 Y, TY ⊕ HomY E , TY . Consider the associated decomposition of σ . Since σ ∈ THom | f is not the pull-back of a vector field on Y by Assumption 2.7, we obtain a non-trivial morphism from E to TY . Using that E |C is ample, this implies that TY |C has a positive subsheaf. Miyaoka’s criterion, Theorem 1.1, therefore applies to show that Y is uniruled. This is in clear contradiction to the assumptions made in Theorem 2.1 and therefore ends the proof.


153

2.2 Further Results Questions 2.2.1 Refinement of Stein Factorisation The methods used to prove Theorem 2.1 show more than claimed above. With a little more work, the technique using the Harder–Narasimhan filtration of f∗ OX can be used to show that there exists a canonically defined refinement of Stein factorisation for any surjective morphism. The following definition summarises its main properties: Definition 2.11 (Maximally étale factorisation). Let f : X → Y be a surjective morphism between normal projective varieties, and assume we are given a factorisation f

X

Z

α

β

Y.

(3)

where β is quasi-étale. We say that the factorisation (3) is maximally étale if the following universal property holds: for any factorisation f = β ◦ α , where β : Z → Y is quasi-étale, there exists a morphism γ : Z → Z such that the following diagram commutes: f

X

Z

α

β

Y

γ

X

α

Z

β

Y.

f

Theorem 2.12 (Existence of a maximally étale factorisation, [20, Theorem 1.4]). Let f : X → Y be a surjective morphism between normal projective varieties. Then there exists a maximally étale factorisation. Remark 2.13. The universal properties of the maximally étale factorisation immediately imply that the maximally étale factorisation is unique up to unique isomorphism and behaves extremely well under deformations of f ; see [20, Sect. 1.B, 4]. The natural refinement of Stein factorisation mentioned above is now an immediate corollary.

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Corollary 2.14 (Refinement of Stein factorisation, [20, Sect. 1.A]). Let f : X → Y be a surjective morphism between normal projective varieties. Then there exists a canonical refinement of Stein factorisation as follows: f

X

α1 , conn. fibres

W

α2 , finite

Z

β , quasi-étale

Y,

where β comes from the maximally étale factorisation of f .

2.2.2 Deformations of Morphisms to Uniruled Varieties If f : X → Y is a surjective morphism onto a rationally connected manifold, there is usually little we can say about the associated connected component of the Homscheme; partial results are found in [10, 11, 13, 14]. With some extra work one can show, however, that the MRC quotient of Y induces a decomposition of Hom f (X,Y ) into an Abelian variety and a space that parametrises deformations over the MRC quotient. We refer to [20, Sect. 1.C] for a precise formulation of the somewhat involved result.

2.2.3 Open Problems We conjecture that Theorem 2.1 and its corollaries hold true when Y is a compact Kähler manifold of non-negative Kodaira dimension. Our proof needs the projectivity assumption because it employs Miyaoka’s characterisation of uniruledness, Theorem 1.1.

3 Applications to Uniruled Manifolds: Partial Rational Quotients Roughly speaking, the uniruledness criteria of Mori and Miyaoka can be summarised as “positivity properties of TX imply the existence of rational curves on X”. However, the precise relation between positivity and the geometric properties of the rational curves found by these criteria remains unclear. • Does “more positivity” give “more rational curves”? • If the tangent bundle contains a particularly positive subsheaf F ⊆ T , can we find rational curves whose geometry relates to F ? If so, how many?


155

Building on work of Miyaoka and Bogomolov–McQuillan, the present section aims to clarify at least some aspects of this relation. Given a projective manifold X, we will see that the terms in the Harder–Narasimhan filtration of TX induce a canonically defined sequence of partial rational quotients. As of today, a precise geometric description of these partial rational quotients and their dependence on the choice of the polarisation is missing. We discuss some evidence which points to a strong connection between the partial rational quotients and the minimal model program.

3.1 Rationally Connected Foliations The key result of this section is a uniruledness criterion for foliated varieties. The following definition will be used: Definition 3.1 (Foliation, singular foliation). Let X be a normal variety and F a coherent subsheaf of the tangent sheaf TX . Let X ◦ ⊆ X be the maximal open set where X is smooth and F is a sub-vectorbundle of TX . We call F a (singular) foliation if the following two conditions hold: 1. The sheaf F is a saturated subsheaf of TX . In other words, the quotient TX /F is torsion-free. 2. The sheaf F is integrable, that is, the sub-vectorbundle F |X ◦ ⊆ TX |X ◦ is closed under Lie-bracket. The foliation F is regular if X ◦ = X. A leaf of F is a connected, locally closed holomorphic submanifold L ⊂ X ◦ such that TL = F |L . A leaf is called algebraic if it is open in its Zariski closure. Remark 3.2. In the setting of Definition 3.1, let L ⊂ X ◦ be an algebraic leaf and L ⊂ X ◦ be its Zariski closure. Then L is again a leaf. The main result of this section asserts that positivity properties of F imply algebraicity of the leaves and rational connectedness of their closures. In particular, it gives a criterion for a manifold to be covered by rational curves. Theorem 3.3 (Rationally connected foliations, [3, 22]). Let X be a normal complex projective variety, C ⊂ X a complete curve which is entirely contained in the smooth locus Xreg , and F ⊂ TX a (possibly singular) foliation which is regular in a neighbourhood of C. The restriction F |C is then a vector bundle on C. If F |C is ample, and if x ∈ C is any point, then any leaf through x is algebraic. If x ∈ C is general, the Zariski closure of any leaf through x is a rationally connected subvariety of X.

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The statement appeared first in the preprint [3] by Bogomolov and McQuillan, and the first full proof was given in [22]. Methods used include a criterion of Hartshorne for a foliation to have algebraic leaves, the result of Graber–Harris– Starr, and bend-and-break arguments relying on a vanishing theorem in positive characteristic. Remark 3.4. In Theorem 3.3, if x ∈ C is any point, it is not generally true that the closure of a leaf through x is rationally connected. This was wrongly claimed in [3] and in the first preprint versions of [22]. The classical Reeb stability theorem for foliations [4, Theorem IV.3], the fact that rationally connected manifolds are simply connected [6, Corollary 4.18], and the openness of rational connectedness [19, Corollary 2.4] immediately yield the following2: Theorem 3.5 (Rationally connected regular foliations, [22, Theorem 2]). In the setup of Theorem 3.3, if F is regular and L ⊂ X any leaf, then L is algebraic and its closure is a rationally connected submanifold. Remark 3.6. In fact, a stronger statement holds, guaranteeing that most leaves are algebraic and rationally connected if there exists a single leaf whose closure does not intersect the singular locus of F ; see [22, Theorem 28]. The following characterisation of rational connectedness is a straightforward corollary of Theorem 3.3. Corollary 3.7 (Criterion for rational connectedness). Let X be a complex projective variety and let f : C → X be a curve whose image is contained in the smooth locus of X. If TX |C is ample, then X is rationally connected.

3.2 Producing Foliations Using the Harder–Narasimhan Filtration of TX The usefulness of Theorem 3.3 depends on our ability to construct geometrically relevant foliations to which the theorem can be applied. In his work on uniruledness criteria and deformations along a foliation, Miyaoka noted that the subsheaves of TX which appear in the Harder–Narasimhan filtrations often satisfy this property. Proposition 3.8 (Foliations from Harder–Narasimhan filtrations). Let X be a normal n-dimensional projective variety and H = {H1 , . . . , Hn−1 } ∈ Pic(X)

2 Höring

has independently obtained similar results, [12].


157

a polarisation by ample line bundles. Consider the associated Harder–Narasimhan filtration of TX , 0 = F0 F1 F2 · · · Fk−1 Fk = TX and set

imax = max 0 < i < k | μH Fi Fi−1 > 0 ∪ {0}.

Assume that imax > 0. For any index 0 < i ≤ imax , Fi is a foliation in the sense of Definition 3.1. Proof (Idea of proof). The sheaf morphism induced by the Lie-bracket, Fi × Fi → TX , is by no means OX -bilinear. An elementary computation shows, however, that the induced map to the quotient, N : Fi × Fi → TX Fi , is in fact bilinear. The claim then quickly follows from the well-known fact that in characteristic zero, semistability and slope are well behaved under tensor product and that there is no morphism from a semistable sheaf of high slope to one of lower slope. The map N must thus be trivial. Remark 3.9. If X is Q-Fano, then imax = k. Corollary 3.10 (Rational connectedness of foliations coming from HNFs). In the setting of Proposition 3.8, the leaves of the foliation Fi are algebraic. The general leaf is rationally connected. Proof. Let C ⊂ X be a general complete intersection curve for the polarisation H. Then C is smooth and entirely contained in the locus where both X is smooth and Fi is regular. The restriction Fi |C is thus an ample vector bundle on C. Theorem 3.3 applies and yields both algebraicity and rational connectedness of leaves that intersect C. The claim holds for all leaves because deformations of C dominate X.

3.3 Applications: Sequences of Partial Rational Quotients Corollary 3.8 allows to construct a rational map X Chow(X) by mapping general points of X to the closures of the associated leaves. In summary, we see that every polarised manifold is canonically equipped with a sequence of increasingly fine partial rational quotients.

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Corollary 3.11 (Partial rational quotients associated to a polarisation). In the setting of Proposition 3.8, there exists a commutative diagram of rational maps, X q1

X

···

Q2

···

Qimax

(4)

X

qimax

q2

Q1

X

MRC Quotient

Q,

with the following additional property: if x ∈ X is a general point and Fi the closure of the qi -fibre through x, then Fi is rationally connected and its tangent space at x is exactly TFi |x = Fi |x . While many people working in the field share the feeling that the Harder– Narasimhan filtrations should measure “density of rational curves with respect to the given polarisation”, no convincing results have been obtained in this direction. The geometric meaning of the canonically given diagram (4) is not fully understood. Section 3.4 discusses the known results and poses a few natural questions and conjectures.

3.4 Interpretation of the Partial Rational Quotients. Open Problems The first questions that came to our mind when we first saw Corollary 3.11 and Diagram (4) were probably the following: Question 3.12. Let X be a uniruled projective manifold or variety, equipped with a polarisation. Is the MRC fibration equal to the map qimax ? Question 3.13. Do the canonically defined morphisms qi carry a deeper geometric meaning? Is Diagram (4) characterised by universal properties? To what extent does it depend on the polarisation chosen? Today these questions can be answered in special cases. We survey the known results in the remaining part of this section. 3.4.1 The MRC Quotient as a Rationally Connected Foliation Perhaps somewhat surprisingly, the relative tangent sheaf of the MRC quotient does generally not appear as one of the terms of the Harder–Narasimhan filtration, unless X is a surface and the polarisation is particularly well chosen. Theorem 3.14 (MRC quotient not always equal to qimax , [7, Sect. 3]). There exist elementary examples of polarised surfaces where the MRC fibration is not equal to the map qimax .


159

Theorem 3.15 (MRC quotient equals qimax for good surface polarisation, [25, Theorem 3.8]). If X is a uniruled surface, then there exists a polarisation such that the MRC fibration equals the map qimax . In higher dimensions, we do not expect an analogue of Theorem 3.15 to hold true. There are, however, positive results when one is willing to generalise the notion of “polarisation” to include “polarisations by movable curve classes”, as defined below. Fact 3.16 (Polarisations by movable curve classes, [26, Sect. 3]). Let X be a projective manifold and α ∈ Mov(X) ⊂ N 1 (X)R a non-trivial numerical curve class, contained in the closure of the movable cone. Define the slope of a coherent sheaf F as α .[det F ] ∈ R. μα (F ) := rank F With this definition, a Harder–Narasimhan filtration exists exactly as in the case of an ample polarisation. The obvious analogue of Proposition 3.8 holds. Warning 3.17. As far as we know, there is no analogue of the Mehta–Ramanathan theorem in the setting of Fact 3.16. Theorem 3.18 (MRC fibration equals qimax for good movable polarisation, [28, Theorem 1.1]). Let X be a uniruled complex projective manifold, and let F ⊆ TX denote the foliation associated with its MRC fibration. Then there exists a numerical curve class α , contained in the interior of the movable cone such that the following holds: 1. The class α is represented by a reduced movable curve C such that F |C is ample. 2. The sheaf F appears as a term in the Harder–Narasimhan filtration of TX with respect to α . Remark 3.19. If X is a surface, then the interior of the movable cone equals the cone of general complete intersection curves. Question 3.20. Let X be a uniruled projective manifold. Is there an ample polarisation such that the MRC fibration equals the map qimax ? Question 3.21. Is there an analogue of Corollary 3.10, “Rational connectedness of foliations coming from HNFs”, when using movable curve classes to define a Harder–Narasimhan filtration? What if X is singular?

3.4.2 Decomposition of the Cone of Movable Curve Classes Let X be a projective manifold. Given a non-trivial numerical curve class α ∈ Mov(X) ⊂ N 1 (X)R , we are interested in the set of classes whose induced Harder– Narasimhan filtration of the tangent bundle agrees with that of α , Δα := β ∈ Mov(X) | HNF(α , TX ) = HNF(β , TX ) .

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The decomposition of the movable cone Mov(X) into disjoint subsets of the form Δα , called “destabilising chambers”, was studied in the 2010 Freiburg thesis of Sebastian Neumann. He obtained the following two results. Theorem 3.22 (Decomposition of the moving cone, [26, Theorem 3.3.4, Proposition 3.3.5]). Let X be a projective manifold. The destabilising chambers are convex cones whose closures are locally polyhedral in the interior of Mov(X). The decomposition of the moving cone is locally finite in the interior of Mov(X). If we assume additionally that the cone of movable curves is polyhedral, then the chamber structure is finite. Remark 3.23 (Movable cone of Fano manifolds, [1, Cor. 1.2]). If X is a Fano manifold, then the closed cone of movable curves is polyhedral. Theorem 3.24 (Relation to the minimal model program, [26, Theorem 4.1]). Let X be a Fano manifold of dimension three and α ∈ Mov(X) ⊂ N 1 (X)R a nontrivial numerical curve class with associated Harder–Narasimhan filtration 0 = F0 F1 F2 · · · Fk−1 Fk = TX . Then each term Fi is the relative tangent sheaf of a (not necessarily elementary) Mori fibration. Remark 3.25. The proof of Theorem 3.24 relies on the fine classification of Fano threefolds. It would be very interesting to understand the relation between the Harder–Narasimhan filtrations and minimal model theory in much greater detail. Acknowledgements The author would like to thank the organisers and the Simons Foundation for an unusually fruitful conference week. He would like to thank the other participants for countless discussions and exchange of ideas. Clemens Jörder and Patrick Graf have kindly read a first version of this paper. The author was supported in part by the DFG-Forschergruppe 790 “Classification of Algebraic Surfaces and Compact Complex Manifolds”.

References 1. CAROLINA ARAUJO , The cone of pseudo-effective divisors of log varieties after Batyrev, Math. Z. 264, no. 1, 179–193, (2010). ˘ , AND THOMAS PETER 2. SEBASTIEN BOUCKSOM, JEAN -PIERRE DEMAILLY, M IHAI PAUN NELL, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, May 2004, arXiv:math/0405285. 3. FEDOR A. BOGOMOLOV AND M ICHAEL L. M C QUILLAN , Rational curves on foliated varieties, February 2001, IHES Preprint. 4. CÉSAR CAMACHO AND ALCIDES LINS NETO , Geometric theory of foliations, Birkhäuser Boston Inc., Boston, MA, 1985, Translated from the Portuguese by Sue E. Goodman. 5. FRÉDÉRIC CAMPANA AND THOMAS PETERNELL, Geometric stability of the cotangent bundle and the universal cover of a projective manifold, Bull. Soc. Math. France 139, no. 1, 41–74, (2011).


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6. OLIVIER DEBARRE, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. 7. THOMAS ECKL, Lower bounds for Seshadri constants, Math. Nachr. 281, no. 8, 1119–1128, (2008). 8. DANIEL GREB , STEFAN KEBEKUS, AND THOMAS PETERNELL, Singular spaces with trivial canonical class, arXiv:1110.5250, October 2011. To appear in Minimal models and extremal rays – proceedings of the conference in honor of Shigefumi Mori’s 60th birthday, Advanced Studies in Pure Mathematics, Kinokuniya Publishing House, Tokyo. 9. JUN -M UK HWANG , STEFAN KEBEKUS, AND THOMAS PETERNELL, Holomorphic maps onto varieties of non-negative Kodaira dimension, J. Algebraic Geom. 15, no. 3, 551–561, (2006). 10. JUN -M UK HWANG AND NGAIMING M OK , Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles, J. Algebraic Geom. 12, no. 4, 627–651, (2003). , Birationality of the tangent map for minimal rational curves, Asian J. Math. 8, no. 1, 11. 51–63, (2004). 12. ANDREAS HÖRING , Uniruled varieties with split tangent bundle, Math. Z. 256, no. 3, 465–479, (2007). 13. JUN -M UK HWANG , Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1, Ann. Inst. Fourier (Grenoble) 57, no. 3, 815–823, (2007). , Deformation of holomorphic maps onto the blow-up of the projective plane, Ann. Sci. 14. École Norm. Sup. (4) 40, no. 1, 179–189, (2007). 15. CLEMENS JÖRDER AND STEFAN KEBEKUS, Deformations along subsheaves II, Impanga lecture notes Contributions of Algebraic Geometry. Edited by Piotr Pragacz. EMS Series of Congress Reports, EMS Publishing House, 2012. DOI:10.4171/114. arXiv:1103.5630. 16. STEFAN KEBEKUS, Holomorphe Abbildungen auf Mannigfaltigkeiten mit nicht-negativer Kodaira-Dimension, Mathematisches Institut Georg-August-Universität Göttingen Seminars 2003/2004 (Y. Tschinkel, ed.), Universitätsverlag der Georg-August-Universität Göttingen, 2004, 157–166. , Differential forms on singular spaces, the minimal model program, and hyperbolicity 17. of moduli stacks, preprint arXiv:1107.4239. To appear in the “Handbook of Moduli, in honour of David Mumford”, to be published by International press, editors Gavril Farkas and Ian Morrison, July 2011. 18. STEFAN KEBEKUS, STAVROS KOUSIDIS, AND DANIEL LOHMANN , Deformations along subsheaves, L’Enseignement MathÃl’matique (2) 56, no. 3–4, 287–313, (2010). arXiv: 0905.2749. 19. JÁNOS KOLLÁR , YOICHI M IYAOKA , AND SHIGEFUMI M ORI , Rational connectedness and boundedness of Fano manifolds, J. Differential Geom. 36, no. 3, 765–779, (1992). 20. STEFAN KEBEKUS AND THOMAS PETERNELL, A refinement of Stein factorization and deformations of surjective morphisms, Asian J. Math. 12, no. 3, 365–389, (2008). 21. STEFAN KEBEKUS AND LUIS SOLÁ CONDE, Existence of rational curves on algebraic varieties, minimal rational tangents, and applications, Global aspects of complex geometry, Springer, Berlin, 2006, 359–416. 22. STEFAN KEBEKUS, LUIS SOLÁ CONDE, AND M ATEI TOMA , Rationally connected foliations after Bogomolov and McQuillan, J. Algebraic Geom. 16, no. 1, 65–81, (2007). 23. ROBERT LAZARSFELD , A Barth-type theorem for branched coverings of projective space, Math. Ann. 249, no. 2, 153–162, (1980). 24. YOICHI M IYAOKA , Deformations of a morphism along a foliation and applications, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 245–268. 25. SEBASTIAN NEUMANN , Rationally connected foliations on surfaces, Doc. Math. 14, 157–165, (2009). , A decomposition of the moving cone of a projective manifold according to the harder26. narasimhan filtration of the tangent bundle, Ph.D. thesis, Albert-Ludwigs-UniversitÃd’t Freiburg, March 2010, http://www.freidok.uni-freiburg.de/volltexte/7287,

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The Cone of Curves of K3 Surfaces Revisited Sándor J. Kovács

1 Introduction The following theorem was proved in [4] over the complex numbers. It turns out that the proof given there works with very small adjustments in arbitrary characteristic. The main difference is that while in the original article we worked in a real homology group of the surface in question, here everything takes place in the group of R-cycles modulo numerical equivalence. The arguments are essentially the same. As already noted the purpose of this note is to verify the above statement, that is, to prove the following. Theorem 1.1. Let X be a K3 surface of Picard number at least three over an algebraically closed field of arbitrary characteristic. Then one of the following mutually exclusive conditions are satisfied: (1.1.1) X does not contain any curve of negative self-intersection. (1.1.2) NE(X) = ∑ R+ where the sum runs over the classes of all smooth rational curves on X. Remark 1.2. To cover all cases one would also need to consider K3 surfaces with Picard number less than 3. The case of Picard number 1 is trivial and the case of Picard number 2 is handled in Corollary 3.2. DEFINITIONS AND NOTATION 1.3. Let k be an algebraically closed field of arbitrary characteristic. Everything will be defined over k. A K3 surface is a smooth projective surface X such that ωX OX and h1 (X, OX ) = 0.

S.J. Kovács () Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, USA ˜ e-mail: [email protected]; http://www.math.washington.edu/kovacs F. Bogomolov et al. (eds.), Birational Geometry, Rational Curves, and Arithmetic, DOI 10.1007/978-1-4614-6482-2__8, © Springer Science+Business Media New York 2013

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Let Z1 (X) denote the free abelian group generated by the irreducible and reduced 1-dimensional subvarieties of X. Elements of this group are called 1-cycles on X. Two 1-cycles C1 ,C2 ∈ Z1 (X) are called numerically equivalent if for any Cartier divisor D on X, the intersection numbers D · C1 and D · C2 agree. This relationship is denoted by C1 ≡ C2 . All 1-cycles numerically equivalent to the 0-cycle form a subgroup of Z1 (X), and the quotient is denoted by N1 (X)Z . By extension of scalars we define N1 (X) := N1 (X)Z ⊗Z R. The effective 1-cycles in N1 (X) generate a subsemigroup denoted by NE(X) ⊆ N1 (X). This subsemigroup is called the cone of effective curves. The closed cone of effective curves is the closure of this in N1 (X): NE(X) := NE(X) ⊆ N1 (X). For more details about the construction and basic properties of this, one should consult [2, II.4]. From now on X is assumed to be a smooth projective surface defined over k. A class ξ ∈ N1 (X) is called integral if it can be represented by a divisor on X. It is called effective (respectively ample) if it is integral and can be represented by an effective (respectively ample) divisor. A class ξ ∈ NE(X) is called extremal if it cannot be written as the sum of two incomparable classes in NE(X). The class of a smooth rational curve is called a nodal class. The set of all nodal classes is denoted by N (X). Let h be an ample class and define Q(X) := {ξ ∈ N1 (X) | ξ · h > 0, ξ · ξ = 0}. Note that by the positivity condition this is just half of a quadric cone. The convex hull of a set will be denoted by Conv. An open subset of the boundary of the cone is called circular if the cone is not locally finitely generated at any point in the open set.

2 Simple Facts Let us start with an easy, well-known consequence of the Riemann-Roch theorem: Lemma 2.1. Let a ∈ N1 (X) be an integral class. Then a · a is an even integer. If furthermore a · a ≥ −2, then either a or −a is effective. Proof. As a is integral, there exists a line bundle L representing a. By RiemannRoch 1 h0 (X, L ) − h1(X, L ) + h0(X, L ∨ ) = a · a + 2 (2.1.1) 2 and hence a · a has to be even.

The Cone of Curves of K3 Surfaces Revisited

165

If a · a ≥ −2, then the right hand side of Eq. (2.1.1) is positive, so either h0 (X, L ) > 0 or h0 (X, L ∨ ) > 0, so either a or −a is effective. Corollary 2.2. Q(X) ⊂ NE(X) and for any e ∈ Q(X) the hyperplane (e· {ξ ∈ N1 (X) | e · ξ = 0} is a supporting hyperplane of Conv(Q(X)).

= 0) =

Proof. The first statement follows directly from Lemma 2.1 and since for any ξ ∈ Conv(Q(X)) (effective) irreducible class, ξ · e ≥ 0, this also implies the second statement. Corollary 2.3. Let e, d ∈ NE(X) such that e · e = 0 and d · d > 0. Then d is in the interior of NE(X) and e · d > 0. Proof. Since d is in the interior of Q(X) and Q(X) ⊂ NE(X), it follows that d is in the interior of NE(X). Then since (e · = 0) is a supporting hyperplane, d cannot be contained in it and hence e · d > 0. Next we establish that nodal rays can only accumulate along the cone generated by Q(X). Lemma 2.4. Let {n } ⊂ N (X) be an infinite sequence of nodal classes such that R+ n

n→+∞

R+ ξ for some ξ ∈ NE(X). Then ξ · ξ = 0.

Proof. Let ∈ N (X). Then for infinitely many n ∈ N, n = , so n · ≥ 0 and hence ξ · ≥ 0. Applying this with = n yields that ξ · ξ ≥ 0. On the other hand since n · n < 0, it follows that ξ · ξ ≤ 0 as well and so we must have ξ · ξ = 0. Corollary 2.5. Let h ∈ NE(X) be an ample class and ε > 0 a real number. Define Qε (X) := {ξ ∈ N1 (X) | ξ · h = 1, ξ · ξ ≥ −ε }. Then the number of nodal classes not contained in Conv(Qε (X)) is finite. Proof. The set {ξ ∈ NE(X) | ξ · h = 1} is compact by Kleiman’s criterion [3, 1.18] and hence any infinite set contained in it has an accumulation point. By Lemma 2.4 all accumulation points have to be contained in Qε (X), which implies the desired statement. Corollary 2.6. Let ξ ∈ NE(X) be an extremal class which is not a multiple of a nodal class. Then ξ · ξ = 0. Proof. It follows from Riemann-Roch and [2, II.4.14] that NE(X) =

∑

∈N (X)

R+ + Conv(Q(X)),

(2.6.1)

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in particular ξ · ξ ≤ 0, and then for any ε > 0, NE(X) =

∑

R+ + (NE(X) ∩ Conv(Qε (X))),

(2.6.2)

∈ N (X ) ∈ Conv(Qε (X ))

where the above sum is finite by Corollary 2.5. Suppose that ξ · ξ < 0. Without loss of generality we may assume that ξ · h = 1 and choose an ε > 0 such that ξ ∈ Conv(Qε (X)). Then ξ is a multiple of a nodal class by Eq. (2.6.2).

3 Subcones Generated by Two Elements The following is a simple, but important computation. Lemma 3.1. Let e, d be effective classes such that e is indecomposable and d · d > 0. Let L be the 2-dimensional linear subspace generated by e and d in N1 (X) and C = L ∩ NE(X). Then: (3.1.1) If e · e = 0, then there exists an f ∈ C such that f is effective, e · f > 0, e and f are on opposite sides of d, and f · f = 0. (3.1.2) If e · e = −2, then there exists an f ∈ C such that f is effective, e · f > 0, e and f are on opposite sides of d, and f · f = 0 or f · f = −2. Proof. Let A = d · d, B = e · d, and C = e · e. Set f = α d − β e. Then f · f = Aα 2 − 2Bαβ + Cβ 2 . If C = e · e = 0, then B = e · d > 0 by Corollary 2.3 and hence the equation 0 = Aα 2 − 2Bαβ = α (Aα − 2Bβ ) has a positive integer solution, α = 2B, β = A such that the class f = α d − β e has f · f = 0. Since α d = f + β e, f and e lie on opposite sides of d and it also follows that f is effective by Lemma 2.1 and since e · f > 0. If C = e · e = −2, then set x = Bα /2 + β , y = α /2, and N = 2A + B2. Then ! f · f = Aα 2 −2Bαβ −2β 2 = −2 (Bα /2+β )2 −(2A+B2 )(α /2)2 = −2(x2 − Ny2 ). Now if N is a square, then as above there are two effective solutions for f · f = 0 and they are on opposite sides of d and hence one of them is on the side of d opposite to e. If N is not a square then finding an f with f · f = −2 is equivalent to solving Pell’s equation x2 − Ny2 = 1 [1, 17.5.2]. One may choose a solution with both x, y > 0 which again ensures that e and f are on opposite sides of d and that e · f > 0. This completes the proof.


167

Corollary 3.2. If ρ (X) = 2, let NE(X) = R+ ξ + R+ η . Then one of the following mutually exclusive cases hold: (3.2.1) Neither R+ ξ nor R+ η contain any effective classes. (3.2.2) Both R+ ξ and R+ η contain an effective class of 0 or −2 self-intersection. Proof. If N (X) = 0, / then the decomposition in Eq. (2.6.1) implies that NE(X) = Conv(Q(X)). If there exists an integral (equivalently, effective) class in Q(X), then by (3.1.1) we are in case (3.2.2). If there are no integral classes in Q(X) then we are in case (3.2.1). If N (X) = 0, / then by the decomposition in Eq. (2.6.1) and (3.1.2) we are in case (3.2.2).

4 K3 Surfaces Containing a Smooth Rational Curve Theorem 4.1. Let X be a K3 surface and ξ ∈ NE(X) an extremal vector. Assume that ρ (X) ≥ 3 and X contains a smooth rational curve. Then: (4.1.1) NE(X) has no circular part. (4.1.2) There exists a sequence n ∈ N (X) such that R+ n

n→+∞

R+ ξ .

(4.1.3) NE(X) = ∑∈N (X) R+ . Proof. Let denote the class of a smooth rational curve, guaranteed by the assumption, and let σ : X → X be the morphism contracting to a point. Recall that X is still projective and let h1 and h2 be two linearly independent ample classes on X . Let di = σ ∗ hi for i = 1, 2 and observe that d1 and d2 are effective classes on X such that · d1 = · d2 = 0 and , d1 , d2 are linearly independent. Suppose there exists U ⊂ ∂ NE(X) a non-empty open subset of ∂ NE(X) such that R+U is a circular part (i.e., nowhere locally finitely generated) of NE(X). By Lemma 2.4 it follows that U ⊆ Q(X) and hence in a neighbourhood of R+U every effective class has non-negative self-intersection. Let h be an arbitrary ample class and observe that Lemma 3.1 implies that in the 2-dimensional linear subspace generated by h and there is an effective class f with either f · f = 0 or f · f = −2 and such that f is on the side of h opposite to . We may repeat the same procedure with replaced by f and h replaced by another ample class and find that these classes are all over near the boundary of NE(X). In particular, we can find an ample class h ∈ NE(X) and an effective class f with either f · f = 0 or f · f = −2 such that the 2-dimensional linear subspace generated by h and f intersects U non-trivially. Then applying Lemma 3.1 again and combining it by the observation above we obtain that there exists an effective class e ∈ U such that e · e = 0. Next let d be one of d1 and d2 such that e, d, are linearly independent. Let A = d · d, B = e · d, and C = e · . Recall that by the choice of d we have d · = 0.

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Claim 4.1.4. 2B2 = AC2 Proof. Suppose 2B2 = AC2 and let f = ACe − BCd + B2 . Then f · f = B2 (AC2 − 2B2 ) = 0 and f · e = 0. Then by Lemma 2.1 f or − f is effective. However, since e ∈ U which is a circular part of ∂ NE(X), the only effective classes contained in the hyperplane (e · = 0) are multiples of e. This implies that BC = B2 = 0, so B = 0. Applying the same argument for d, it would follow that d is a multiple of e which is impossible by the choice of d and e. Therefore we reached a contradiction and hence the claim is proven. Continuation of the proof of Theorem 4.1. Next let n ∈ N and for n ≥ 3 define ±dn = (2(2B2 − AC2)Cn2 − 4Bn)e + (2C2n)d + (1 − 2BCn). Then dn · dn = −2 and by Lemma 2.1 either dn or −dn is effective. Choose dn to be effective. By Claim 4.1.4 2B2 − AC2 = 0 and hence R+ dn → R+ e, but this contradicts the observation that in a neighbourhood of U every effective class has non-negative self-intersection. Therefore Theorem 4.1.1 is proven. Now let ξ ∈ NE(X) extremal. If no multiple of ξ is in N (X), then ξ ∈ Q(X) by Corollary 2.6. If ξ were not contained in the closure of the convex cone generated by N (X), then by [4, 2.6] NE(X) would have a circular part, so (4.1.2) follows from (4.1.1). Finally, since every class in NE(X) may be written as a sum of finitely many extremal classes, (4.1.3) follows from (4.1.2). Corollary 4.2. Let X be a K3 surface of Picard number at least three over an algebraically closed field of arbitrary characteristic. Then one of the following mutually exclusive conditions are satisfied: NE(X) = Conv(Q(X)) NE(X) =

∑

∈N (X)

R+ .

(4.2.1) (4.2.2)

Proof. If X does not contain any curve of negative self-intersection, then (4.2.1) follows from Eq. (2.6.1). Otherwise (4.2.2) follows from Theorem 4.1. Remark 4.3. Clearly Corollary 4.2 is equivalent to Theorem 1.1. Acknowledgements This paper owes its existence to Max Lieblich who suggested that perhaps the results of [4] also hold in arbitrary characteristic. The author was supported in part by NSF Grant DMS-0856185 and the Craig McKibben and Sarah Merner Endowed Professorship in Mathematics at the University of Washington.


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References 1. K. IRELAND AND M. ROSEN , A classical introduction to modern number theory, second ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. 2. J. KOLLÁR , Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 32, Springer-Verlag, Berlin, 1996. 3. J. KOLLÁR AND S. M ORI , Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998, With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. 4. S. J. KOVÁCS, The cone of curves of a K3 surface, Math. Ann. 300, no. 4, 681–691, (1994).

Around and Beyond the Canonical Class Vladimir Lazić

Mathematics Subject Classification codes (2000): 14E30

1 Introduction This survey is an invitation to recent techniques related to the Minimal Model Program (MMP). My goal is to persuade you that the MMP, at least in some of its parts, is not a subject to be afraid of any more and that it can be swallowed by a hungry postgraduate student within one (advanced) course. Indeed, in writing this paper, I had in particular such a student in mind. I deliberately tried not to be too pedantic, so that the material can be widely accessible and that the exposition can be as clear as possible. Until recently, the proofs of foundational results in the MMP were of such technical complexity that they remained opaque to all but a handful of experts. This state of affair is changing due to the emergence of a new outlook on the subject. This new outlook is the topic of this paper. The MMP has seen tremendous progress in the last decade, which is measurable both in scope of the results achieved, as well as in the depth of our understanding of the subject. The seminal paper [1], building on earlier results of Mori, Reid, Kawamata, Kollár, Shokurov, Siu, Corti, Nakayama, and many others, settled many results and advanced hugely our knowledge of the theory. The paper [1] builds upon, in some sense, classical theory, starting with the Cone theorem on our preferred variety, and employing a complicated induction to construct a sequence of surgery operations, which is then shown to terminate and yield a desired birational

V. Lazić () Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany e-mail: [email protected] F. Bogomolov et al. (eds.), Birational Geometry, Rational Curves, and Arithmetic, DOI 10.1007/978-1-4614-6482-2__9, © Springer Science+Business Media New York 2013

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model which has exceptional properties. This, in turn, provided a proof of one of the most influential conjectures in Algebraic Geometry in the last 50 years, the finite generation of the canonical ring, posed in Zariski’s famous paper [35]. A more general version of this result is Theorem 2.3 below. On the other hand, it has recently become clear that we can look at the picture the other way round. In [3,16], Theorem 2.3 was proved directly and without the MMP, only by using induction on the dimension and the Kawamata–Viehweg vanishing. The proof of this result is not the topic here, as it was clearly surveyed in [2, 5]. In this paper, I take Theorem 2.3 as a black box, and build upon it. The moral of the story is that this result, together with the right tools which are developed in Sect. 2, implies (almost) everything we know about the MMP in a clearer and quicker way. This was worked out in [4] and is the content of Sect. 3 below. This section forms the basis for the discussion in the remainder of the paper, and it is important both from the motivational viewpoint and in the scope of the techniques used. Moreover, we will see below in Sect. 5 that the new outlook gives the right perspective to think about some other problems in the field. One of them is a (possibly more philosophical) question: what makes the canonical sheaf ωX special, say on a smooth projective variety X. Ever since Riemann’s work on curves in the ninetieth century, the importance of ωX has been realised: in part because of the Riemann–Roch theorem and in part because often it is very difficult to find reasonable and useful divisors on X. Of course, in the twentieth century, it was understood further that this line bundle is important because of Serre duality, Kodaira vanishing, and so on. Therefore, it is logical to concentrate on ωX as the centre point of classification, i.e. the MMP. The class of varieties where the classical MMP works is huge—in particular, all smooth varieties are covered. However, there are many singular varieties where the results cannot apply. Indeed, Example 5.1 gives a normal projective variety for which no reasonable definition of the MMP attached to ωX works. On the other hand, there are varieties, called Mori Dream Spaces, which possess a rich birational geometry similar to the classical MMP, but they need not necessarily fall into the class of singularities allowed by the classical MMP. I survey this type of varieties in Sect. 4, drawing parallels to Sect. 3, and this motivates what happens in Sect. 5. This begs the question whether we can formulate a framework which contains both the classical MMP and Mori Dream Spaces and which constitutes, in some sense, the maximal class where a “reasonable” birational geometry can be performed. Indeed, this was done in [13] by extending the techniques from [4], and as I try to convince you in Sect. 5, the result is surprisingly simple and appealing. Finally, I close the paper with a discussion of a particular conjecture which aims to describe various cones in the space of divisors on Calabi–Yau manifolds. The Cone conjecture, due to Morrison and Kawamata, is a still pretty mysterious prediction, but I argue that it is consistent with probably the most important outstanding conjecture in birational geometry, the Abundance conjecture. On the way, we will see how the material from Sect. 2 applies nicely to show that parts of these cones have a particularly good shape.

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Throughout the paper, all varieties are normal and projective, and everything happens over the complex numbers. I follow notation and conventions from [15], and anything which is not explicitly defined here can be found there.

2 Graded Rings of Higher Rank In this section I make a brief introduction to divisorial rings, with particular accent on the higher rank case. It has only recently become clear that, even though at first they seem more complicated than rings graded by N, once you are ready to make a brave step and develop (or are just simply willing to accept) the necessary theory, then most proofs become much easier and more conceptual. To start with, let X be a Q-factorial projective variety, and let D be a Q-divisor on X. Then we define the global sections of D by H 0 (X, D) = { f ∈ k(X) | div f + D ≥ 0}. Note that, even though D might not be an integral divisor, this makes perfect sense and that H 0 (X, D) = H 0 (X, D!), where the latter H 0 is the vector space of global sections of the standard divisorial sheaf OX ( D!). This is compatible with taking sums: in other words, there is a well-defined multiplication map H 0 (X, D1 ) ⊗ H 0(X, D2 ) → H 0 (X, D1 + D2 ). Therefore, if we are given a bunch of Q-divisors D1 , . . . , Dr on X, we can define the corresponding divisorial ring as R = R(X; D1 , . . . , Dr ) =

H 0 (X, n1 D1 + · · · + nr Dr ).

(n1 ,...,nr )∈Nr

When r = 1, then we usually say that the ring R(X, D1 ) is the section ring of D1 . Throughout this paper, there is a recurring assumption that the rings that we study are finitely generated, and we will see that this assumption alone has far-reaching consequences. So say that we have a divisorial ring R as above, and assume that it is finitely generated. Then we have a corresponding cone C = ∑ R+ Di which sits in the space of R-divisors DivR (X). Inside C, there is another, much more important cone— the support of R. This cone, SuppR, is defined as the convex hull of all integral divisors D ∈ C which have sections, i.e. H 0 (X, D) = 0. It is easily seen that SuppR is a rational polyhedral cone: indeed, pick generators fi of R, and let Ei ∈ C be the divisors such that fi ∈ H 0 (X, Ei ). Then clearly Supp R = ∑ R+ Ei . Example 2.1. The first example when R is finitely generated is when all Di are semiample divisors: indeed, this is an old result of Zariski [35].

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On the other hand, even on curves there are divisorial rings which are not finitely generated. Indeed, let E be an elliptic curve, let D be a non-torsion divisor of degree 0, and let A be an ample divisor on E. Then the N2 -graded ring R(E; D, A) =

Ri, j

(i, j)∈N2

is not finitely generated: it is easy to see that the support of this ring is equal to the set (R+ D + R+A) \ R>0D, and hence it is not a rational polyhedral cone. This immediately yields a surface Y and a line bundle M on Y whose section ring is not finitely generated: set Y = P(OE (D) ⊕ OE (A)) and M = OY (1). Then " H 0 (Y, M ⊗k ) i+ j=k Ri, j , hence the section ring R(Y, M) is not finitely generated by the argument above. The following lemma summarises the main tools when operating with finite generation of divisorial rings. The proof can be found in [3, Sect. 2.4]. Lemma 2.2. Let X be a Q-factorial projective variety, and let D1 , . . . , Dr be Qdivisors such that the ring R(X; D1 , . . . , Dr ) is finitely generated. (1) If p1 , . . . , pr ∈ Q+ , then the ring R(X; p1 D1 , . . . , pr Dr ) is finitely generated. (2) Let G1 , . . . , G be Q-divisors such that Gi ∈ ∑ R+ Di for all i. Then the ring R(X; G1 , . . . , G ) is finitely generated.

2.1 An Important Example It has become clear in the last several decades that sometimes varieties are not the right objects to look at—often, it is much more convenient to look at pairs (X, Δ ), where X is a normal projective variety and Δ is a Weil Q-divisor on X such that KX + Δ is Q-Cartier. There are plenty of reasons for looking at these objects: they obviously generalise the concept of a (Q-Gorenstein) variety (by taking Δ = 0), they are suitable for induction because of adjunction formula, they are closely related to open varieties X \ Supp Δ , and so on. It is difficult to name all the advantages of working in this setting, especially since the idea of pairs and their singularities had brewed for a very long time; a good place to find a thorough explanation of all this is [14]. Not all pairs are useful for us. We concentrate on a special kind of pairs, those that have klt singularities. This means the following. First note that if f : Y → X is a log resolution of the pair (X, Δ ), that is, Y is a smooth variety and the support of the set f∗−1 Δ ∪ Exc f is a simple normal crossings divisor, then there exists a Q-divisor R on Y such that KY = f ∗ (KX + Δ ) + R.


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Then we say that (X, Δ ) is klt, or that it has klt singularities, if all the coefficients of R are bigger than −1. It can be shown that this does not depend on the choice of the resolution f . This looks like a very mysterious condition. However, a good way to think about it is to assume from the start that X is smooth, that Supp Δ has simple normal crossings, and that all coefficients of Δ lie in the open interval (0, 1). It is a fun exercise to prove that such a pair indeed has klt singularities. In particular, smooth varieties X, viewed as pairs (X, 0), have klt singularities. Also of importance for us is that this is an open condition, in the following sense. Say you have at hand a klt pair (X, Δ ) with X being Q-factorial and that you have an effective Q-divisor D on X. Then for all rational 0 ≤ ε " 1, the pair (X, Δ + ε D) is again klt. This is easy to see from the definition. Therefore, divisors of the form KX + Δ are of special importance for us, and they are called adjoint divisors. A special case of the divisorial ring above is when all Di are (multiples of) adjoint divisors—we then say that the ring R is an adjoint ring. Now we are ready to state the most important example of a finitely generated divisorial ring. Theorem 2.3. Let X be a Q-factorial projective variety, and let Δ1 , . . . , Δr be big Q-divisors such that all pairs (X, Δi ) are klt. Then the adjoint ring R(X; KX + Δ1 , . . . , KX + Δr ) is finitely generated. This was first proved in [1] by employing the full machinery of the classical MMP: the idea is to prove that a certain version of the MMP works and then to deduce the finite generation as a consequence. However, as mentioned in the introduction, of importance here for us is that Theorem 2.3 can be proved without the MMP, and this was done in [3, 16].

2.2 Asymptotic Valuations We will see that finite generation of a divisorial ring R has important consequences on the convex geometry of the cone SuppR. We would like to relate the ring R to the behaviour of linear systems |D| for integral divisors D ∈ Supp R. The way to achieve this is via asymptotic geometric valuations. Let X be a Q-factorial projective variety. Then each prime divisor Γ on X gives a valuation on the ring of rational function k(X) as the order of vanishing at the generic point of Γ . This is not sufficient, as the behaviour of elements of k(X) depends also on the higher codimension points.

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Therefore a geometric valuation Γ on X is any valuation on k(X) which is given by the order of vanishing at the generic point of a prime divisor on some birational model Y → X, and we denote the value of this valuation on a Q-divisor D by multΓ D; in other words, we take into account exceptional divisors as well. Now, if D is an effective Q-Cartier divisor, then the asymptotic order of vanishing of D along Γ is oΓ (D) = inf{multΓ D | D ∼Q D ≥ 0}; put differently, if multΓ |kD| is the valuation at Γ of a general element of the linear system |kD|, then 1 oΓ (D) = inf multΓ |kD| k over all k sufficiently divisible. It is straightforward to see that each oΓ is a homogeneous function of degree 1, that oΓ (D + D ) ≤ oΓ (D) + oΓ (D ) for every two effective Q-divisors D and D , and that oΓ (A) = 0 for every semiample divisor A. The following is a basic result [23]: Lemma 2.4. Let X be a Q-factorial projective variety, and let D and D be two big Q-divisors on X such that D ≡ D . Then oΓ (D) = oΓ (D ). Proof. I first claim that for any ample Q-divisor A, we have oΓ (D) = lim oΓ (D + ε A). ε ↓0

To this end, note that by Kodaira’s trick we can write D ∼Q δ A+ E for some rational δ > 0 and an effective Q-divisor E. Therefore (1 + ε )oΓ (D) = oΓ (D + εδ A + ε E) ≤ oΓ (D + εδ A) + ε oΓ (E) ≤ oΓ (D) + ε oΓ (E), and we obtain the claim by letting ε ↓ 0. Now, fix an ample divisor A and a rational number ε > 0. Since the divisor D − D + ε A is numerically equivalent to ε A, and thus ample, we have oΓ (D + ε A) = oΓ D + (D − D + ε A) ≤ oΓ (D ). Letting ε ↓ 0 and applying the claim, we get oΓ (D) ≤ oΓ (D ). The reverse inequality is analogous. Now we have all the theory needed to state the result which gives us the main relation between finite generation and the behaviour of linear systems.


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Theorem 2.5. Let X be a Q-factorial projective variety, and let D1 , . . . , Dr be Qdivisors on X. Assume that the ring R = R(X; D1 , . . . , Dr ) is finitely generated. Then: (1) Supp R is a rational polyhedral cone. (2) If Supp R contains a big divisor, then all pseudo-effective divisors in ∑ R+ Di are in fact effective. (3) There is a finite rational polyhedral subdivision SuppR = Ci into cones of maximal dimension, such that oΓ is linear on Ci for every geometric valuation Γ over X. (4) There exists a positive integer k such that oΓ (kD) = multΓ |kD| for every integral divisor D ∈ Supp R. We already saw (1) before, and the proof of (2) is also very easy. I omit the proof of (3) and (4), but it is not too difficult once one sets up a good basis of algebra and convex geometry. This important result is contained in the proof of [6, Theorem 4.1] and is merely extracted verbatim in [4, Theorem 3.6]. A simple, but as we will see important consequence is the following. Lemma 2.6. Let X be a normal projective variety and let D be an effective Q-divisor on X. Then D is semiample if and only if R(X, D) is finitely generated and oΓ (D) = 0 for all geometric valuations Γ over X. The proof is very simple: if D is semiample, then we conclude by Example 2.1. Conversely, for every point x ∈ X, Theorem 2.5 implies that x does not belong to the base locus of the linear system |mD| for m sufficiently divisible. As a demonstration of the previous two results, we will see immediately how inside the cone Supp R, all the cones that we can imagine behave nicely. Recall first that the movable cone Mov(X) is the closure of the cone spanned by movable divisors—these are integral divisors whose base loci are of codimension at least 2. The following is effectively the proof of Mori’s Cone theorem, see Sect. 3. I borrow the proof from [4, 13]. Proposition 2.7. Let X be a Q-factorial projective variety and let D1 , . . . , Dr be Q-divisors on X. Assume that the ring R = R(X; D1 , . . . , Dr ) is finitely generated, and denote by π : DivR (X) → N 1 (X)R the natural projection. Then: (1) The cone Supp R ∩ π −1 Mov(X) is rational polyhedral. (2) If Supp R contains an ample divisor, then the cone Supp R ∩ π −1 Nef(X) is rational polyhedral, and every element of this cone is semiample. Proof. For every prime divisor Γ on X, denote by CΓ the span of the set ofall Q-divisors D ∈ Supp R such that oΓ (D) = 0. Note that Supp R ∩ π −1 Mov(X) is the intersection of all CΓ by Theorem 2.5(4). Let Supp R = Ci be a finite rational polyhedral subdivision as in Theorem 2.5. We may add all the faces of all the cones Ci to the subdivision. To show (1), it is enough to prove that each CΓ is a union of some of Ci . But this follows once we

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Mov(X) Nef(X)

Supp

notice that if CΓ intersects the relative interior of some C , then C ⊆ CΓ since oΓ is a linear non-negative function on C . For (2), if the relative interior of C contains an ample divisor, then oΓ |C ≡ 0 for every Γ as above. Hence, every element of C is semiample by Lemmas 2.2 and 2.6, and so C ⊆ Supp R ∩ π −1 Nef(X) . Therefore, the cone SuppR ∩ π −1 Nef(X) is equal to the union of some Ci , which suffices. We also note the following crucial consequence of Theorem 2.5 [13, Theorem 4.2]. It shows that the chamber decomposition as in Theorem 2.5 gives canonically birational contractions from our variety. Theorem 2.8. Let X be a Q-factorial projective variety, and let D1 , . . . , Dr be Q-divisors on X. Assume that the ring R = R(X; D 1 , . . . , Dr ) is finitely generated and that Supp R contains a big divisor. Let Supp R = Ci be a finite rational polyhedral decomposition as in Theorem 2.5, and let F j be all the codimension 1 faces of the cones Ci . (1) For each i, let Di be a Cartier divisor in the interior of Ci , and let Xi = Proj R(X, Di ). Then the variety Xi and the birational map ϕi : X Xi do not depend on the choice of Di (up to isomorphism). The map ϕi is a contraction. (2) For each j, let G j be a Cartier divisor in the relative interior of F j , and let Y j = Proj R(X, G j ). If F j contains a big divisor, then the variety Y j and the birational map θ j : X Y j do not depend on the choice of G j (up to isomorphism). The map θ j is a contraction. (3) If F j ⊆ Ci , then there is a birational morphism ρi j : Xi → Y j such that the diagram ϕi

X

Xi ρi j

θj

Yj commutes.


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Proof. I will only show (1) and (3), as the proof of (2) is analogous to that of (1). Theorem 2.5 implies that we can find a resolution f : X˜ → X and a positive integer d such that Mob f ∗ (dD) is basepoint free for every Cartier divisor D ∈ Supp R. Denote Mi = Mob f ∗ (dDi ). Then we have the induced birational morphism ψi : X˜ → Xi , which is just the Iitaka fibration associated to Mi . A result of Reid [28, Proposition 1.2] shows that the divisor Fix | f ∗ (dDi )| (and also any f -exceptional divisor) is contracted by ψi —in other words, ϕi is a contraction. Let us show that the definition of ϕi does not depend on the choice of Di . Indeed, pick any other Cartier divisor Di in the interior of Ci , and let ψi : X˜ → Proj R(X, Di ) be the corresponding map. There exists a Cartier divisor Di in the interior of Ci , together with positive integers ri , ri , ri such that ri Di = ri Di + ri Di . Denoting Mi = Mob f ∗ (dDi ) and Mi = Mob f ∗ (dDi ), then we have ri Mi = ri Mi + ri Mi

(1)

(since all functions oΓ are linear on Ci ), and the divisors Mi , Mi , Mi are basepoint free. For any curve C on X˜ contracted by ψi we have Mi · C = 0, hence Eq. (1) implies Mi · C = 0, and so C is contracted by ψi . Reversing the roles of Di and Di , we obtain that ψi and ψi contract the same curves; therefore they are the same map up to isomorphism. The same method proves (3). I finish this section with a simple consequence of Lemma 2.4 and Theorem 2.5, which will be crucial in Sect. 5. Lemma 2.9. Let X be a Q-factorial projective variety, and let D1 and D2 be big Qdivisors such that D1 ≡ D2 . Assume that the rings R(X, Di ) are finitely generated, and consider the maps ϕi : X Proj R(X, Di ). Then there exists an isomorphism η : Proj R(X, D1 ) −→ Proj R(X, D2 ) such that ϕ2 = η ◦ ϕ1 . Proof. By passing to a resolution and by Theorem 2.5, we may assume that there is a positive integer k such that Mob(kDi ) are basepoint free and that each ϕi is given by the linear system | Mob(kDi )|. By Lemma 2.4 we have Mob(kD1 ) ≡ Mob(kD2 ), hence ϕ1 and ϕ2 contract the same curves.

3 Picture 1: Classification I review briefly the “classical” Minimal Model Program, concentrating on parts which are important in what follows. There are many well-written surveys and books on the topic, and if needed, you can consult [14] and references therein. It is important to point out that many concepts which are related or grew out of the MMP,

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and that we consider natural or given because they fit beautifully into many corners of algebraic geometry (such as nefness, the canonical ring, minimal and canonical models and so on), took a long time to conceive. In order to fully appreciate this formative process, I urge you to read the wonderful semi-autobiographical survey [29] and also [20, Sect. 9]. For the sake of simplicity and clarity, I state all the results for smooth varieties, but note that with minor changes they hold for pairs with klt singularities. So say you have at hand a smooth n-dimensional projective variety X. As mentioned in the introduction, it is reasonable to concentrate on the canonical divisor KX as the central object of our study. On the other hand, having ample divisors on a projective variety X is extremely important: they give embeddings of X into some projective space, and their cohomological and numerical properties are as nice as you can hope for. Assume that KX is pseudo-effective. Then, a reasonable question to pose is: Is there a birational map f : X Y such that the divisor f∗ KX is ample? Here the map f should not be just any birational map, but a birational contraction— in other words, f −1 should not contract divisors. This is an important condition since the variety Y should be in almost every way simpler than X; in particular, some of its main invariants, such as the Picard number, should not increase. Likewise, we would like to have KY = f∗ KX , and this will almost never happen if f extracts divisors (take, for instance, an inverse of almost any blowup). Further, we impose that f should preserve sections of all positive multiples of KX . This is also important, since global sections are something we definitely want to keep track of, if we want the divisor KY = f∗ KX to bear any connection with KX . Another way to state this is as follows. Consider the canonical ring of X: R(X, KX ) =

H 0 (X, mKX ).

m∈N

Then we require that f induces an isomorphism between R(X, KX ) and R(Y, KY ). Apart from the relation to Zariski’s conjecture that was mentioned in the introduction, this is also fundamental in the construction of the moduli space of canonically polarised varieties; for an introduction to this beautiful topic, upon which I do not touch any more in these notes, see [8, Part III]. We immediately see that the answer to the question above is in general “no”— the condition would imply that KX is a big divisor. Nevertheless, we can settle for something weaker, but still sufficient for our purposes: we require that the divisor KY is semiample. This then still produces an Iitaka fibration g : Y → Z and an ample divisor A such that KY = g∗ A, and the composite map X Z, which is now not necessarily birational, gives an isomorphism of section rings R(X, KX ) and R(Z, A). Historically, by the influence of the classification of surfaces on the way we think about higher dimensional classification, this splits into two problems: finding a birational map f : X Y such that the divisor KY = f∗ KX is nef; and then proving that the nef divisor KY is semiample. This last part—the Abundance conjecture—is


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one of main open problems in higher dimensional geometry, in dimensions at least 4. We know it holds in dimensions up to 3, or when the canonical divisor is big, but very little is known in general. Thus, hopefully by now it is clear that the main classification criterion is whether the canonical divisor KX is nef. If KX is nef, we are done, at least with the first part of the programme above. Life gets much tougher, but also much more interesting when the answer is no.

3.1 The Cone and Contraction Theorems Indeed, let NE(X) ⊆ N1 (X)R denote the closure of the cone spanned by effective curves; note that the nef cone Nef(X) is dual to NE(X) by Nakai’s criterion, with respect to the intersection pairing. Since KX is not nef, the hyperplane KX⊥ = {C ∈ N1 (X)R | KX ·C = 0} ⊆ N1 (X)R must cut the cone NE(X) into two parts; let us denote the two pieces by NE(X)KX ≥0 and NE(X)KX 0 be a rational number such that all pairs (X, ε Bi ) are klt, and notice that KX + ε Bi ∼Q ε Bi . Hence, the ring R(X; B1 , . . . , Bq ) is finitely generated by Theorem 2.3 and by Lemma 2.2.

We immediately obtain the following. Theorem 6.9. Let X be a variety of Calabi–Yau type. (1) The cone Nef(X) ∩ Big(X) is locally rational polyhedral in Big(X), and every element of Nef(X) ∩ Big(X) is semiample. (2) The cone Mov(X) ∩ Big(X) is locally rational polyhedral in Big(X). Proof. Part (1) was first proved in [11, Theorem 5.7]. The problem of finding the shape of Mov(X) ∩ Big(X) was posed in [11, Problem 5.10]. This was solved in [12, Corollary 2.7] for threefolds and in [13, Theorem 3.8] in general. Let V be a relatively compact subset of the boundary of Nef(X) ∩ Big(X), and denote by π : DivR (X) −→ N 1 (X)R the natural projection. Then we can choose finitely many big Q-divisors B1 , . . . , Bq such that V ⊆ π (∑qi=1 R+ Bi ). Theorem 6.8 implies that the ring R = R(X; B1 , . . . , Bq ) is finitely generated, and hence π −1 Nef(X) ∩ Supp R is a rational polyhedral cone and its every element is semiample by Corollary 2.7. But then V is contained in finitely many rational hyperplanes. This shows (1), and the proof of (2) is similar. Let us recall the following known conjecture which generalises Theorem 2.3. Conjecture 6.10. Let X be a Q-factorial projective variety, and let Δ1 , . . . , Δr be Qdivisors such that all pairs (X, Δi ) are klt. Then the adjoint ring R(X; KX + Δ1 , . . . , KX + Δr ) is finitely generated.


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This conjecture is implied by the full force of the MMP [30], including termination of any sequence of flips and Abundance, although a priori it is weaker than the MMP. In particular, the conjecture is a theorem in dimensions up to 3. Finally, the following result shows that Conjecture 6.10 and the Cone conjecture are, in some sense, consistent. Proposition 6.11. Let X be an n-dimensional variety of Calabi–Yau type. Assume either Conjecture 6.10 in dimension n or the Cone conjecture in dimension n. Then the cones Nef(X) ∩ Eff(X) and Mov(X) ∩ Eff(X) are spanned by rational divisors. Proof. I only show the statements for Nef(X) ∩ Eff(X), the rest is analogous. Assume Conjecture 6.10 in dimension n. Let D be an R-divisor whose class is in Nef(X) ∩ Eff(X). Then we can write D ≡ ∑ri=1 δi Di for prime divisors Di and positive real numbers δi . Fix an ample Q-divisor A on X. By Theorem 6.8, the ring R(X; D1 , . . . , Dr , A) is finitely generated, and hence, the cone N = π −1 Nef(X) ∩ ∑ R+ Di is rational polyhedral by Proposition 2.7, where π : DivR (X) → N 1 (X)R is the natural map. Since π (D) ∈ N , the result follows. Now assume the Cone conjecture in dimension n. Let D be an R-divisor whose class is in Nef(X) ∩ Eff(X), and let Π be the fundamental domain for the action of A(X) on Nef(X) ∩ Eff(X). Then there exists g ∈ A(X) such that D ∈ gΠ , and the conclusion follows since gΠ is a rational polyhedral cone. Acknowledgements Many thanks to P. Cascini, A. Corti, K. Frantzen, D. Greb, A.-S. Kaloghiros, J. Kollár, A. Küronya, Th. Peternell, and S. Weigl for many useful comments and discussions, and to the referee for many useful remarks. I was supported by the DFG-Forschergruppe 790 “Classification of Algebraic Surfaces and Compact Complex Manifolds”.

References 1. C. BIRKAR , P. CASCINI , C. D. HACON , and J. MC KERNAN , Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23, no. 2, 405–468, (2010). 2. P. CASCINI and V. LAZI C´ , The Minimal Model Program revisited, Contributions to Algebraic Geometry (Piotr Pragacz, ed.), EMS Series of Congress Reports, EMS Publishing House, Zürich, 2012, pp. 169–187. 3. P. CASCINI and V. LAZI C´ , New outlook on the Minimal Model Program, I, Duke Math. J. 161, no. 12, 2415–2467, (2012). 4. A. CORTI and V. LAZI C´ , New outlook on the Minimal Model Program, II, Math. Ann. (2012), DOI:10.1007/s00208-012-0858-1. 5. A. CORTI , Finite generation of adjoint rings after Lazić: an introduction, Classification of Algebraic Varieties, EMS Series of Congress Reports, EMS Publishing House, 2011, 197–220. 6. L. EIN , R. LAZARSFELD , M. M USTA T¸ A˘ , M. NAKAMAYE, and M. POPA , Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble) 56, no. 6, 1701–1734, (2006). 7. Y. HU and S. KEEL, Mori dream spaces and GIT, Michigan Math. J. 48, 331–348, (2000).

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8. C. D. HACON and S. J. KOVÁCS, Classification of higher dimensional algebraic varieties, Oberwolfach Seminars, vol. 41, Birkhäuser Verlag, Basel, 2010. 9. B. HASSETT and Y. TSCHINKEL, Flops on holomorphic symplectic fourfolds and determinantal cubic hypersurfaces, J. Inst. Math. Jussieu 9, 125–153, (2010). 10. J. KOLLÁR ET AL., Flips and abundance for algebraic threefolds, Astérisque 211, Soc. Math. France, Paris, 1992. 11. Y. KAWAMATA , Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2) 127 (1988), no. 1, 93–163. 12. Y. KAWAMATA , On the cone of divisors of Calabi-Yau fiber spaces, Internat. J. Math. 8, 665–687, (1997). 13. A.-S. KALOGHIROS, A. KÜRONYA , and V. LAZI C´ , Finite generation and geography of models, to appear in “Minimal models and extremal rays”, Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Tokyo, arXiv:1202.1164, (2012). 14. J. KOLLÁR and S. M ORI , Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. 15. R. LAZARSFELD , Positivity in algebraic geometry. I, II, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 48, 49, Springer-Verlag, Berlin, 2004. 16. V. LAZI C´ , Adjoint rings are finitely generated, arXiv:0905.2707, (2009). 17. V. LAZI C´ and TH . PETERNELL, On the Cone conjecture for Calabi-Yau manifolds with Picard number two, arXiv:1207.3653, (2012). 18. E. M ARKMAN , A survey of Torelli and monodromy results for holomorphic-symplectic varieties, Complex and Differential Geometry (W. Ebeling, K. Hulek, and K. Smoczyk, eds.), Springer Proceedings in Mathematics, vol. 8, Springer Berlin Heidelberg, 2011, 257–322. 19. J. MC KERNAN , Mori dream spaces, Japan. J. Math. 5, no. 1, 127–151, (2010). 20. S. M ORI , Classification of higher-dimensional varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, 269–331. 21. S. M ORI , Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1, no. 1, 117–253, (1988). 22. D. M ORRISON , Compactifications of moduli spaces inspired by mirror symmetry, Astérisque 218, 243–271, (1993). 23. N. NAKAYAMA , Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. 24. Y. NAMIKAWA , Periods of Enriques surfaces, Math. Ann. 270 (1985), 201–222. 25. K. OGUISO , Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups, arXiv:1107.5862, (2011). 26. K. OGUISO , Automorphism groups of Calabi-Yau manifolds of Picard number two, arXiv:1206.1649, (2012). 27. A. PRENDERGAST-SMITH , The Cone Conjecture for Abelian Varieties, J. Math. Sci. Univ. Tokyo 19, no. 2, 243–261, (2012). 28. M. REID , Canonical 3-folds, Journées de Géométrie Algébrique d’Angers (A. Beauville, ed.), Sijthoof and Nordhoof, Alphen aan den Rijn, 1980, 273–310. 29. M. REID , Twenty-five years of 3-folds–an old person’s view, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 313–343. 30. V. V. SHOKUROV and S. R. CHOI , Geography of log models: theory and applications, Cent. Eur. J. Math. 9 (2011), no. 3, 489–534. 31. A. J. SOMMESE, On the adjunction theoretic structure of projective varieties, Complex analysis and algebraic geometry (Göttingen, 1985), Lecture Notes in Math., vol. 1194, Springer, Berlin, 1986, 175–213. 32. H. STERK , Finiteness results for algebraic K3 surfaces, Math. Z. 189, 507–513, (1985). 33. B. TOTARO , The cone conjecture for Calabi-Yau pairs in dimension two, Duke Math. J. 154, 241–263, (2010).


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On the Ubiquity of Twisted Sheaves Max Lieblich

Mathematics Subject Classification codes (2000): 14F22, 14H10, 11E81

1 Introduction In this paper we describe several results, built up over the last decade or so, that use twisted sheaves to study questions in algebra, arithmetic, and cohomology. The paper is for the most part expository and aims to collect in one place what is known about these methods, their applications, and some questions about their future. It also hopes to be a guide to a chunk of the literature, where the reader can find the details of those things only hinted at here.

2 Classical Questions To warm up, here are some questions that one can attack with the methods we will sketch below. Readers unfamiliar with the Brauer group of a field are referred to the masterful exposition in [45]. Question 2.1. Suppose X is a quasi-compact separated scheme. Do the Brauer group and cohomological Brauer group of X coincide? Question 2.2. Suppose K is a Cd -field. Does every class α ∈ Br(K) satisfy ind(α )| per(α )d−1 ?

M. Lieblich () Department of Mathematics, University of Washington, Seattle, WA, USA e-mail: [email protected] F. Bogomolov et al. (eds.), Birational Geometry, Rational Curves, and Arithmetic, DOI 10.1007/978-1-4614-6482-2__10, © Springer Science+Business Media New York 2013

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Question 2.3 (Colliot-Thélène). Suppose X is a geometrically integral variety over a global field k. Is the Brauer–Manin obstruction the only one for 0-cycles on X of degree 1 over k? Question 2.4. Suppose K is of transcendence degree 1 over a totally imaginary number field. Is the u-invariant of K finite? Question 2.5. What is the Kodaira dimension of the moduli space of polarized supersingular K3 surfaces? Question 2.6. Fix a finite field k. Are there only finitely many K3 surfaces defined over k? This hopefully gives the reader a sense of the broad range of questions and areas that make contact with twisted sheaves. Entirely left out of this exposition is the connection between twisted sheaves and mathematical physics; this is in fact the (modern) birthplace of the theory of twisted sheaves (although traces can be found in France in the 1960s and 1970s [15]). The interested reader is referred to [5, 6, 13, 19, 32, 33, 43, 44, 50, 52, 53] and their references for a start on that story.

3 Twisted Sheaves What are twisted sheaves? They are most profitably approached as sheaves on gerbes (a certain kind of stack), but they also have a “softer” description as sheaves with gluing data that are not quite descent data. We briefly summarize these two descriptions. Fix an algebraic space X and a G-gerbe X → X, where G → Gm is a closed subgroup scheme (so G ∼ = Gm or G ∼ = μ n for some n). (For background on G-gerbes, see [15, 25, 26, 35].) Any sheaf F on X gets a natural right G-action F × G → F in the following way: given an object f : U → X and an element g ∈ G(U), the automorphism g of f induces (by the sheaf property of F ) an isomorphism g∗ FU → FU , giving an action on sections g∗ : F (U) → F (U). Definition 3.1. A sheaf of OX -modules is X -twisted if the left action G×F → F associated to the right action described above is identified with scalar multiplication via the embedding G → Gm and the O-module structure on F . This might seem rather abstract, so let us record the softer definition for the sake of intuition-building. The gerbe X → X represents a class in H2 (X, G). We can also represent such classes by appropriate 2-cocycles for hypercoverings in the fppf topology. Suppose for simplicity that we have an fppf covering U → X and ˇ a two-cocycle a ∈ G(U ×X U ×X U) (i.e., in the associated Cech hypercovering) representing the same cohomology class.

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Definition 3.2. An a-twisted sheaf is a pair (F , ϕ ) where F is an OU -module and ∼ ϕ : F ×X U → U ×X F is an isomorphism whose coboundary ∼

δ ϕ : F ×X U ×X U → F ×X U ×X U is equal to multiplication by a. For simplicity of notation, we think of everything—U, X, U ×X U, F , etc.—as sheaves on the big fppf site of X. This way, for example, F ×X U is a sheaf on U ×X U that is usually written as (pr1 )∗ F . (We will not make much use of this definition or notation in what follows.) Colloquially, an a-twisted sheaf is a sheaf on U with a gluing datum that is obstructed from being a descent datum by the cocycle a. Remark 3.3. A warning to the reader who plans to look at the literature on twisted sheaves, especially that arising from the physics side. Early workers on twisted sheaves made an absolutely disastrous decision: when dealing with twisted skyscraper sheaves over a geometric point x → X (or, more generally, twisted sheaves supported on schemes over which the twisting class is trivial), they used the fact that a is cohomologous to 1 to claim that the “twisted skyscraper sheaf at x” and the (usual) “skyscraper sheaf at x” are “the same” or that the usual skyscraper sheaf could be “given the structure of twisted sheaf.” As is hopefully made clear in Definition 3.1, these are objects in different categories. This terminological mistake has had some negative consequences over the years. I beg you to make the distinction if you ever use these things. Notation 3.4. Write QCohtw (X ) for the category of quasi-coherent X -twisted sheaves and Cohtw (X ) for the subcategory of coherent X -twisted sheaves. One thing Definition 3.2 makes fairly clear is the following change of structure group statement. Lemma 3.5. Let Y → X be the Gm -gerbe associated to the G-gerbe X → X. The natural G-equivariant map X → Y induces by pullback equivalences of abelian categories QCohtw (Y ) → QCohtw (X ) and Cohtw (Y ) → Cohtw (X ). Thus, from a categorical point of view, Gm -gerbes suffice. We will see in a moment that the proper formulation of standard moduli problems is somewhat sensitive to the structure group.

4 The First Appearance of Twisted Sheaves Before we dive deeper into the problems of Sect. 2, let us see the most primitive appearance of twisted sheaves: as Picard obstructions.

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Example 4.1. Recall that a Brauer–Severi variety over a field k is a variety X such that for some n > 0, X ⊗k ∼ = Pn . k

In other words, X is an étale form of projective space. It is a basic result [45, Sect. X.6] that X ∼ = Pnk if and only if X contains a divisor D ⊂ X that is a hyperplane over k if and only if X contains a point defined over k. The canonical example is a plane conic: the conic Z(X 2 + Y 2 + Z 2 = 0) ⊂ P2R does not contain a rational point, hence is not isomorphic to P1 . More generally, a conic is isomorphic to P1 if and only if it has a rational point: the isomorphism is given by projection from that point. (In the general case, we can make a rational isomorphism by projecting from the intersection point of several copies of a hyperplane, and we can make a hyperplane by pulling back a hyperplane along such a rational projection, showing that the last two conditions are equivalent. Once we have a hyperplane, the full linear system it defines gives an isomorphism with projective space, showing that the last two conditions are equivalent to the first!) What is the relative Picard scheme PicX/k ? Well, PicX ⊗ k/k ∼ = Zk and the ample generator defines a section that is Galois-invariant. We conclude that PicX/k ∼ = Zk , the constant group scheme with value Z. What is the relative Picard stack? Since X is geometrically integral, we know that PicX/k is a Gm -gerbe over PicX/k . In particular, the preimage of the section 1 ∈ Zk defines a Gm -gerbe G → Spec k. The restriction of the universal sheaf L ∈ Pic(X × PicX/k ) yields an invertible sheaf on X := X ×Speck G . As a first exercise in unwinding definitions we invite the reader to show that this sheaf L is an X -twisted sheaf. Let us consider the gerbe G itself. Does it admit an invertible twisted sheaf? Suppose it did, say L. Then the sheaf L ⊗ pr∗2 L∨ would be an “untwisted” invertible sheaf on X whose image in PicX/k is 1. In other words, it would cut out a hyperplane, and X would be trivial (i.e., isomorphic to Pnk ). On the other hand, if X is trivial, then there is a hyperplane, so there is an object of PicX/k mapping to 1 and thus a k-map


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Spec k → G . We conclude that G is trivial (isomorphic to the classifying stack BGm ) if and only if X is trivial. Since X has a k(X)-point, we see that Gk(X) should be trivial, and we indeed get a trivialization from the universal sheaf L . This example is less trivial than it may seem. By thinking of Picard obstructions as gerbes with twisted sheaves (geometric objects) instead of cohomology classes, we can study moduli problems associated to these obstructions. These moduli problems play an integral role in all but the first question in Sect. 2, as we will see in a moment.

5 Twisted Sheaves and the Brauer Group Let us briefly summarize some aspects of the relation between twisted sheaves and the Brauer group of a scheme. By analogy with the Brauer group of a field, one can define the Brauer group of a scheme as follows [35, Chap. IV], [16]. Fix a scheme X. Definition 5.1. An Azumaya algebra of degree n on X is a sheaf A of OX -algebras that is étale-locally isomorphic to Mn (OX ). Two Azumaya algebras A and B are Brauer-equivalent if there are locally free sheaves V and W on X with positive rank at every point of X and an isomorphism of OX -algebras A ⊗ End(V ) ∼ = B ⊗ End(W ). The Brauer group of X is the group whose objects are equivalence classes of Azumaya algebras of positive degree on X, with group operation induced by tensor product. Note that to be entirely rigorous, one should take the degree n to be an element of Γ (X, ZX ), and the group structure is defined on the equivalence class of Azumaya algebras with everywhere-positive rank. It is rather surprising upon one’s first exposure to learn that the Brauer group is indeed a group. Since Mn (O) ⊗ Mm (O) ∼ = Mmn (O) as O-algebras and ⊗ is associative up to canonical isomorphism, we see that ⊗ induces an associative operation on equivalence classes. The identity element of this operation is given by the trivial Azumaya algebra O. Inverses are shockingly explicit: given an Azumaya algebra A , let A ◦ denote the opposite algebra. The map

μ : A ⊗ A ◦ → EndO (A ) that sends an elementary tensor a ⊗ b to the O-linear endomorphism x → axb is in fact an isomorphism of Azumaya algebras. One can see this by looking in geometric fibers, whence one may choose an isomorphism A ∼ = Mn (k) and then

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μ is identified with a k-algebra endomorphism of Mn2 (k). Since Mn2 (k) is a finitedimensional simple algebra (i.e., possessing no nontrivial two-sided ideals), μ must be an isomorphism. Thus, A ◦ is the inverse of A . An Azumaya algebra of degree n is an étale form of Mn (OX ). By the SkolemNoether theorem and descent theory, there is a corresponding class [A ] ∈ H1 (X, PGLn ). Combining this with the usual exact sequence 1 → Gm → GLn → PGLn → 1 and Giraud’s sequence of non-abelian cohomology yields a class associated to A in H2 (X, Gm ). In fact, this class is n-torsion, as one sees from the analogous sequence 1 → μ n → SLn → PGLn → 1. The basic result is that the map sending A to [A ] ∈ H2 (X, Gm ) gives a group injection Br(X) → H2 (X, Gm ). The following results describe the well-known relationship between twisted sheaves and the Brauer group. The reader is referred to [26] for more details. Nothing here is deep, but the translation into this terminology turns out to have a number of interesting consequences: • A class α ∈ H2 (X, Gm ) lies in the image of Br(X) if and only if for some (equivalently, any) Gm -gerbe X → X representing α there is a locally free X twisted sheaf of everywhere-positive rank. • A Gm -gerbe X → X represents the trivial Brauer class if and only if there is an invertible X -twisted sheaf. • Suppose X = Spec K is a field and X → X is a Gm -gerbe representing a Brauer class α . The index of α is the minimal positive rank of an X -twisted sheaf. • (Generalization) Suppose X is an integral quasi-compact separated scheme with function field K and X → X is a Gm -gerbe representing a Brauer class α ∈ Br(X). The index of the restriction αK equals the gcd of the ranks of quasicoherent X -twisted sheaves of finite rank. Equivalently, the index of αK equals the minimal positive rank of a perfect complex of X -twisted sheaves. √ Recall that the index of a Brauer class α ∈ Br(k), k a field, is defined to be dimk D, with D a finite-dimensional central division algebra over k. Equivalently, if X is a Brauer–Severi variety with class α , the index of α is the minimal positive kdegree of a 0-cycle on X (and this is equal to the gcd of the k-degrees of all 0-cycles on X). The criterion above relates this number to the ranks of twisted sheaves. This small amount of theory already gives us a way of thinking about Question 2.1. Recall that the cohomological Brauer group of X is the torsion subgroup of cohomology Br (X) = H2 (X, Gm )tors .


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We have just seen that there is a natural inclusion (∗) : Br(X) → Br (X). The standard question, first formulated by Grothendieck [17], asks when (∗) is an isomorphism. We can rephrase this in twisted sheaves as follows. Question 5.2. For which schemes X does every μ n -gerbe X → X (for every n) carry a nowhere vanishing locally free twisted sheaf? In this form, the question is an infantile form the famous question on the resolution property [51]: for which spaces/stacks is every quasi-coherent sheaf of finite presentation a quotient of a locally free sheaf? It is well known that this is always true for schemes admitting an ample family of invertible sheaves. The interesting part of Question 5.2 is that one does not have a “trivial sheaf” to start with, even in the presence of ample sheaves for twisting, trying to produce maps, etc. One is simply trying to prove that a nowhere-zero locally free twisted sheaf even exists at all. What should one expect? In this thesis [14], Gabber proved that (∗) is an isomorphism when X is a separated union of two affine schemes. He subsequently proved the same for any quasi-compact separated scheme admitting an ample invertible sheaf (i.e., an invertible sheaf L such that the non-vanishing loci of sections of tensor powers of L generate the Zariski topology of X). De Jong devised a proof of this more general statement [12] that uses twisted sheaves. The idea is this: first one can assume that X is of finite type over Z. Gabber tells us that there are locally free twisted sheaves over every affine, and these can be extended to coherent twisted sheaves. De Jong’s argument cleverly leverages these local sheaves to inductively shrink the nonlocally free locus (while possibly extending the base ring to a finite flat Z-algebra, something that is immaterial for the result at hand). The key point? This is “easy” to do locally by taking maps between two well-chosen coherent twisted sheaves. To globalize this, one must produce global maps with good local properties, and this is something one can do by tensoring a global Hom-sheaf with a sufficiently ample invertible sheaf and taking sections. As the reader may gather from the verbiage, this is a highly geometric proof of a statement that arises directly from cohomology and K-theory. (Note that it is much easier to show that there is a perfect complex of twisted sheaves of positive rank; the geometry in de Jong’s proof might be interpreted as enabling the transition from perfect complexes to locally free sheaves, but this statement is perhaps too fuzzy to be meaningful.)

6 Results on Moduli of Twisted Sheaves Let us start our brief discussion of the moduli of twisted sheaves by reflecting further on Example 4.1. Let E be a curve of genus 1 over a field K with a rational point P ∈ E(K), endowed with its canonical structure of elliptic curve (where P is the

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identity). The Leray spectral sequence for Gm on the morphism f : E → Spec K yields an edge isomorphism H1 (K, E ∨ ) → H1 (K, R1 f∗ Gm ) → H2 (E, Gm )/ Br(K). Geometrically, we can realize this as follows: given a torsor under E ∨ , i.e., a form X of E ∨ , we get a gerbe EX → E (and hence a Brauer class on E) by forming the Picard stack Pic0X/K → Pic0X/K of degree 0 invertible sheaves. If we are a bit careful, we can reverse this construction by making a moduli space of twisted sheaves. Here is how it works. Fix a μ n -gerbe E → E that has trivial cohomology class in the fiber over P. (One can use Gm -gerbes instead if one is willing to work slightly harder, but for ease of exposition we will not pursue this.) An invertible E -twisted sheaf L has a degree: the sheaf L ⊗ n is the pullback of a unique invertible sheaf L on E and we can define deg L =

1 deg L. n

Define a stack ME as follows: the objects over a K-scheme T are invertible E × T twisted sheaves L whose restriction to each geometric fiber Et has degree 0. Lemma 6.1. The stack ME is a trivial Gm -gerbe over an étale form ME of E ∨ . Moreover, the constructions X → EX and E → ME are inverse. Sketch of proof. The stack ME is algebraic as a consequence of Artin’s theorem. We know that ME is a Gm -gerbe over an algebraic space ME because it parametrizes geometrically simple sheaves. Moreover, since E is trivial over P, restricting a universal sheaf along a map SpecK → E ⊗ κ (P) trivializes the gerbe ME → ME . The formation of ME is also compatible with base change. Tsen’s theorem tells us that Br(EK ) = 0, which allows one to reduce to the case in which E → E is trivial. It follows that ME is a form of E ∨ , as claimed. Lemma 6.1 has a geometric interpretation: given an elliptic fibration, we can produce new elliptic fibration by taking moduli spaces of twisted sheaves with respect to a suitable Brauer class. This is of course fraught with difficulties (singular fibers, the absence of sections of the fibration, etc.), but it turns out to be surprisingly fruitful, as we will see in Sect. 9. More generally, we can make forms of the moduli space of sheaves on any curve, something that will prove useful in Sect. 7. Proposition 6.2. Let C/K be a curve over a field with a rational point P ∈ C(K). Fix a μ n -gerbe C → C (with n · 1 ∈ K × ) whose cohomology class restricts to 0 in both H2 (Spec κ (P), μ n ) and H2 (CK , μ n ). Then: 1. For any integer d, the stack of geometrically stable C -twisted sheaves of rank n and degree d is a Gm -gerbe over an étale form MC /K (n, d) of the moduli space MC/K (n, d) of geometrically stable sheaves on C of rank n and degree d.


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d 2. The determinant defines a map δ : MC /K (n, d) → PicC/K which is an étale form d . In particular, the geometric of the classical determinant map MC/K → PicC/K fibers of δ are unirational. 3. For a given invertible sheaf L on C, write MC/K (n, L) for the fiber of the determinant map over L (i.e., geometrically stable twisted sheaves with determinants indentified with L), and MC/K (n, L) for its coarse space. The Brauer class

o ∈ Br(MC/K (n, L)) associated to the gerbe MC/K (n, L) → MC/K (n, L) generates the Brauer group of MC/K (n, L). Sketch of proof. This works as above: under the assumption that n is invertible in K, the cohomology class of C splits over the separable closure of K, which reduces us to the case in which the class of C → C in H2 (C, μ n ) is trivial. Assuming the class is trivial, there is an isomorphism C ∼ = C × Bμ n . Letting Λ denote the pullback of the invertible sheaf on B μ n corresponding to the scalar multiplication character μ n → Gm , tensoring by Λ sets up the isomorphism of MC/K → MC /K . The unirationality was proven by Serre and can be found in [37, Chap. 5]. The last result can be found in [4]. So much for curves. What about higher-dimensional base varieties? There are several theorems to describe surfaces and then a large sinkhole opens up. As we will illustrate below, one can in fact use theorems in algebra to prove a meta-theorem: the structure must be worse for threefolds. Proposition 6.3. Let X → X be a μ n -gerbe on a smooth geometrically connected projective surface over a field K. Fix an invertible sheaf L ∈ Pic(X). Given an integer c, let MX (n, L, c) be the stack of geometrically stable locally free X twisted sheaves of rank n with determinant L and second Chern class c. Then: 1. The stack MX (L, c) is a μ n -gerbe over an algebraic space MX (n, L, c) of finite type over K. 2. For any C, there is c > C such that MX (n, L, c) is nonempty. 3. For all sufficiently large c, the space MX (n, L, c) is geometrically integral when it is nonempty. This is usually described as “asymptotic irreducibility” of the moduli space. The proof, while nontrivial, is very similar in spirit (and often in detail) to that coming from the classical theory of stable vector bundles and is described in [25]. The classical version, first proven in [38], is expertly described in [18], especially in Chap. 9 (and, in particular, Sect. 9.4). For more general kinds of surfaces, there is a similar theorem. Let us state it as a vague principle, to be expanded upon below.

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Principle 6.4. Let X → X be a smooth proper geometrically connected Deligne– Mumford stack of dimension 2 over a field K that is a μ n -gerbe over another Deligne–Mumford stack. The stack N of torsion free X -twisted sheaves of rank N often contains a geometrically integral locally closed substack. The word “often” and the vague integer N make this a principle rather than a proposition. The basic structure of an instance of Principle 6.4 works as follows. Let us assume that K is perfect for simplicity of exposition (and the typical applications): 1. Show that the stack N is nonempty. Note: this is surprisingly hard and relies heavily on the properties of X, N, and X → X. 2. Show that the unobstructed locus of N (i.e., the points [V ] whose equideterminantal obstruction space Ext2 (V,V )0 is 0) is nonempty. This is usually a straightforward deformation theory problem, but depends crucially on the fact that X is a surface. We will now replace N by the unobstructed locus and assume that N consists of unobstructed sheaves (and is thus smooth). 3. Now comes a trick originally due to O’Grady (see [18, Sect. 9.4]) that has also appeared in work of de Jong and Starr (see [11, Theorem 13.1]): N is naturally organized in a hierarchy via the formation of (partial) reflexive hulls. If a sheaf V is not locally free at a point p, we can locally form V ∨∨ and glue it to V away from p to form a new sheaf (Note: this may be obstructed and thus not be in N ); conversely, we can start with V and form a subsheaf V by taking the kernel of any map V → κ , where κ is a twisted skyscraper sheaf (now V does stay unobstructed). The observation of O’Grady is this: given two connected components N1 and N2 of N and a general point [Vi ] of Ni , we can make two cofinite subsheaves V1 ⊂ V1 and V2 ⊂ V2 that lie in a flat family over a connected base. Moreover, the construction [Vi ] → [Vi ] gives a well-defined Galois-equivariant map between connected components! 4. Now, to produce a geometrically integral locally closed substack of N , we argue as follows: it is enough to find a geometrically connected open substack (as N is smooth). By bounding, for example, regularity or adding a stability condition, we may replace N with an open substack of finite type over K, hence possessing only finitely many connected components, say ξ1 , . . . , ξm . This gives a continuous Galois action on the set {ξ1 , . . . , ξm }. Applying the trick from the previous step, we can find another Galois set S of connected components of N and an equivariant map {ξi } → S whose image is a singleton. In other words, we have found a Galois fixed point in the set of connected components of N , which gives a geometrically connected open substack, as desired. What about higher-dimensional ambient spaces? Here the situation is much more mysterious. Things like the Hilbert scheme of finite-length closed subschemes are already a disaster. Question 6.5. Is it impossible for anything like Principle 6.4 to hold for higherdimensional ambient spaces X in any kind of reasonable generality? Does “any” nonempty moduli space of stable sheaves on a threefold contain a geometrically integral locally closed subspace?


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While this would not necessarily help us understand things for twisting classes on varieties over algebraically closed fields, it would certainly imply interesting things about unramified classes on varieties over finite fields.

7 Applications to Algebra The modest amount of moduli theory described in Sect. 6 has a rather outsized payoff in its applications to algebra, in particular to Questions 2.2–2.4. Let us start with a reminder of a classical problem on algebra known as the period-index problem. This is carefully explained in [26] (and goes back to [21, 23] in a slightly different form), so we will be brief. Fix a Brauer class α over a field K. We can associate to α a finite-dimensional central division algebra D and also a Brauer–Severi variety X. Recall from above that the index of α is ind(α ) = dimK D = min{[L : K]|X(L) = 0}. / The period of α is its order in the Brauer group; in terms of the geometry of X, the period is the smallest positive integer d such that X contains a divisor D ⊂ X of ∼ degree d via any (equivalently: some) identification X ⊗ K → PnK . Colloquially: the index is the smallest degree of a point in X and the period is the smallest degree of an effective divisor in X. Using basic Galois cohomology, one can show that per(α )| ind(α ) and that they have the same prime factors, which means that there is some integer α such that ind(α )| per(α )α . The obvious question, which turns out to be surprisingly deep, is how one can characterize this integer α . • To what extent does α depend on α ? • To what extent can α be made to depend only on K or some property of K? Colliot-Thélène asked if the following is true. Question 7.1. If K is a Cd -field then must it be true that α = d − 1 always works? In other words, do we always know that ind(α )| per(α )d−1 ? The answer is known to be “yes” in a shockingly small number of cases. Theorem 7.2 (Omnibus period-index theorem). The answer to Question 7.1 is “yes” in the following cases: 1. 2. 3. 4.

K is algebraically closed (Gauss). K is C1 (Lang). K is of transcendence degree 1 over a finite field (Brauer-Hasse-Noether). K is of transcendence degree 2 over an algebraically closed field (de Jong, de Jong-Starr, Lieblich [10, 11, 26]). 5. K is of transcendence degree 2 over a finite field [28].

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6. K is of transcendence degree at most 1 over a higher local field with residue field of characteristic 0 [27]. Sinkhole 7.3. It is remarkable to note that it is not known if one can produce any bound at all for the Brauer group of any chosen threefold. For example, we do not 200000000000 know if there is some (even 210 ) such that for all α ∈ Br(C(x, y, z)) we have ind(α )| per(α ) . This is deeply disturbing. (However, the result quoted above for surfaces over a finite field does at least give an example of a naturally arising geometric class of C3 -fields for which the answer to Question 7.1 is “yes.”) Note that we do know that the period and index need not be equal for classes over this field [9]. As it turns out, the proof of each statement of Theorem 7.2 can be deduced from the geometry of a suitable moduli space of twisted sheaves. Let us sketch the main ideas, giving references to the literature where appropriate. Proof (Omnibus sketch of proof of Theorem 7.2). The idea is this: given a field K and a class α as in the statement of Theorem 7.2, attach a gerbe X → X on some kind of nice algebraic space or stack of dimension at most two over a small field k ⊂ K. To bound the index of α by an integer N, one needs to find a coherent X twisted sheaf of rank N. Stupidly, one is looking for a point on the moduli stack M of such sheaves. Strategy 7.4. The fundamental paradigm of algebraic geometry gives us this strategy: • First, study M ⊗ k. Important: show that it is nonempty! If possible, show that it is integral. (If necessary, restrict to a locally closed substack of M and start over.) • Now hope that M is nice enough and k is small enough that we can establish that M has a 0-cycle of degree 1. For example, if k = C(t), it is enough for M to be unirational. Similarly, if k is finite (or, more generally, pseudo-algebraically closed), it is enough that M be geometrically integral. • Even better: try to package this inductively by fibering X and studying bundles of moduli spaces for lower-dimensional problems. This strategy works perfectly for the cases enumerated in Theorem 7.2 and then fails miserably (whence the abrupt drop-off in our understanding). Let us illustrate with the first nontrivial case: de Jong’s theorem that period and index are equal for Brauer classes over function fields of surfaces (over algebraically closed fields k). Let us suppose X is a surface and α ∈ Br(X) is a Brauer class. (What about classes over k(X) instead of X? This is a very important question, and we will return to it soon.) Choosing a pencil of very ample divisors and blowing up its base locus, we may assume that X admits a generically smooth proper flat morphism


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π : X → P1 with at least one section σ : P1 → X. Suppose per(α ) = n. Choose a μ n -gerbe X → X representing α and let M → P1 be the relative moduli stack parameterizing geometrically stable twisted sheaves of rank n on the fibers of π with determinant identified with O(σ (P1 )). We know that M → M is a μ n -gerbe over a smooth algebraic space that is proper over the locus U ⊂ P1 parameterizing smooth fibers of π . Claim. The fibration MU → U has a section. Proof. Proposition 6.2 tells us that M has unirational fibers over U. By the GraberHarris-Starr theorem, M has a rational section, i.e., a rational point P over k(t). On the other hand, M → M is a μ n -gerbe, so the obstruction to lifting P to an object of M (and thus a proof that ind(α ) = per(α )) lies in H2 (Spec k(t), μ n ) ⊂ Br(k(t)). By Tsen’s theorem, the latter group is 0. This establishes the proof. (And—in spite of being the “simplest” case—it already includes all steps of Strategy 7.4: fibering, taking advantage of nice moduli, and using induction on dimensions!) What if

α ∈ Br(k(X)) \ Br(X)? has the geometry escaped? There is a special property of working over an algebraically closed field that saves the day. Claim. Let H be a general sufficiently ample divisor on X such that D + H is ample. Let X˜

X

P1 be a general pencil of elements of |D + H| with generic fiber C → Speck(t). Then the class α lies in the subgroup Br(C) ⊂ Br(k(X)) = Br(k(t)(C)). Idea of proof. The proof relies on the theory of ramification of Brauer classes, as described in, for example, [2, Sect. 3]. The crux of the matter is this: there is a smallest divisor D ⊂ X such that α lies in the subgroup Br(X \ D) ⊂ Br(k(X)).

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Moreover, there is a cyclic extension of each generic point of D that measures the failure of α to extend across that divisor, and these cyclic extensions can themselves only ramify over singular points of D. Even better, this whole setup is mildly functorial, in the following sense: given another regular scheme with a morphism f :Z→X such that f −1 (D) is a Cartier divisor, we have that the ramification extensions over f −1 (D) are the preimages of the ramification extensions over D. A general pencil of divisors in |D + H| will have base locus intersecting D in smooth locus D. In particular, the preimage of D in the incidence correspondence X˜ → X consists of a union of D and several sections of X˜ → P1 . Moreover, mild functoriality dictates that the ramification extensions on these P1 s are the preimage of the ramification extension of D at the corresponding basepoint, but then they are trivial (as k is algebraically closed). Restricting to C, we see that the ramification is trivial. We can now use the same argument on the relative moduli space to arrive at a proof that the period and index coincide. Now let us follow Strategy 7.4 and consider an unramified class α ∈ Br(Z), where Z is a smooth projective surface over a finite field. Let Z → Z be a μ n gerbe representing α and let M → M be the μ n -gerbe parameterizing stable locally free Z -twisted sheaves of rank n, determinant O, and sufficiently large second Chern class. By Proposition 6.3, the space M is eventually geometrically integral. But then the Lang-Weil estimates [22] tell us that M has a 0-cycle of degree 1, and Wedderburn’s theorem (induction again!) tells us that we can lift this 0-cycle to M . Since we really interested in the K-theory of Z , it is enough to find such a 0-cycle to establish that again per(α ) = ind(α ). (In other words, we can cook up a perfect complex of Z -twisted sheaves that has rank n using the 0-cycle on M . Of course, after the fact one can simply make a twisted sheaf of rank n.) Comparing the two situations shows us that there is a basic interplay at work: Interplay 7.5. In the modular interpretation of the period-index problem, we can arrive at affirmative answers by simple geometry over larger fields or hard geometry over smaller fields. Given the nature of the proof and Principle 6.4, one might think that per(α ) = ind(α ) for any class α in the Brauer group of a field of transcendence degree 2 over a finite field! Unfortunately, this is not true: as proven in [9], for any odd prime power q and any element × 2 a ∈ F× q \ (Fq ) ,


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the class (x, y) ⊗(a, 1 − x) (tensor product of quaternion algebras over Fq (x, y)) has period 2 (it is a sum of classes of period 2!) and index 4 (it is itself a division algebra). But this class is ramified: it does not lie in Br(P2Fq ) = 0. Let us think more about this. Fix a class α in the Brauer group of the function field of a surface X over a finite field. Interpreting the theory of ramification in a certain way tells us that while the class α might not live in Br(X), there is a stacky surface X with the same function field and such that α ∈ Br(X ). (The Brauer group of a stack is defined using Azumaya algebras in precisely the same way as for schemes.) More precisely, let us suppose that there is a strict normal crossing divisor D = D1 + · · · + Dm ⊂ X such that α is ramified precisely along D. Let us suppose in addition that n = is a prime invertible on X. Then we have the following: there is a stack X → X of “th roots of each Di .” Its étale-local structure around a point x ∈ X is the same as: 1. Spec k[s,t] if x ∈ X \ D 2. [Spec k[s,t]/ μ ] with ζ · (s,t) = (ζ s,t) if x lies in precisely one component of D 3. [Spec k[s,t]/ μ × μ ] with (ζ , η ) · (s,t) = (ζ s, η t) if x is a singular point of D A sheaf on a stack has representations of stabilizer groups as fibers (rather than simply vector spaces). Roughly speaking, we can think about twisted sheaves the same way. In particular, we must specify the representations that occur as part of any moduli problem of twisted sheaves. As it turns out, the only consistent way to do this that works for all Brauer classes requires a multiple of the “regular representation” at points of type (2) above, and for certain points of type (3) above, this is only possible along both branches if the sheaves have rank 2 (essentially, the representations are given by tensoring the regular representation of each factor group). This is a stacky interpretation of Saltman’s well-known meteorology of ramification points [42] and is explained in [27, 28]. In other words, there is a local obstruction to the equality of period and index for ramified classes on surfaces over finite fields. Once the local obstruction has been erased by a suitable definition of the moduli problem, we can indeed follow Principle 6.4 (and the usual proofs work, for the most part).

8 Applications to Arithmetic Applications to algebra are amusing, but the usefulness of twisted sheaves does not stop there. There is an interesting interaction between the moduli theory of twisted sheaves, Colliot-Thélène’s conjectures on 0-cycles and the Brauer–Manin obstruction, and various properties of finitely generated fields. There is also an interplay between the derived category of twisted sheaves on a K3 surface, FourierMukai equivalences, and the Tate conjecture for K3 surfaces over finite fields.

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Let us start by recalling the statement of the (mildest form of) Colliot-Thélène conjecture. Conjecture 8.1 (Colliot-Thélène). If X is a smooth projective geometrically rational variety over a global field k, then the Brauer–Manin obstruction is the only obstruction to the existence of 0-cycles on X of degree 1 over k. The Brauer–Manin obstruction is described in [46]. In [24] (part of which improves results of [27]), the reader will find proofs of the following results. Proposition 8.2. Let k be a totally imaginary global field with scheme of integers V and K/k be a finitely generated field extension of transcendence degree 1. Suppose X → V is regular proper model of K over V . If Colliot-Thélène’s conjecture holds over extensions of k, then: 1. Every Brauer class α ∈ Br(X) satisfies ind(α )| per(α )2 . 2. Every class α ∈ Br(K) satisfies ind(α )| per(α )5 . While we already know (more than) this when k has positive characteristic (as explained in the previous section), this is totally foreign over number fields. The powerful techniques of the last section—compactifying, transforming a curve over a function field into a surface, etc.—are unavailable over a number field. The crux of the proof is an analysis of ramification of p-torsion classes at places lying over p. This turns out to be surprisingly subtle, reliant on the work of Bloch and Kato (especially [20], which is beautifully described in [8]), and special to the case of surfaces. Low dimensionality again rears its ugly (beguiling?) head. Idea of proof. We proceed as in Sect. 7: let C/k be the proper smooth curve with function field K. After passing to the algebraic closure of k in K, we may assume that C is geometrically connected. (Note: this is the only place where we need to make a nontrivial extension of k.) Write M (resp. M) for the stack (resp. coarse space) of geometrically stable twisted sheaves on C of rank and trivial determinant. Assume α is unramified on all of X. Given a place v of k with complete local ring Ov , we know that Br(XOv ) = 0; it follows from the deformation theory of twisted sheaves that there is a stable locally free twisted sheaf with trivial determinant and rank . In other words, M (Ak ) = 0. / On the other hand, M → M generates Br(M), so the resulting elements of M(Ak ) lie in M(Ak )Br . Choosing a smooth compactification of M, we find that the same holds. Applying Colliot-Thélène’s conjecture and a simple moving argument, we see that there is a 0-cycle of degree 1 in M. Since the obstruction to lifting this to a 0-cycle in M is in the Brauer group of k (and its finite extensions), we see that this introduces at most one more factor of into the estimate: after making an extension of the 0-cycle of degree at most we can lift to the stack. This shows the desired statement.


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To get to the ramified case one can split the ramification by making a suitable field extension. The methods involved are beyond the scope of this survey; the reader is referred to [24] for more details. Building on this result, one can also prove the following. Recall that the u-invariant of a field K is the largest dimension of an anisotropic quadratic form over K. In general, questions about the u-invariant turn out to be quite subtle. For example, the u-invariant can take any power of 2 as a value, but not 3, 5, or 7. Famous results of Merkurjev (resp. Izhboldin) give a field with u-invariant 6 (resp. 9). It is not clear how the u-invariant should behave in field extensions (unlike, say, the Cd -property). Theorem 8.3. Suppose K/k is a field extension of transcendence degree 1 over a totally imaginary number field. If Conjecture 8.1 holds then the u-invariant of K is finite. Idea of proof. Parimala and Suresh proved [40] that the u-invariant of K will be finite if there is some N such that HN (K, μ 2 ) = 0 and for each i < N there is a number λi such that the elements of Hi (K, μ 2 ) are sums of at most λi symbols. Since K has transcendence degree 1 over a totally imaginary number field, one has that H4 (K, μ 2 ) = 0. The works of Suresh [47] and Arason [1] then show that if λ2 exists then λ3 = λ2 . In other words, one is reduced to bounding the number of symbols it takes to write Brauer classes of order 2. Theorem 8.2 shows that any Brauer class of period 2 has index dividing 32. But the generic central simple algebra of degree 32 over K has an expansion as a sum of (finitely many!) symbols by Merkurjev’s theorem; by specialization this provides an upper bound for all algebras of index 2 over K, showing that λ2 exists. There is another amusing application of the theory of twisted sheaves to a certain kind of arithmetic question: given a finite field k, are there only finitely many K3 surfaces over k? This is a simple question in a long line of similar questions (are there only finitely many curves of genus g? finitely many abelian varieties of dimension g? etc.). It turns out that this question is intimately related to the Tate conjecture. Theorem 8.4 ([31]). Fix a finite field k. The Tate conjecture holds for all K3 surfaces over finite extensions of k if and only if there are only finitely many isomorphism classes of K3 surfaces over each finite extension of k. Idea of proof. The relevant direction of this theorem is how one might deduce that the Tate conjecture holds for K3 surfaces if there are only finitely many K3 surfaces over each finite field. (The other direction is more believable: the Tate conjecture hands control over the Picard group to the Galois action on the étale and crystalline cohomology. Using semisimplicity of Frobenius and a bit of elbow grease, one can thus control the degrees of divisors, bounding the degree of an ample divisor. This embeds all of the K3s over k into a fixed projective space, so there can be only finitely many.)

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If one starts with a K3 surface X over k with infinite Brauer group Br(X), one would like to show that there are infinitely many K3 surfaces over k. If Br(X) is infinite, then in fact there is a prime and a sequence of classes αn such that αn = αn−1 and αn has order exactly n . From this sequence we can construct infinitely many K3 surfaces as moduli spaces of stable twisted sheaves! This is familiar from the classical theory of Mukai [36]: given a K3 surface, one can make many new K3 surfaces M as moduli spaces of sheaves on X. Moreover, there are equivalences of derived categories of coherent sheaves D(X) ∼ = D(M). The idea is to apply precisely the same “numerical technology” to show that many moduli spaces of twisted sheaves are themselves also K3 surfaces. So, associated to αn , we get Mn and equivalences of derived categories D(Mn ) ∼ = D(Cohtw (Xn )), where Xn → X is a gerbe representing αn . (In the existing literature, the latter category is often written D(X, αn ).) If there are finitely many K3s over k, then infinitely many of the Mn coincide. Flipping this around, there is one K3 surface with infinitely many “twisted partners.” By carefully lifting to characteristic 0 and using a twisted variant of the classical Torelli theorem, one can show that this is impossible. The reader is referred to [31] for details. Remark 8.5. One could compare this kind of argument to Tate’s proof of the Tate conjecture for abelian varieties over finite fields [48] or the Artin–SwinnertonDyer proof of the Tate conjecture for elliptic K3 surfaces over finite fields [3]. In each case, an argument of a similar flavor appears: one has a geometric finiteness statement (for abelian varieties with a given dimension, for K3 surfaces with a polarization of bounded degree) that, combined with an assumption about the infinitude of the Brauer group, leads to a contradiction by the pigeon-hole principle. Remark 8.6. The Tate conjecture for K3 surfaces over finite fields of characteristic at least five has recently been proved: [7, 34, 41]. In particular, Theorem 8.4 implies that there are only finitely many K3 surfaces over any finite field of characteristic at least five. It would be amusing to find a proof that makes use of the opposite implication in Theorem 8.4; this is not entirely far-fetched, as the existing proofs have the quasi-finiteness of the Kuga–Satake morphism as a key aspect.

9 Applications to Geometry Before concluding, let us briefly mention a recent application of the ideas found in Sects. 6 and 8 to the geometry of the moduli space of supersingular K3 surfaces in characteristic p. These are K3 surfaces with Picard number 22. They also have a remarkable property: some very large, positive-dimensional, cohomology with finite coefficients. This is a theorem first proven by Artin, but the proof seems to have escaped publication. A proof will appear in [29], along with the details of everything else in this section.


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Proposition 9.1 (Artin). Let f : X → Speck be a supersingular K3 surface. The fppf sheaf R2 f∗ μ p is representable by a smooth group scheme locally of finite type over k whose connected component containing 0 is isomorphic to Ga . Suppose k is algebraically closed. Playing with the Leray spectral sequence, one finds that there is a universal class

α ∈ H2 (X × A1 , μ p ) giving rise to the immersion Ga → R2 f∗ μ p . We can choose a μ p -gerbe X → X × A1 representing α . This gives a continuous ptorsion deformation of the trivial Brauer class over A1 . (In fact, the existence of such a deformation characterizes supersingular K3 surfaces.) Using moduli spaces of twisted sheaves and equivalences between their derived categories, we can transform this moving Brauer class into a moving surface. While there are many ways to do this, there is one in particular that links this back up nicely to Example 6.1. Suppose given an elliptic pencil X → P1 (supersingular K3s always have infinitely many of these, although they form finitely many orbits under the automorphism group [30]). Definition 9.2. Let M → P1 × A1 be the moduli space of stable sheaves of rank 1 and degree 1 on the fibers of X × A1 → P 1 × A1 (with respect to a fixed sufficiently general polarization). This construction is very natural and geometric. Even better, it is precisely a geometrization of the Artin–Tate isomorphism [49] between Br(X) and III(Jac(Xη )), where Xη is the generic fiber of the elliptic pencil: Proposition 9.3. Given a field K and a point Spec K → A1 , let β ∈ Br(XK ) denote the pullback of α . Then the fibration MK → P1K is an étale form of XK → P1K and there is an isomorphism of Q-lattices ∼

H(X) ⊗ Q → H(MK ) ⊗ Q. Moreover, it is precisely the translate of the constant form by the element of III (Xη ) associated to β . In particular, if β is not defined over k, neither is MK . Here, the lattice H(X) is formed by taking the Chow theory CH(X) = CH0 (X) ⊕ CH1 (X) ⊕ CH2 (X) and imposing a different intersection form, namely, that (a, b, c) · (a , b , c ) = bb − ac − ac ∈ CH2 (X) = Z.

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One can in fact think of this lattice isomorphism as a reflection of something motivic, and one can see it in the derived category in the form of an equivalence between two suitable derived categories of twisted sheaves. The upshot of Proposition 9.3 is that the formation of the space of stable twisted sheaves transforms the curve Ga in cohomology into a family of K3 surfaces over A1 whose fiber at 0 is X and whose geometric generic fiber cannot be defined over k. In other words, it is a truly moving family of K3 surfaces. We will now see how to make this a curve in a moduli space. In his deep study of the crystalline cohomology of supersingular K3 surfaces [39], Ogus produced a period domain, analogous to the classical period domain coming from Hodge theory for complex K3 surfaces. This space parametrizes pairs (X, ι : N → Pic(X)), where X is a supersingular K3 surface, N is one of ten lattices, and ι is a lattice embedding into the Picard group. Of particular interest is the generic case (so-called Artin invariant 10). Here, N has discriminant −p20 and ι is an isomorphism. Ogus showed that this defines a dense open subspace of the period space. Let us call this subspace U. The theorem we have proven is following. Theorem 9.4. For any point (X, ν ) of U, the construction of Proposition 9.3 yields a nonconstant morphism A1 → U sending 0 to (X, ν ). All that remains to explain is why the marking should deform along with the surface. This comes from the isomorphism between lattices: since H(X) is invariant under field extension, there can be no monodromy action on H(M) ⊗ Q (i.e., no nontrivial action of the Galois group of k(t) on the Picard group Pic(Mk(t) ) ⊗ Q) and thus none on H(M). Of course, the space U is highly non-separated and generally pathological. If instead one looks at the supersingular locus in any moduli space of polarized K3 surfaces, one will find precisely the same phenomenon—a covering by rational curves—since one can dominate these loci by U in all cases. Acknowledgements During the writing of this paper, the author was partially supported by a Sloan Fellowship and NSF CAREER grant DMS-1056129. He thanks the referee for helpful comments and the Simons Foundation for its generous support of the Simons Symposia.

References 1. JÓN KR . ARASON , Cohomologische invarianten quadratischer Formen, J. Algebra, 36(3): 448–491, (1975). 2. M. ARTIN and D. M UMFORD , Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3), 25:75–95, (1972). 3. M. ARTIN and H. P. F. SWINNERTON -DYER , The Shafarevich-Tate conjecture for pencils of elliptic curves on K3 surfaces, Invent. Math., 20:249–266, (1973).


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27. M AX LIEBLICH , Period and index in the Brauer group of an arithmetic surface, J. Reine Angew. Math. 659:1–41, (2011). With an appendix by Daniel Krashen. The period-index problem for fields of transcendence degree 2, 28. M AX LIEBLICH , arXiv:0909.4345, (2009). 29. M AX LIEBLICH , The moduli space of supersingular K3 surfaces is uniruled, (2012). Preprint in preparation. 30. M AX LIEBLICH and DAVESH M AULIK , A note on the cone conjecture for K3 surfaces in positive characteristic, (2011). 31. M AX LIEBLICH , DAVESH M AULIK , and ANDREW SNOWDEN , Finiteness of K3 surfaces and the Tate conjecture, (2012). 32. SHOUHEI M A , On the 0-dimensional cusps of the Kähler moduli of a K3 surface, Math. Ann. 348(1):57–80, (2010). 33. DIMITRI M ARKUSHEVICH , Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces, Manuscripta Math. 120(2):131–150, (2006). 34. DAVESH M AULIK , Supersingular K3 surfaces for large primes. arXiv:1203.2889, (2012). 35. JAMES S. M ILNE, Étale cohomology, volume 33 of Princeton Mathematical Series, Princeton University Press, Princeton, N.J., 1980. 36. S. M UKAI , On the moduli space of bundles on K3 surfaces. I, In Vector bundles on algebraic varieties (Bombay, 1984), volume 11 of Tata Inst. Fund. Res. Stud. Math., 341–413. Tata Inst. Fund. Res., Bombay, 1987. 37. P. E. NEWSTEAD , Introduction to moduli problems and orbit spaces, volume 51 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Tata Institute of Fundamental Research, Bombay, 1978. 38. KIERAN G. O’GRADY , Moduli of vector bundles on projective surfaces: some basic results, Invent. Math. 123(1):141–207, (1996). 39. ARTHUR OGUS, A crystalline Torelli theorem for supersingular K3 surfaces, In Arithmetic and geometry, Vol. II, volume 36 of Progr. Math., 361–394. Birkhäuser Boston, Boston, MA, 1983. 40. RAMAN PARIMALA and VENAPALLY SURESH , On the length of a quadratic form, In Algebra and number theory, 147–157. Hindustan Book Agency, Delhi, 2005. 41. KEERTHI M ADAPUSI PERA , The Tate conjecture for K3 surfaces on odd characteristic, In preparation. 42. DAVID J. SALTMAN , Division algebras over surfaces, J. Algebra 320(4):1543–1585, (2008). 43. JUSTIN SAWON , Derived equivalence of holomorphic symplectic manifolds, In Algebraic structures and moduli spaces, volume 38 of CRM Proc. Lecture Notes, 193–211. Amer. Math. Soc., Providence, RI, 2004. 44. JUSTIN SAWON , Lagrangian fibrations on Hilbert schemes of points on K3 surfaces, J. Algebraic Geom. 16(3):477–497, (2007). 45. JEAN -PIERRE SERRE, Local fields, volume 67 of Graduate Texts in Mathematics. SpringerVerlag, New York, 1979. Translated from the French by Marvin Jay Greenberg. 46. ALEXEI SKOROBOGATOV , Torsors and rational points, volume 144 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001. 47. VENIPALLY SURESH , Galois cohomology in degree 3 of function fields of curves over number fields, J. Number Theory, 107(1):80–94, 2004. 48. JOHN TATE, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2:134–144, (1966). 49. JOHN TATE, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, In Séminaire Bourbaki, Vol. 9, Exp. No. 306, 415–440. Soc. Math. France, Paris, 1995. 50. YUKINOBU TODA , Deformations and Fourier-Mukai transforms, J. Differential Geom. 81(1):197–224, (2009).


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Algebraic Surfaces in Positive Characteristic Christian Liedtke

Mathematics Subject Classification codes (2010): 14-02, 14J10, 14G17.

1 Introduction These notes grew out of a series of lectures given at Sogang University, Seoul, in October 2009. They were meant for complex geometers, who are not familiar with characteristic-p-geometry but who would like to see similarities, as well as differences, to complex geometry. More precisely, these notes are on algebraic surfaces in positive characteristic and assume familiarity with the complex side of this theory, say, on the level of Beauville’s book [9]. Roughly speaking, the theory of curves in characteristic zero and p > 0 looks very similar, and many fundamental results were already classically known to hold in arbitrary characteristic. Also, curves lift from characteristic p over the Witt ring to characteristic zero, which implies that many “characteristic-p-pathologies” cannot happen. Abelian varieties admit at least formal lifts over the Witt ring, and we refer to Sect. 11 for details and implications. However, from dimension two on, geometry in positive characteristic displays more and more differences to classical complex geometry. In fact, this geometry has long been considered as “pathological” and “exotic,” as even reflected in the titles of a series of articles by Mumford (the first one being [97]). There, the emphasis was more on finding and exploring differences to the classical theory. For a short overview over the main new phenomena for surfaces in positive characteristic, we refer to [59, Sect. 15].

C. Liedtke () Mathematisches Institut, Endenicher Allee 60, D-53115 Bonn, Germany e-mail: [email protected] F. Bogomolov et al. (eds.), Birational Geometry, Rational Curves, and Arithmetic, DOI 10.1007/978-1-4614-6482-2__11, © Springer Science+Business Media New York 2013

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However, in their three fundamental articles [16, 17, 99], Bombieri and Mumford established the Kodaira–Enriques classification of algebraic surfaces in positive characteristic. Together with Artin’s results [1,2] on surface singularities, especially rational and Du Val singularities, as well as work of Ekedahl [37] on pluricanonical systems of surfaces of general type (extending Bombieri’s results to characteristic p), this sets the scene in positive characteristic. It turns out that surface theory in positive characteristic is in many respects not so different from characteristic zero, at least, if one takes the right angle. As over the complex numbers, there are a vast number of examples, counterexamples, and (partial) classification results for special classes of surfaces in positive characteristic. Unfortunately, it was impossible for me to mention all of them in these introductory notes—for example, I could have written much more on K3 surfaces, elliptic surfaces, and (birational) automorphisms of surfaces. These notes are organized as follows: Preparatory Material Section 2 We introduce the various Frobenius morphisms and proceed to basic results on algebraic curves. Finally, we discuss finite, constant, and infinitesimal group schemes, as well as the three group schemes of length p. Section 3 We recall Hodge, étale, and deRham (hyper-)cohomology. Next, we discuss Albanese and Picard schemes, non-closed differential forms, and their relation to (non-)degeneracy of the Frölicher spectral sequence from Hodge to deRham cohomology. Finally, we sketch how crystalline cohomology links all the above mentioned cohomology theories. Classification of Algebraic Surfaces Section 4 We discuss blowups and Castelnuovo’s contraction theorem, introduce minimal models, and describe the structure of rational and birational maps of surfaces. We classify birationally ruled surfaces and state the rationality theorem of Castelnuovo–Zariski. Section 5 We recall elliptic fibrations and discuss the phenomenon of quasi-elliptic fibrations and wild fibers. Then, we state the canonical bundle formula and give the possible degeneration types of fibers in (quasi-)elliptic fibrations. Section 6 We sketch the Kodaira–Enriques classification of algebraic surfaces according to their Kodaira dimension. Section 7 We discuss the four classes of minimal surfaces of Kodaira dimension zero in greater detail. We put an emphasis on the nonclassical classes of Enriques surfaces in characteristic 2, as well as the new classes of quasi-hyperelliptic surfaces in characteristics 2 and 3. Section 8 We start with Ekedahl’s work on pluricanonical maps of surfaces of general type. Then, we continue with what is known about various inequalities (Noether, Castelnuovo, Bogomolov–Miyaoka–Yau) in positive characteristic and end with a couple of results on surfaces of general type with small invariants.

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Special Topics in Positive Characteristic Section 9 We study uniruled surfaces that are not birationally ruled and introduce two notions of supersingularity, due to Artin and Shioda. Then, we discuss these notions for K3 surfaces. Next, we turn to surfaces over finite fields, zeta functions, and the Tate conjecture. Section 10 We explain Jacobson’s correspondence for purely inseparable field extensions. On the geometric level, this corresponds to p-closed foliations. We give applications to global vector fields on K3 surfaces and end by discussing quotients by infinitesimal group schemes. From Positive Characteristic to Characteristic Zero Section 11 We recall the ring of Witt vectors, discuss various notions of what it means to “lift to characteristic zero,” and discuss what each type of liftings implies. We end by giving examples and counterexamples. Section 12 As an application of characteristic-p-geometry, we establish infinitely many rational curves on complex projective K3 surfaces of odd Picard rank using reduction modulo p and special characteristic-p-features. Finally, we advise the reader who is interested in learning surface theory over algebraically closed ground fields of arbitrary characteristic from scratch (including proofs) to have a look at Badescu’s excellent textbook [6]. From there, the reader can proceed to more advanced topics, including the original articles by Bombieri and Mumford mentioned above, as well as the literature given in these notes.

2 Frobenius, Curves, and Group Schemes Before dealing with surfaces, we first shortly review a little bit of background material. Of course, the omnipresent Frobenius morphism has to be mentioned first—in many cases, when a characteristic zero argument breaks down in positive characteristic, inseparable morphisms and inseparable field extensions are responsible. The prototype of an inseparable morphism is the Frobenius morphism, and in many situations it also provides the solution to a problem. Next, we give a very short overview over curves and group scheme actions. We have chosen our material in view of what we need for the classification and description of surfaces later on.

2.1 Frobenius Let us recall that a field k of positive characteristic p is called perfect, if its Frobenius morphism x → x p is surjective, i.e., if every element in k has a pth root in k.

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For example, finite fields and algebraically closed fields are perfect. On the other hand, function fields of varieties in positive characteristic are almost never perfect. Let X be an n-dimensional variety over a field k with structure morphism f : X → Spec k. Then, the morphism that is the identity on points of X and is x → x p on the structure sheaf OX is called the absolute Frobenius morphism Fabs of X. However, the absolute Frobenius morphism is not “geometric”: namely, it acts as x → x p on the ground field k, which is nontrivial except for k = F p . To obtain a morphism over k, we first form the pullback pr

2 Spec k X (p) := X ×Spec k Spec k −→

with respect to the structure map f : X → Spec k and with respect to the absolute Frobenius Fabs : Spec k → Spec k. This Frobenius pullback f (p) := pr2 : X (p) → Spec k is a new variety over k. If k is perfect then X and X (p) are abstractly isomorphic as schemes, but not as varieties over k (see below). Now, by the universal property of pullbacks, we obtain a morphism of varieties over k, the k-linear Frobenius morphism F : X → X (p) X Fabs F

X (p)

X

f f (p)

f Fabs

Spec k

Spec k

In more down to earth terms and for affine space this simply means k[x1 , . . . , xn ] → k[x1 , . . . , xn ] absolute Frobenius : f (x1 , . . . , xn ) → ( f (x1 , . . . , xn )) p k-linear Frobenius : c → c if c ∈ k. → xip xi When dealing with varieties over finite fields there are even more Frobenius morphisms: over the field Fq with q = pn elements one has a Frobenius morphism Fq : x → xq , and for technical reasons sometimes its inverse has to be considered (see, e.g., [48, Appendix C.4]). Depending on author and context all these morphisms and various base changes are called “Frobenius” and so, a little care is needed. Next, if X is n-dimensional over k, then the k-linear Frobenius F : X → X (p) is a finite morphism of degree pn . Moreover, if k is perfect, then, on the level of function fields, this morphism corresponds to the inclusion k(X (p) ) = k(X) p ⊆ k(X) .


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Note that k(X) p , the set of pth powers of k(X), is in fact a field: it is not only closed under multiplication but also under addition since x p + y p = (x+ y) p in characteristic p. Let us also fix an algebraic closure K of K = k(X). For every integer i ≥ 0 we define Kp

−i

i

:= { x ∈ K | x p ∈ K } −i

and note that these sets are in fact fields. The field K p is a finite and purely −∞ inseparable extension of K of degree pni . Their union K p as i tends to infinity is called the perfect closure of K in K, as it is the smallest subfield of K that is perfect and contains K. Definition 2.1. Let L be a finite and purely inseparable extension of K. The height −i of L over K is defined to be the minimal i such that K ⊆ L ⊆ K p . Similarly, if ϕ : Y → X is a purely inseparable and generically finite morphism of varieties over a perfect field k, then the height of ϕ is defined to be the height of the extension of function fields k(Y )/k(X). For example, the k-linear Frobenius morphism is of height one. For more on inseparable morphisms, we refer to Sect. 10.

2.2 Curves Most of the results of this section are classical, and we refer to [48, Chap. IV] or [86] for details, specialized topics, and further references. Let C be a smooth projective curve over an algebraically closed field of characteristic p ≥ 0. Then its geometric genus is defined to be g(C) := h0 (C, ωC ) = h1 (C, OC ), where ωC denotes the dualizing sheaf. The second equality follows from Serre duality. Since C is smooth over k, the sheaf ωC is isomorphic to the sheaf of Kähler differentials ΩC/k . Let ϕ : C → D be a finite morphism between smooth curves. Let us also assume that ϕ is separable, i.e., the induced field extension k(D) ⊂ k(C) is separable. Then, the Riemann–Hurwitz formula states that there is a linear equivalence of divisors on C KC ∼ ϕ ∗ (KD ) +

∑ length(ΩC/D )P · P.

P∈C

Here, ΩC/D is the sheaf of relative Kähler differentials. Since ϕ is separable, it is generically étale, and thus, ΩC/D is a torsion sheaf supported in finitely many points. By definition, the points in the support of this sheaf are called ramification points. In case ϕ is inseparable, ΩC/D is nontrivial in every point, and every point would count as ramification point.

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For a point P ∈ C with image Q = ϕ (P), and still assuming ϕ to be separable, we choose a local parameter t ∈ OD,Q and define the ramification index eP of ϕ at P to be the valuation of ϕ # (t) in OC,P (see [48, Sect. IV.2]). Then, the ramification at a ramification point P is called tame, if eP is not divisible by p = char(k), and it is called wild otherwise. We have if P is tame, = eP − 1 length(ΩC/D )P > eP − 1 if P is wild. In general, it is very difficult to bound eP whenever the ramification is wild (see the example below). An important case where one can say more about wild ramification is in case ϕ is a Galois morphism; then, one can define for every wild ramification point P certain subgroups of the Galois group, the so-called higher ramification groups, that control the length of ΩC/D at P (cf. [128, Chap. IV.1]). Example 2.2. Galois covers with Galois group Z/pZ are called Artin–Schreier covers. An example is ϕ : C → P1 given by the projective closure and normalization of the affine equation z p − z = t hp−1 . This cover is branched only over t = ∞, the ramification is wild of index e∞ = p, and we have length(ΩC/P1 )∞ = p(p − 1)h. Thus, C is a curve of genus 1 − p + 12 p (p − 1)h and there are (p − 1)h nontrivial higher ramification groups. In particular, ϕ defines a nontrivial étale cover of A1 , which implies that the affine line A1 is not algebraically simply connected. In fact, by Raynaud’s theorem (formerly Abhyankar’s conjecture), every finite group that is generated by its pSylow subgroups occurs as quotient of π1e´ t (A1 ). We refer to [20] for an overview and references. However, it is still true that every étale cover of P1 is trivial, i.e., P1 is algebraically simply connected [44, Chap. XI.1]. If ϕ is purely inseparable, there still exists a sort of Riemann–Hurwitz formula. We refer to [36] or [94, Lecture III] for more information on ΩC/D in this case. In this case, the “ramification divisor” is defined only up to linear equivalence. On the other hand, the structure of purely inseparable morphisms between curves is simple: namely, every such morphism is just the composite of k-linear Frobenius morphisms (Proposition 10.3). However, from dimension two on, inseparable morphisms become more complicated. We will come back to this in Sect. 10. Let us now give a rough classification of curves: If a smooth projective curve over a field k has genus zero, then ωC∨ is very ample and embeds C as a quadric in P2k . Moreover, a quadric with a k-rational point is isomorphic to P1k over any field. Moreover, the Riemann–Hurwitz formula implies that every curve that is dominated by a curve of genus zero also has genus zero (Lüroth’s theorem). Thus, since we assumed k to be algebraically closed, we find Theorem 2.3. If g(C) = 0 then C ∼ = P1k , i.e., C is rational. Moreover, every unirational curve, i.e., a curve that is dominated by P1k , is rational.


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Although unirational surfaces are rational in characteristic zero by Castelnuovo’s theorem, this is false in positive characteristic (see Theorem 9.3). For curves of genus one, we refer to [48, Chap. IV.4] or [135]. Their classification is as follows: Theorem 2.4. Let C be a smooth projective curve of genus g(C) = 1 over an algebraically closed field k of characteristic p ≥ 0. Then: 1. After choosing a point O ∈ C there exists the structure of an Abelian group on the points of C, i.e., C is an Abelian variety of dimension one, an elliptic curve. 2. The linear system |2O| defines a finite morphism of degree 2

ϕ : C → P1k . 3. There exists a j-invariant j(C) ∈ k such that two genus one curves are isomorphic if and only if they have the same j-invariant. 4. If p = 2 then ϕ is branched over four points and j can be computed from the cross ratio of these four points. 5. The linear system |3O| embeds C as a cubic curve into P2k . Moreover, if p = 2, 3 then C can be given by an affine equation (Weierstraß equation) y2 = x3 + ax + b for some a, b ∈ k. We note that the description of complex elliptic curves as quotients of C by lattices also has an analog in positive characteristic. This leads to the theory of Drinfel’d modules, which is parallel to the theory of elliptic curves but has not so much to do with the theory of curves (see [47, Chap. 4]). A curve C of genus g ≥ 2 is called hyperelliptic if there exists a separable morphism ϕ : C → P1k of degree 2. If p = char(k) = 2 then ϕ is branched over 2g + 2 points. (We also note that a morphism of degree 2 in characteristic = 2 is automatically separable.) On the other hand, if p = 2 then every ramification point of ϕ is wildly ramified and thus, there are at most g + 1 branch points. In any characteristic, curves of genus g = 2 are hyperelliptic and the generic curve of genus g ≥ 3 is not hyperelliptic. Theorem 2.5. If g ≥ 2 then ωC⊗2 is very ample if and only if C is not hyperelliptic. In any case, ωC⊗3 is very ample for all curves with g ≥ 3 and ωC⊗4 is very ample for all curves of genus g ≥ 2. Thus, curves of genus g ≥ 2 embed into some fixed projective space that depends on g only. This is the first step towards constructing their moduli spaces. More precisely, Deligne and Mumford [29] showed the existence of a Deligne–Mumford stack, flat and of dimension 3g − 3 over Spec Z that parametrizes curves of genus g. Thus, the moduli space of curves in positive characteristics arises by reducing the one over Spec Z modulo p.

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Let us finally mention a couple of facts concerning automorphism groups: 1. If p = 2, 3 then the automorphism group of an elliptic curve, i.e., automorphisms fixing the neutral element O, has order 2, 4, or 6. 2. However, the elliptic curve with j = 0 has 12 automorphisms if p = 3 and even 24 automorphisms if p = 2 (see [135, Theorem III.10.1]). 3. The automorphism group of a curve of genus g ≥ 2 is finite. However, the Hurwitz bound 84(g − 1) on its in characteristic zero can be violated. We refer to [48, Chap. IV.2, Exercise 2.5] for details and further references. Let us note that some classes of surfaces arise as quotients (C1 × C2 )/G, where C1 , C2 are curves with G-actions. Now, in positive characteristic larger automorphism groups may show up and thus, new possibilities have to be considered. For example, we will see in Sect. 7.4 that hyperelliptic surfaces arise as quotients of products of elliptic curves in any characteristic. It is remarkable that no new classes arise in characteristics 2 and 3 from larger automorphism groups of elliptic curves with j = 0.

2.3 Group Schemes Constructions with groups are ubiquitous in geometry. Instead of finite groups we will consider finite and flat group schemes G over a ground field k, which we assume to be algebraically closed of characteristic p ≥ 0. We refer to [147] or [143] for overview, details, and references. Thus, G = Spec A for some finite-dimensional k-algebra A, and there exist morphisms O : Spec k → G

and

m : G × G → G,

where m stands for multiplication and O for the neutral element. These morphisms have to fulfill certain axioms that encode that G is a group object in the category of schemes. We refer to [147, Chap. I] for the precise definition and note that it amounts to saying that A is a Hopf algebra. The dimension dimk A is called the length, or order, of the group scheme G. The following construction associates to every finite group a finite flat group scheme: for a finite group or order n with elements g1 , . . . , gn we take a disjoint union of n copies of Spec k, one representing each gi , and define m via the multiplication in the group we started with. This defines the constant group scheme associated to a finite group. Conversely, we have Theorem 2.6. A finite flat group scheme G of length prime to p over an algebraically closed field is a constant group scheme. In particular, over an algebraically closed field of characteristic zero, we obtain an equivalence between the categories of finite groups and finite flat group schemes.


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One feature of constant group schemes is that the structure morphism G → Speck is étale, i.e., A = H 0 (G, OG ) is a separable k-algebra. For example, consider the constant group scheme Z/pZ, which is of length p. As an algebra, A is isomorphic to k p with componentwise addition and multiplication, and thus étale over k. On the other hand, we will see below that there are two different structures of group schemes on Spec k[x]/(x p ), which is not reduced—these are examples of infinitesimal group schemes. Let us first note that infinitesimal group schemes are a particular characteristic p phenomenon: Theorem 2.7 (Cartier). Group schemes over fields of characteristic zero are smooth and, thus, reduced. To give examples of infinitesimal group schemes, we consider Ga and Gm . Here, Ga denotes the group scheme corresponding to the additive group, i.e., (Ga (k), ◦) = (k, +). Similarly, Gm denotes the group scheme corresponding to the multiplicative group of k, i.e., (Gm (k), ◦) = (k× , ·) (see [147, Chap. I.2])—these group schemes are affine but not finite over k. Then, the first example of an infinitesimal group scheme is μ p , the group scheme of pth roots of unity. Namely, there exists a short exact sequence of group schemes (in the flat topology) x→x p

0 → μ p → Gm −→ Gm → 0 . We note that the kernel μ p is infinitesimal because of the equality x p − 1 = (x − 1) p in characteristic p. The second example is α p , the kernel of Frobenius on Ga , i.e., we have a short exact sequence F

0 → α p → Ga −→ Ga → 0 . Both group schemes, α p and μ p , are isomorphic to Spec k[x]/(x p ) as schemes and are thus infinitesimal (non-reduced) but have different multiplication maps. Together with Z/pZ these are all group schemes of length p: Theorem 2.8 (Oort–Tate). A finite flat group scheme of length p over an algebraically closed field of characteristic p is isomorphic to Z/pZ, μ p , or α p . For more general results we refer to [108, 109]. Let us also mention that there exist non-Abelian group schemes of order p2 . Thus, in positive characteristic, the theory of finite flat group schemes is richer than the theory of finite groups, already over an algebraically closed field. For example, if E is an elliptic curve in characteristic p, then multiplication by p induces a morphism E → E, whose kernel E[p] is a finite flat group scheme of length p2 (as expected from characteristic zero). More precisely, and still assuming k to be algebraically closed, ⎧ ⎨ either μ p ⊕ (Z/pZ) and E is called ordinary, E[p] ∼ a non-split extension of α p by itself or M2 = ⎩ and E is called supersingular.

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Looking at k-rational points, we find E[p](k) = Z/pZ if E is ordinary, and E[p](k) = 0 if E is supersingular. Thus, k-rational points do not suffice to see the full p-torsion, and the theory of finite flat group schemes is really needed. As the name suggests, the generic elliptic curve is ordinary. More precisely, a theorem of Deuring states that there exist approximately p/12 supersingular elliptic curves in characteristic p (see [135, Theorem V.4.1]). In classical algebraic geometry, one often constructs interesting and new varieties as Galois covers or quotients by finite groups of “well-understood” varieties. In positive characteristic, one very successful way to construct a “pathological characteristic-p” example is via purely inseparable covers, or via quotients by infinitesimal group schemes. The role of Galois covers is often played by torsors under α p and μ p . We come back to this in Sect. 10.5.

3 Cohomological Tools and Invariants This section circles around algebraic versions of Betti and Hodge numbers and deRham cohomology. Especially towards the end, the subjects get deeper, our exposition becomes sketchier and we advise the reader, interested in surface theory only, to skip all but the first three paragraphs. In this section, X will be a smooth and projective variety of arbitrary dimension over an algebraically closed field k of characteristic p ≥ 0.

3.1 Hodge Numbers As usual, we define the Hodge numbers of X to be hi, j (X) := dimk H j (X, ΩXi ) . We note that Serre duality holds for projective Cohen–Macaulay schemes over any field [48, Chap. III.7], and in particular we find hi, j (X) = hn−i,n− j (X),

where

n = dim(X) .

Over the complex numbers, complex conjugation induces the Hodge symmetry hi, j = h j,i (see, e.g., [43, Chap. 0.7]). However, even for a smooth projective surface in positive characteristic, the numbers h0,1 and h1,0 may be different. For example, in [78, Theorem 8.3] we constructed a sequence {Xi }i∈N of surfaces with fixed Pic0red in characteristic 2, where h1,0 (Xi ) − h0,1(Xi ) tends to infinity.


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3.2 Betti Numbers An algebraic replacement for singular cohomology is -adic cohomology, whose construction is due to Grothendieck. We refer to [48, Appendix C] for motivation, as well as to [92] for a complete treatment. Let us only describe here its basic properties: let be a prime number different from p and let Q be the field of adic numbers, i.e., the completion of Q with respect to the -adic valuation (see also Sect. 11.1). Then: 1. The -adic cohomology groups Heít (X, Ql ) are finite-dimensional Q -vector spaces. 2. They are zero for i < 0 and i > 2 dim(X). 3. The dimension of Heít (X, Q ) is independent of (here, = p is crucial), and we denote it by bi (X), the ith Betti number. 4. He´∗t (X, Q ) satisfies Poincaré duality. If k = C, then the so-called comparison theorems show that these Betti numbers coincide with the topological ones. Let us also mention the following feature: if k is not algebraically closed, then the absolute Galois group Gal(k/k) acts on the -adic cohomology groups of Xk , which gives rise to interesting representations of Gal(k/k). For the following two classes of varieties, -adic cohomology and Hodge invariants are precisely as one would expect them from complex geometry: 1. If C is a smooth and projective curve over k, then b0 = b2 = 1, b1 = 2g, and h1,0 = h0,1 = g. 2. If A is an Abelian variety of dimension g over k, then b0 = b2g = 1, b1 = 2g, and h0,1 = h1,0 = g. Moreover, there exists an isomorphism

Λ i He´1t (A, Ql ) ∼ = Heít (A, Q ) giving—among many other things—the expected Betti numbers. However, for more general classes of smooth and projective varieties, the relations between Betti numbers, Hodge invariants, deRham cohomology, and the Frölicher spectral sequence in positive characteristic are more subtle than the complex numbers, as we shall see below. Let us first discuss h1,0 , h0,1 , and b1 in more detail, since this is important for the classification of surfaces. Also, these numbers can be treated fairly elementary.

3.3 Picard Scheme and Albanese Variety If X is smooth and proper over a field k, then there exists an Abelian variety Alb(X) over k, the Albanese variety of X, and an Albanese morphism: albX : X → Alb(X) .

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The pair (Alb(X), albX ) is characterized by the universal property that every morphism from X to an Abelian variety factors over albX . For a purely algebraic construction, we refer to [125]. Next, the Picard functor, which classifies invertible sheaves on X, is representable by a group scheme, the Picard scheme Pic(X), whose neutral element is [OX ] (see [45]). We denote by Pic0 (X) the identity component of Pic(X). Deformation theory provides us with a natural isomorphism: T Pic0 (X) ∼ = H 1 (X, OX ) , where T Pic0 (X) denotes the Zariski tangent space at [OX ]. Now, group schemes over fields of positive characteristic may be non-reduced (the group schemes μ p and α p from Sect. 2.3 are examples), but the reduction of Pic0 (X) is still an Abelian variety, which is the dual Abelian variety of Alb(X) (see [6, Chap. 5]). Also, the first Betti number b1 is twice the dimension of Alb(X). Thus, we get 1 b1 (X) = dim Alb(X) = dim Pic0 (X). 2 Since, the dimension of the Zariski tangent space at [OX ] ∈ Pic0 (X) is at least equal to the dimension of Pic0 (X), we find h0,1 (X) = h1 (X, OX ) ≥

1 b1 (X) , 2

with equality if and only if Pic0 (X) is a reduced group scheme, i.e., an Abelian variety. By Cartier’s theorem (Theorem 2.7), group schemes over a field of characteristic zero are reduced. As a corollary, we obtain a purely algebraic proof of the following fact Proposition 3.1. A smooth and proper variety over a field of characteristic zero satisfies b1 (X) = 2h0,1 (X). For curves and Abelian varieties over arbitrary fields, b1 , h1,0 , and h0,1 are precisely as over the complex numbers. On the other hand, over fields of positive characteristic: 1. There do exist surfaces with h0,1 > b1 /2, i.e., with non-reduced Picard schemes (see [53, 124]). In [98, Lecture 27], the non-reducedness of Pic0 (X) is related to non-trivial Bockstein operations βn from subspaces of H 1 (X, OX ) to quotients of H 2 (X, OX ). In particular, a smooth projective variety with h2 (X, OX ) = 0 has a reduced Pic0 (X), which applies, for example, to curves. In the case of surfaces, a quantitative analysis of which classes can have non-reduced Picard schemes has been carried out in [80].


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3.4 Differential One-Forms We shall see in Sect. 9.1 that in positive characteristic, the pullback of a nonzero differential form under a morphism may become zero. However, by a fundamental theorem of Igusa [52], every nontrivial global one-form on Alb(X) pulls back, via albX , to a nonzero global one-form on X. This implies the estimate h1,0 (X) = h0 (X, ΩX1 ) ≥

1 b1 (X) . 2

Moreover, all global one-forms arising as pullback from Alb(X) are d-closed, i.e., closed under the exterior derivative. Despite of Igusa’s theorem, it is still possible that the Albanese morphism in positive characteristic becomes purely inseparable (see Theorem 8.7). We have h1,0 = b1 /2 for curves and Abelian varieties over arbitrary fields and their global one-forms are d-closed. On the other hand, over fields of positive characteristic: 1. There do exist surfaces with h1,0 > b1 /2, i.e., with “too many” global one-forms (see [53]) 2. There do exist surfaces with global one-forms that are not d-closed (see [40, 97]). These forms give rise to a nonzero differential in their Frölicher spectral sequences, which thus do not degenerate at E1 . We refer to [55, Proposition II.5.16] for more results and to [55, Sect. II.6.9] for the connection to Oda’s subspace in first deRham cohomology.

3.5 Igusa’s Inequality We denote by ρ the rank of the Néron–Severi group NS(X), which is always finite. More precisely, Igusa’s theorem [54] states

ρ (X) ≤ b2 (X) . This follows from the existence of a Chern map from NS(X) to second -adic or crystalline cohomology. On the other hand, d log induces a “naive” cycle map: d log : NS(X) ⊗Z k → H 1 (X, ΩX1 ) , which is injective in characteristic zero and which then implies the inequality ρ ≤ h1,1 (see, e.g., [6, Exercise 5.5]). However, this map may fail to be injective in positive characteristic, as the example of supersingular Fermat surfaces [131] shows. More precisely, these surfaces satisfy b2 = ρ > h1,1 (see also Sect. 9.4).

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3.6 Kodaira Vanishing Raynaud [114] gave the first counterexamples to the Kodaira vanishing theorem in positive characteristic. However, we mention the following results that tell us that the situation is not too bad: 1. If L is an ample line bundle then L⊗ν , ν 0 fulfills Kodaira vanishing (in fact, this is just Serre vanishing) [48, Theorem III.7.6]. 2. If a smooth projective variety of dimension < p lifts over W2 (k) then Kodaira vanishing holds (see [57, Theorem 5.8], [32, Corollaire 2.11], and Sect. 11.2). Under stronger lifting assumptions, also Kawamata–Viehweg vanishing holds [148]. 3. Kodaira vanishing and even stronger vanishing results hold for the (admittedly rather special) class of Frobenius-split varieties [21, Theorem 1.2.9]. 4. In [37, Sect. II], Ekedahl develops tools to handle possible failures of Kodaira vanishing (see also Sect. 8.1). 5. In [149], Xie shows that all surfaces violating Kodaira–Ramanujam vanishing arise as in [114]. Here, W (k) denotes the ring of Witt vectors and W2 (k) the ring of Witt vectors of length 2 (see Sect. 11.1). Let us just mention that if k is a perfect field then W (k) is a complete discrete valuation ring of characteristic zero with residue field k, and that this ring is in a certain sense minimal and universal.

3.7 Frölicher Spectral Sequence Let ΩXi be the sheaf of (algebraic) differential i-forms. These sheaves, together with the exterior derivative d, form a complex, the (algebraic) deRham complex (ΩX∗ , d). Now, the Zariski topology is too coarse to have a Poincaré lemma. Thus, ∗ (X/k) to be the hypercohomology we define (algebraic) deRham cohomology HdR of this complex. In particular, there always exists a spectral sequence: E1 = H j (X, ΩXi ) ⇒ HdR (X/k), i, j

i+ j

the Frölicher spectral sequence, from Hodge to deRham cohomology. If k = C, and X is proper over k, then these cohomology groups and the spectral sequence coincide with the analytic ones (see [46]). Already the existence of the Frölicher spectral sequence implies for all m ≥ 0 the inequality

∑

h j (X, ΩXi ) ≥ hm dR (X/k) .

i+ j=m

Equality for all m is equivalent to the degeneration of this spectral sequence at E1 . Over the complex numbers, degeneration at E1 is true—however, the classical proof uses methods from differential geometry, functional analysis, and partial


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differential equations (see [43, Chap. 0.7]). On the other hand, if a variety of positive characteristic admits a lift over W2 (k), then we have the following result from [32] (but see [57] for an expanded version): Theorem 3.2 (Deligne–Illusie). Let X be a smooth and projective variety in characteristic p ≥ dim X and assume that X admits a lift over W2 (k). Then the Frölicher spectral sequence of X degenerates at E1 . The assumptions are fulfilled for curves and Abelian varieties (see Sect. 11). Moreover, if a smooth projective variety X in characteristic zero admits a model over W (k) for some perfect field of characteristic p ≥ dim X it follows from semi-continuity that the Frölicher spectral sequence of X degenerates at E1 in characteristic zero. From this one obtains purely algebraic proofs of the following Theorem 3.3. Degeneration at E1 holds for 1. Smooth projective curves and Abelian varieties over arbitrary fields 2. Smooth projective varieties over fields of characteristic zero We already mentioned above that varieties with global one-forms that are not dclosed, such as Mumford’s surfaces [97], provide examples where degeneration at E1 does not hold.

3.8 Crystalline Cohomology To link deRham, Betti, and Hodge cohomology, we use crystalline cohomology. Its construction, due to Berthelot and Grothendieck, is quite involved [10]. This cohomology theory takes values in the Witt ring W = W (k), which is a discrete valuation ring if k is perfect (see Sect. 11). In case a smooth projective variety lifts to ∗ (X /W (k)). some X /W (k), crystalline cohomology is the deRham cohomology HdR It was Grothendieck’s insight, and the starting point of crystalline cohomology, that this deRham cohomology does not depend on the choice of lift X . One of the main technical difficulties to overcome defining crystalline cohomology for arbitrary smooth and proper varieties is that they usually do not lift over W (k). If X is a smooth projective variety over a perfect field k then: i (X/W ) are finitely generated W -modules. The groups Hcris They are zero for i < 0 and i > 2 dim(X), i (X/W ). There are actions of Frobenius and Verschiebung on Hcris ∗ Hcris (X/W ) ⊗W K satisfies Poincaré duality, where K denotes the field of fractions of W . 5. If X lifts over W (k) then crystalline cohomology is isomorphic to the deRham cohomology of a lift.

1. 2. 3. 4.

We remind the reader that in order to get the “right” Betti numbers from the -adic cohomology groups, we had to assume = p. Crystalline cohomology takes values

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in W (k) (recall W (F p ) ∼ = Z p with field of fractions Q p ), and this is the “right” cohomology theory for = p. In fact, i (X/W ) bi (X) = dimQ Heít (X, Q ) = rankW Hcris

for all

= p,

i.e., the Betti numbers of X are encoded in the rank of crystalline cohomology. ∗ (X/W ) are W -modules, there may be nontrivial torsion— However, since the Hcris and this is precisely the explanation for the differences between Hodge and Betti numbers. More precisely, there is a universal coefficient formula, and for all m ≥ 0 there are short exact sequences: m m m+1 0 → Hcris (X/W ) ⊗W k → HdR (X/k) → TorW 1 (Hcris (X/W ), k) → 0 .

(In view of what we already said in case X admits a lift over W , it should be plausible that there is a connection between crystalline and deRham cohomology.) In particular, Betti and deRham numbers coincide if and only if all crystalline cohomology groups are torsion-free W -modules.

3.9 Hodge–Witt Cohomology In [55], Illusie constructed the deRham–Witt complex W ΩX∗ and studied its cohomology groups H j (X,W ΩXi ), the Hodge–Witt cohomology groups. For i = 0, these coincide with Serre’s Witt vector cohomology groups introduced in [124]. The Hodge–Witt cohomology groups are W -modules, whose torsion may not be finitely generated. In any case, there exists a spectral sequence, the slope spectral sequence: i+ j E1i, j = H j (X,W ΩXi ) ⇒ Hcris (X/W ) ,

which degenerates at E1 modulo torsion. We refer to [55, Sect. II.7] for computations and further results. Finally, using Hodge–Witt cohomology and slopes on crystalline cohomology, Ekedahl [35, p. 85] (but see also [56]) proposed new invariants of smooth projective varieties: slope numbers, dominoes, and Hodge–Witt numbers. It is not yet clear what role they will eventually play in characteristic-p-geometry. We refer to [64] for some results.

4 Birational Geometry of Surfaces From this section on, we study smooth surfaces. To start with, we discuss their birational geometry, which turns out to be “basically the same” as over the complex numbers. Unless otherwise stated, results and proofs can be found in [6], and we refer to [15] for an overview different from ours.


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4.1 Riemann–Roch Let S be a smooth projective surface over an algebraically closed field k of characteristic p ≥ 0. Actually, asking for properness would be enough: by a theorem of Zariski and Goodman, a surface that is smooth and proper over a field is automatically projective (see [6, Theorem 1.28]). For every locally free sheaf E, Grothendieck constructed Chern classes ci (E) that take values in Chow-groups, -adic, or crystalline cohomology. As usual, for a smooth variety X with tangent sheaf ΘX , we set ci (X) := ci (ΘX ). We have Noether’s formula:

χ (OS ) =

1 2 c1 (S) + c2 (S) . 12

Moreover, if L is an invertible sheaf on S, we have the Riemann–Roch formula: 1 χ (L) = χ (OS ) + L · (L ⊗ ωS∨) . 2 We note that Serre duality holds for Cohen–Macaulay schemes that are of finite type over a field. Thus, we have an equality hi (S, L) = h2−i (S, ωS ⊗ L∨ ) for surfaces. However, we have seen in Sect. 3.6 that Kodaira vanishing may not hold. Finally, if D is an effective divisor on S, then D is a Gorenstein curve and the adjunction formula yields ωD ∼ = (ωS ⊗ OS (D)) |D , where ωD and ωS denote the respective dualizing sheaves. In particular, if D is reduced and irreducible, we obtain 2pa (D) − 2 = D2 + KS · D , where pa denotes the arithmetic genus of D. We refer to [48, Chap. V.1], [48, Appendix A], [6, Chap. 5], [41, 92] for details and further references.

4.2 Blowing Up and Down First of all, blowing up a point on a smooth surface over an algebraically closed field has the same effect as over the complex numbers. Proposition 4.1. Let f : S˜ → S be the blowup in a closed point and denote by E the ˜ Then exceptional divisor on S. E ∼ = P1k ,

E 2 = −1,

and

KS˜ · E = −1 .

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Moreover, the equalities ˜ = b2 (S) + 1 b2 (S)

and

˜ = ρ (S) + 1 ρ (S)

hold true. As in the complex case, we call such a curve E with E 2 = −1 and E ∼ = P1 an exceptional (−1)-curve. A surface that does not contain exceptional (−1)-curves is called minimal. Conversely, exceptional (−1)-curves can be contracted and the proof (modifying a suitable hyperplane section) is basically the same as in characteristic zero (cf. [6, Theorem 3.30] or [48, Theorem V.5.7]): Theorem 4.2 (Castelnuovo). Let E be an exceptional (−1)-curve on a smooth surface S. Then, there exists a smooth surface S and a morphism f : S → S , such that f is the blowup of S in a closed point with exceptional divisor E. Since b2 drops every time one contracts an exceptional (−1)-curve, Castelnuovo’s theorem implies that for every surface S there exists a sequence of blowdowns S → S onto a minimal surface S . In this case, S is called a minimal model of S.

4.3 Resolution of Indeterminacy As in characteristic zero, a rational map from a surface extends to a morphism after a finite number of blowups in closed points, which gives resolution of indeterminacy of a rational map. Moreover, every birational (rational) map can be factored as a sequence of blowups and Castelnuovo blowdowns (see, e.g., [48, Chap. V.5]).

4.4 Kodaira Dimension As over the complex numbers, the following two notions are crucial in the Kodaira– Enriques classification of surfaces: first, the nth plurigenus Pn (X) of a smooth projective variety X is defined to be Pn (X) := h0 (X, ωX⊗n ). Second, the Kodaira dimension κ (X) is defined to be κ (X) = −∞ if Pn (X) = 0 for all n ≥ 1, or else

κ (X) := max { dim φn (X) } n∈N


247

P (X)−1

where φn : X Pk n denotes the nth pluricanonical (possibly only rational) map. This recalled, we have the following important result (cf. [6, Corollary 10.22]), which is already nontrivial in characteristic zero: Theorem 4.3. Let S be a smooth projective surface with κ (S) ≥ 0. Then, S possesses a unique minimal model.

4.5 Birationally Ruled Surfaces We recall that a surface S is called birationally ruled if it is birational to P1 × C for some smooth curve C. Such surfaces are easily seen to satisfy Pn (S) = 0 for all n ≥ 1, i.e., they are of Kodaira dimension κ (S) = −∞. Conversely, one can show (see, e.g., [6, Theorem 13.2]) that such surfaces with κ (S) = −∞ possess a smooth rational curve that moves: Theorem 4.4. If S is birationally ruled, then κ (S) = −∞. Conversely, if κ (S) = −∞ then S is birationally ruled, i.e., birational to P1 × C, and h1 (S, OS ) =

1 b1 (S) = g(C) , 2

where g(C) denotes the genus of C. As in the complex case, minimal models for surfaces with κ (S) = −∞ are not unique. More precisely, we have Nagata’s result. Theorem 4.5. Let S be a minimal surface with κ (S) = −∞. 1. If h1 (S, OS ) ≥ 1, then the image C of the Albanese map is a smooth curve. Moreover, there exists a rank two vector bundle E on C such that albS : S → C is isomorphic to P(E) → C. 2. If h1 (S, OS ) = 0, then S is isomorphic to P2 or a Hirzebruch surface Fd := P(OP1 ⊕ OP1 (d)) → P1 with d = 1. Also, Castelnuovo’s cohomological characterization of rational surfaces holds true. The proof in positive characteristic is due to Zariski [152], but see also the discussion in [15, Part 4]: Theorem 4.6 (Castelnuovo–Zariski). For a smooth projective surface S, the following are equivalent: 1. S is rational, i.e., birational to P2 . 2. h1 (S, OS ) = P2 (S) = 0. 3. b1 (S) = P2 (S) = 0.

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So far, things look pretty much the same as over the complex numbers. However, one has to be a little bit careful with the notion of uniruledness: we will see in Sect. 9 below that unirationality (resp. uniruledness) does not imply rationality (resp. ruledness).

4.6 Del Pezzo Surfaces A surface S is called del Pezzo, or Fano, if ωS∨ is ample. In every characteristic, these surfaces are rational. More precisely, they are isomorphic to P1 × P1 , P2 , or P2 blown up in at most 8 points in general position. We refer to Várilly–Alvarado’s lecture notes in this volume for details.

5 (Quasi-)Elliptic Fibrations For the classification of surfaces of special type in characteristic zero, elliptic fibrations play an important role. In positive characteristic, wild fibers and quasielliptic fibrations are new features that show up. We refer to [6, Sect. 7] for an introduction, [16, 17] for more details, [123] for an overview article with many examples, and [27, Chap. V] for more advanced topics.

5.1 Quasi-Elliptic Fibrations Given a dominant morphism from a smooth surface S onto a curve in any characteristic, we may pass to its Stein factorization and obtain a fibration S → B (cf. [48, Corollary III.11.5]). Then, its generic fiber Sη is an integral curve, i.e., reduced and irreducible [6, Theorem 7.1]. Moreover, in characteristic zero, Bertini’s theorem implies that Sη is in fact smooth over the function field k(B). Now, if char(k) = p > 0, then it is still true that the generic fiber is a regular curve, i.e., all local rings are regular local rings. However, this does not necessarily imply that Sη is smooth over k(B). Note that since Sη is one-dimensional, regularity is the same as normality. We refer to [87, Chap. 11.28] for a discussion of smoothness versus regularity. Suppose Sη is not smooth over k(B) and denote by k(B) the algebraic closure of k(B). Then Sη := Sη ×Spec k(B) Spec k(B) is still reduced and irreducible [6, Theorem 7.1] but no longer regular and we denote by Sη → Sη its normalization. Then, Tate’s theorem on genus change in inseparable extensions [139] (see [122] for a modern treatment) states


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Theorem 5.1 (Tate). Under the previous assumptions, the normalization map Sη → Sη is a homeomorphism, i.e., Sη has only unibranch singularities (“cusps”). Moreover, if p ≥ 3, then the arithmetic genus of every cusp of Sη is divisible by (p − 1)/2. If the generic fiber Sη has arithmetic genus one and the fiber is not smooth over k(B), then the normalization of Sη is P1 . Also, Sη can have only one singularity, which, by Tate’s theorem, must be a cusp of arithmetic genus one. Since (p − 1)/2 divides this genus if p ≥ 3, we find p = 3 as only solution. Thus, Corollary 5.2. Let f : S → B be a fibration from a smooth surface whose generic fiber Sη is a curve of arithmetic genus one, i.e., h1 (Sη , OSη ) = 1. Then: 1. Either Sη is smooth over k(B) 2. Or Sη is a singular rational curve with one cusp The second case can happen in characteristics 2 and 3 only. Definition 5.3. If the generic fiber of a fibration S → B is a smooth curve of genus one, the fibration is called elliptic. If the generic fiber is a curve that is not smooth over k(B), the fibration is called quasi-elliptic, which can exist in characteristics 2 and 3 only. Some authors require elliptic fibrations to have a section, which we do not. The literature is not consistent. We refer to [16] and [6, Exercises 7.5 and 7.6] for examples of quasi-elliptic fibrations and to [17] for more on the geometry of quasi-elliptic fibrations. For results on quasi-elliptic fibrations in characteristic 3, see [69]. We note that quasi-elliptic surfaces are always uniruled, but may not be birationally ruled, and refer to Sect. 9.3, where we discuss this in greater detail. We also note that the situation gets more complicated in higher dimensions: Mori and Saito [96] constructed Fano 3-folds X in characteristic 2 together with fibrations X → S, whose generic fibers are conics in P2k(S) that become non-reduced over k(S). Such fibrations are called wild conic bundles.

5.2 Canonical Bundle Formula Let S be a smooth surface and f : S → B be an elliptic or quasi-elliptic fibration. Since B is smooth, we obtain a decomposition R1 f∗ OS = L ⊕ T , where L is an invertible sheaf and T is a torsion sheaf on B. In characteristic zero, the torsion sheaf T is always trivial.

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Definition 5.4. Let b ∈ B a point of the support of T . Then, the fiber of f above b is called a wild fiber or an exceptional fiber. Proposition 5.5. Let f : S → B be a (quasi-)elliptic fibration, b ∈ B and Fb the fiber above b. Then the following are equivalent: 1. b ∈ Supp(T ), i.e., Fb is a wild fiber. 2. h1 (Fb , OFb ) ≥ 2. 3. h0 (Fb , OFb ) ≥ 2. In particular, wild fibers are multiple fibers. Moreover, if Fb is a wild fiber, then its multiplicity is divisible by p, and we have h1 (S, OS ) ≥ 1. The canonical bundle formula for relatively minimal (quasi-)elliptic fibrations has been proved in [16]—as usual, relatively minimal means that there are no exceptional (−1)-curves in the fibers of the fibration. Theorem 5.6 (Canonical bundle formula). Let f : S → B be a relatively minimal (quasi-)elliptic fibration from a smooth surface. Then, ∗ ∨ ωS ∼ = f (ωB ⊗ L ) ⊗ OS ∑ ai Pi , i

where 1. 2. 3. 4.

mi Pi = Fi are the multiple fibers of f 0 ≤ a i < mi ai = mi − 1 if Fi is not a wild fiber deg(ωS ⊗ L∨ ) = 2g(B) − 2 + χ (OS) + length(T )

For more results on the ai ’s we refer to [27, Proposition V.5.1.5], as well as to [66] for more details on wild fibers.

5.3 Degenerate Fibers of (Quasi-)Elliptic Fibrations Usually, an elliptic fibration has fibers that are not smooth and the possible cases have been classified by Kodaira and Néron. The list in positive characteristic is the same as in characteristic zero (cf. [27, Chap. V, Sect. 1] and [136, Theorem IV.8.2]). This is not such a surprise, as the classification of degenerate fibers rests on the adjunction formula and on matrices of intersection numbers, and these numerics do not depend on the characteristic of the ground field. Let us recall that the possible singular fibers together with their Kodaira symbols are as follows (after reduction): 1. 2. 3. 4.

An irreducible rational curve with a node as singularity (I1 ). A cycle of n ≥ 2 rational curves (In ). An irreducible rational curve with a cusp as singularity (II). A configuration of rational curves forming a root system of type A∗2 (III), A3 (IV), E˜6 (IV∗ ), E˜7 (III∗ ), E˜8 (II∗ ), or D˜ n (I∗n−4).


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In the first two cases, the reduction is called multiplicative or semi-stable, whereas in the last two cases, it is called additive or unstable. The latter names come from the theory of Néron models (see [136, Chap. IV] or [19]). The former names are explained by the fact that semi-stable reduction remains semi-stable after pullback, whereas unstable reduction, may become semi-stable after pullback. In fact, for every fiber with unstable reduction there exists a pullback, whose reduction is semistable [136, Proposition IV.10.3]. For an elliptic fibration S → B from a smooth surface, the second Chern class (Euler number) can be expressed in terms of the singular fibers by Ogg’s formula c2 (S) =

∑ ν (Δi ), i

where i runs through the singular fibers, Δi is the minimal discriminant of the singular fiber, and ν denotes its valuation. If a fiber has n irreducible components, then this minimal discriminant is as follows: 1 + (n − 1) if the reduction is multiplicative, i.e., of type In , ν (Δ ) = 2 + (n − 1) + δ if the reduction is additive. Here, δ is the Swan conductor or wild part of the conductor of the fiber, which is zero if p = 2, 3. We refer to [136, Chap. IV, Sect. 10] for details and to [27, Proposition 5.1.6] for a version for quasi-elliptic fibrations. In a quasi-elliptic fibration, all fibers are additive and the geometric generic fiber is of type II, i.e., an irreducible rational curve with one cusp. After an inseparable base change of the base curve B → B, the normalized pullback yields a fibration whose generic fiber is of genus zero (see also Theorem 9.4). All fibers are reduced or have multiplicity equal to the characteristic p = 2, 3. The list of possible geometric fibers is as follows [27, Corollary 5.2.4]: p = 3 : II, IV, IV∗ and II∗ p = 2 : II, III, III∗ , II∗ and I∗n Finally, we mention that if a (quasi-)elliptic fibration from a surface has a section, then there exists a Weierstraß model [27, Chap. 5, Sect. 5], which is more involved in characteristics 2, 3 than in the other characteristics.

6 Enriques–Kodaira Classification We now come to the Kodaira–Enriques classification of surfaces. In positive characteristic, it is due to Bombieri and Mumford (see [16, 17, 99]). Let S be a smooth projective surface of Kodaira dimension κ (S).

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6.1 Negative Kodaira Dimension First, let us recall and repeat Theorem 4.4: Theorem 6.1. If κ (S) = −∞, then S is birationally ruled. In fact, κ (S) = −∞ is equivalent to p12 (S) = 0, where p12 is the 12th plurigenus [6, Theorem 9.8]. Moreover, although their minimal models are not unique, they have the same structure as in characteristic zero by Theorem 4.5. In Sect. 9, we shall see that uniruled surfaces in positive characteristic may not fulfill κ = −∞.

6.2 Positive Kodaira Dimension We recall that the canonical ring of a smooth and proper variety X is defined to be Rcan (X) := H 0 X, ωX⊗n . n≥0

This said, we have the following fundamental result. Theorem 6.2 (Zariski–Mumford). The canonical ring Rcan (S) of a smooth projective surface is a finitely generated k-algebra. If κ (S) ≥ 0, then Rcan (S) has transcendence degree 1 + κ (S) over k. We refer to [16] and [6, Corollary 9.10]. More generally, we refer to [6, Chap. 14] for a discussion of Zariski decompositions and finite generation of the more general rings R(S, D) for a Q-divisor D on S. For a surface with κ (S) ≥ 1 one studies the Iitaka fibration: S Proj Rcan (S) . By the theorem of Zariski–Mumford just mentioned, the right-hand side is a projective variety of dimension κ (S). Theorem 6.3. Let S be a minimal surface with κ (S) = 1. Then, (the Stein factorization of) the Iitaka fibration is a morphism, which is a relatively minimal elliptic or quasi-elliptic fibration. If κ (S) = 1 and p = 2, 3, then the fibration is elliptic and unique, |mKS | for m ≥ 14 defines this fibration, and 14 is the optimal bound [66]. The main difficulties dealt with by Katsura and Ueno [66] are related to wild fibers. We note that their bound m ≥ 14 is better than Iitaka’s bound m ≥ 86 over the complex numbers, since over the complex numbers also analytic surfaces that are not algebraic are taken into account. Although we will discuss surfaces with κ (S) = 2, i.e., surfaces of general type, in Sect. 8, let us already anticipate Theorem 8.1:


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Theorem 6.4. Let S be a minimal surface with κ (S) = 2. Then, the Iitaka fibration is a birational morphism that contracts all rational (−2)-curves and nothing more.

7 Kodaira Dimension Zero As in the complex case, surfaces in positive characteristic that are of Kodaira dimension zero fall into four classes. However, there are new subclasses of Enriques surfaces in characteristic 2 and new subclasses of hyperelliptic surfaces, so-called quasi-hyperelliptic surfaces, in characteristics 2 and 3. In particular, there are no fundamentally new classes in characteristic p ≥ 5. We start with a result that follows from the classification, especially from the explicit classification of (quasi-)hyperelliptic surfaces: Theorem 7.1. Let S be a minimal surface with κ (S) = 0. Then, ωS⊗12 ∼ = OS , and in particular, p12 (S) = 1. The key to the classification of minimal surfaces with κ (S) = 0 is to use KS2 = 0 to rewrite Noether’s formula (see Sect. 4.1) as follows: 10 + 12pg = 8h0,1 + 2Δ + b2 , where Δ := 2h0,1 − b1 measures the defect of smoothness of Pic0 (S). We have 0 ≤ Δ ≤ 2pg in general, and pg ≤ 1, since κ (S) = 0. Also, all terms in the above formula are nonnegative, which gives a finite list of possibilities, leading eventually to four classes (see the introduction of [16] or [6, Chap. 10]). Let us now discuss these classes in detail.

7.1 Abelian Surfaces These are two-dimensional Abelian varieties. Their main invariants are as in characteristic zero:

ωS ∼ = OS pg = 1 h0,1 = 2 h1,0 = 2, χ (OS ) = 1 c2 = 0 b1 = 4 b2 = 2. Abelian surfaces are usually studied within the framework of Abelian varieties of arbitrary dimension. There exists a huge amount of literature on Abelian varieties and their moduli spaces, both in characteristic zero and in positive characteristic (see, e.g., [101]). By an (unpublished) result of Grothendieck [58, Theorem 5.23], Abelian varieties lift formally to characteristic zero (see also Sect. 11.3). For an Abelian variety A of dimension g, multiplication by p is a finite morphism. Its kernel A[p] is a finite and flat group scheme of length p2g , and refer to Sect. 2.2,

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where we already discussed elliptic curves (g = 1). The identity component A[p]0 is infinitesimal of length at least g. The quotient A[p]/A[p]0 is an étale group scheme isomorphic to (Z/pZ)r , for some 0 ≤ r ≤ g. This quantity r is called the p-rank of A. For Abelian varieties of dimension at most two, the p-rank can be detected by the Frobenius action F : H 1 (A, OA ) → H 1 (A, OA ). Definition 7.2. An Abelian surface A is called 1. Ordinary if r = 2. Equivalently, F acts bijectively on H 1 (A, OA ). 2. Supersingular if r = 0. Equivalently, F is zero on H 1 (A, OA ). We remark that the image of the Albanese morphism of a uniruled surface is at most one-dimensional. Thus, an Abelian surface cannot be uniruled. We note this in view of Shioda’s notion of supersingularity and its connection with unirationality discussed in Sect. 9.7.

7.2 K3 Surfaces These surfaces have the following invariants:

ωS ∼ pg = 1 h0,1 = 0 h1,0 = 0, = OS χ (OS ) = 2 c2 = 24 b1 = 0 b2 = 22. Their formal deformation spaces are smooth W (k)-algebras in any characteristic, i.e., the Bogomolov–Tian–Todorov unobstructedness theorem for K3 surfaces in positive characteristic is true [116]: Theorem 7.3 (Rudakov–Shafarevich). A K3 surface has no global vector fields. Thus, SD

H 2 (S, ΘS ) ∼ = H 2 (S, ΩS1 ⊗ ωS ) ∼ = H 0 (S, ΘS )∨ = 0, where SD denotes Serre duality. In particular, deformations of K3 surfaces are unobstructed. For K3 surfaces over arbitrary fields, we have h2 (ΘS ) = h1,2 = h1,0 by Serre duality and h0,1 = 0 by our list of invariants. Over the complex numbers, vanishing of the former then follows easily from the Hodge symmetry h1,0 = h0,1 , which is induced by complex conjugation and, thus, may not hold over arbitrary ground fields. The proof in positive characteristic of [116] makes heavy use of purely characteristic-ptechniques (see Sect. 10.4). We note that over fields of positive characteristic and in dimension three the Bogomolov–Tian–Todorov unobstructedness theorem for Calabi–Yau varieties may fail (cf. [50, 120]). The vanishing H 2 (S, ΘS ) = 0 implies that K3 surfaces possess formal lifts over the Witt ring. Deligne [30] showed in fact (see Sect. 11 for more on lifts): Theorem 7.4 (Deligne). K3 surfaces lift projectively to characteristic zero.


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The moduli space of polarized K3 surfaces in positive and mixed characteristic exists by a result of Rizov [115]. However, it is still open, whether these moduli spaces are irreducible. What makes moduli spaces of (polarized) K3 surfaces so difficult to come by is that no local or global Torelli theorems are known (except for supersingular K3 surfaces; see Sect. 9.8). We come back to K3 surfaces in Sect. 9, where we discuss arithmetic conjectures and conjectural characterizations of unirational K3 surfaces.

7.3 Enriques Surfaces In characteristic p = 2 these surfaces have the following invariants:

ωS ∼ = OS ωS⊗2 ∼ = OS pg = 0 h0,1 = 0, χ (OS ) = 1 c2 = 12 b1 = 0 b2 = 10. Moreover, the canonical sheaf ωS defines an étale double cover S˜ → S, where S˜ is a K3 surface. Also, there always exist elliptic or quasi-elliptic fibrations. Every such fibration has precisely two multiple fibers, both of which are not wild. The most challenging case is characteristic 2, where Enriques surfaces are characterized by (here, ≡ denotes numerical equivalence)

ωS ≡ OS χ (OS ) = 1 c2 = 12 b1 = 0 b2 = 10. It turns out that pg = h0,1 ≤ 1 (see [17]). Since b1 = 0, we conclude that the Picard scheme of an Enriques surface with h0,1 = 1 is not smooth. In this case, Frobenius induces a map F : H 1 (S, OS ) → H 1 (S, OS ), which is either zero or a bijection. We thus obtain three possibilities: Definition 7.5. An Enriques surface (in characteristic 2) is called: 1. Classical if h0,1 = pg = 0, 2. Nonclassical if h0,1 = pg = 1. Such a surface is called: (a) Ordinary if Frobenius acts bijectively on H 1 (S, OS ) (b) Supersingular if Frobenius is zero on H 1 (S, OS ) All three types exist [17]. We note that the terminology is inspired by Abelian surfaces (see Definition 7.2). In any characteristic, every Enriques surface possesses elliptic or quasi-elliptic fibrations. Such a fibration always has multiple fibers. Moreover, if S is classical, then every (quasi-)elliptic fibration has precisely two multiple fibers, both of multiplicity two and neither of them is wild. If S nonclassical, then there is only one multiple fiber, which is wild with multiplicity two. Finally, if S is nonclassical and ordinary it does not possess quasi-elliptic fibrations. We refer to [27, Chap. V.7] for details.

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As explained in [17, Sect. 3], every Enriques surface possesses a finite and flat morphism of degree two: ϕ : S˜ → S ˜ O ˜ ) = h2 (S, ˜ O ˜ ) = 1, and h1 (S, ˜ O ˜ ) = 0, i.e., S˜ is “K3such that ωS˜ ∼ = OS˜ , h0 (S, S S S like.” More precisely, in characteristic = 2, or, if S is nonclassical and ordinary, then ϕ is étale of degree two and S˜ is in fact a K3 surface. However, in the remaining cases in characteristic 2, S˜ is only an integral Gorenstein surface that need not even be normal, since ϕ is a torsor under an infinitesimal group scheme. In any case and any characteristic, S˜ is birational to the complete intersection of three quadrics in P5 (see [83]), which generalizes results of Cossec [26] and Verra [146]. Moreover, the moduli space of Enriques surfaces in characteristic p = 2 is irreducible, unirational, smooth, and ten-dimensional. In characteristic 2, it consists of two irreducible, unirational, and ten-dimensional components, whose intersection is nine-dimensional. This intersection corresponds to nonclassical supersingular surfaces, and outside their intersection the ten-dimensional components parametrize nonclassical ordinary and classical Enriques surfaces, respectively. We refer to [83] for details, as well as [38] for a complementary approach. We refer to [17, 72, 73, 83], and, of course, [27] for more details and partial classification results.

7.4 (Quasi-)Hyperelliptic Surfaces In characteristic p = 2, 3, these surfaces have the following invariants:

ωS ∼ OS ωS⊗12 ∼ = = OS pg = 0 h0,1 = 1, χ (OS ) = 0 c2 = 0 b1 = 2 b2 = 2. Moreover, these surfaces are equipped with two elliptic fibrations: one is the Albanese fibration S → E, where E is an elliptic curve, and then, there exists a second fibration S → P1 . It turns out that all these surfaces arise as quotients S = (E × F)/G , where E and F are elliptic curves and G is a group acting faithfully on both, E and F. The quotient yielding S is via the diagonal action. In particular, the classical list of Bagnera–DeFranchis (see [6, List 10.27]) gives all classes. The more complicated classes arise in characteristic 2 and 3. First, these surfaces have invariants (again, ≡ denotes numerical equivalence):

ωS ≡ OS χ (OS ) = 0 c2 = 0 b1 = 2 b2 = 2. It turns out that 1 ≤ pg + 1 = h0,1 ≤ 2 and that surfaces with h0,1 = 1 are precisely those with non-smooth Picard schemes [17].


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In any case, the Albanese morphism S → Alb(S) is onto an elliptic curve, and its generic fiber is a curve of genus one. This motivates the following. Definition 7.6. The surface is called hyperelliptic, if S → Alb(S) is an elliptic fibration, and quasi-hyperelliptic if this fibration is quasi-elliptic. In both cases, there exists a second fibration S → P1 , which is always elliptic. Finally, for every (quasi-)hyperelliptic surface S, there exists: 1. An elliptic curve E 2. A curve C of arithmetic genus one, which is smooth if S is hyperelliptic, or rational with a cusp if S is quasi-hyperelliptic 3. A finite and flat group scheme G (possibly non-reduced), together with embeddings G → Aut(C) and G → Aut(E), where G acts by translations on E such that S is isomorphic to S∼ = (E × C)/G . The Albanese map arises as projection onto E/G with fiber C, and the other fibration onto C/G ∼ = P1 is elliptic with fiber E. It turns out that G may contain infinitesimal subgroups, which gives rise to new cases even for hyperelliptic surfaces. In any case, it turns out that G is Abelian. This implies that the especially large and non-Abelian automorphism groups of elliptic curves with j = 0 in characteristics 2 and 3 (see Sect. 2.2) do not give rise to new classes. We refer to [16] for the complete classification of hyperelliptic surfaces and to [17] for the classification of quasi-hyperelliptic surfaces. An interesting feature in characteristics 2 and 3 is the possibility that G acts trivially on ωE×C , and thus, the canonical sheaf on S is trivial. In this case, we find pg = 1, h0,1 = 2, and the Picard scheme of S is not reduced.

8 General Type In this section we discuss surfaces of general type, and refer to [7, Chap. VII] and the references given there for the corresponding results over the complex numbers.

8.1 Pluricanonical Maps Let S be a minimal surface of general type. Clearly, KS2 > 0 since some pluricanonical map has a two-dimensional image. However, we shall see below that Castelnuovo’s inequality c2 > 0 may fail. Let us recall that a rational (−2)-curve is a curve C on a surface with C ∼ = P1k and C2 = −2.

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Theorem 8.1. Let S be a minimal surface of general type. Then, the (a priori) rational Iitaka fibration to the canonical model S Scan := Proj Rcan (S) = Proj

H 0 (S, ωS⊗n )

n≥0

is a birational morphism that contracts all rational (−2)-curves and nothing more. Rational (−2)-curves form configurations, whose intersection matrices are negative definite. These matrices are Cartan matrices and correspond to Dynkin diagrams of type A, D, and E. In particular, the morphism S → Scan contracts these curves to DuVal singularities (also known as canonical singularities or rational double points) (see [6, Chap. 3], as well as Artin’s original papers [1, 2]). Bombieri’s results on pluricanonical systems were extended to positive characteristic in [37] and refined in [22,129], and we refer to these articles for more results. We give a hint of how to modify the classical proofs below. Also, the reader who is puzzled by the possibility of purely inseparably uniruled surfaces of general type in the statements below might want to look at Sect. 9 first. Theorem 8.2 (Ekedahl, Shepherd–Barron). Let S be a minimal surface of general type and consider the linear system |mKS | on the canonical model Scan : 1. 2. 3. 4.

It is ample for m ≥ 5 or if m = 4 and KS2 ≥ 2 or m = 3 and KS2 ≥ 3. It is base-point free for m ≥ 4 or if m = 3 and KS2 ≥ 2. It is base-point free for m = 2 if KS2 ≥ 5 and p ≥ 11 or p ≥ 3 and S is not uniruled. It defines a birational morphism for m = 2 if KS2 ≥ 10, S has no pencil of genus 2 curves and p ≥ 11 or p ≥ 5 and S is not uniruled.

Next, we have the following version of Ramanujam vanishing (see [37, Theorem II.1.6] for the complete statement): Theorem 8.3 (Ekedahl). Let S be a minimal surface of general type and let L be an invertible sheaf that is numerically equivalent to ωS⊗i for some i ≥ 1. Then, H 1 (S, L∨ ) = 0 except possibly for certain surfaces in characteristic 2 with χ (OS ) ≤ 1. On the other hand, minimal surfaces of general type with H 1 (S, ωS∨ ) = 0 in characteristic 2 do exist [37, Proposition I.2.14]. Bombieri’s proof of the above results over the complex numbers is based on vanishing theorems H 1 (S, L) = 0 for certain more or less negative invertible sheaves. However, these vanishing results may fail in positive characteristic (see [114] or Sect. 3.6). Ekedahl [37] overcomes this problem as follows: he considers an invertible sheaf L and its Frobenius pullback F ∗ (L) ∼ = L⊗p as group schemes over S. Then, Frobenius induces a short exact sequence of group schemes (for the flat topology on S): 0 → αL → L → F ∗ (L) → 0 . F

(1)


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By definition, αL is the kernel of F (see also the definition of the group scheme α p in Sect. 2.3). This αL is an infinitesimal group scheme over S and can be thought of as a possibly non trivial family of α p ’s over S. Now, if L∨ is ample, then H 1 (S, L⊗ν ) = 0 for ν 0 (Serre vanishing; see [48, Theorem III.7.6]). In order to get vanishing of H 1 (S, L), we assume that this is not the case and replace L by some L⊗ν such that H 1 (S, L) = 0 and H 1 (S, L⊗p ) = 0. Then, the long exact sequence in cohomology for Eq. (1) yields Hfl1 (S, αL ) = 0. Such a cohomology class corresponds to an αL -torsor, which implies that there exists a purely inseparable morphism of degree p: π

Y → S, where Y is an integral Gorenstein surface, whose dualizing sheaf satisfies ωY ∼ = π ∗ (ωS ⊗ L p−1 ). (The subscript fl in the cohomology group above denotes the flat topology, which is needed since αL -torsors are usually only locally trivial with respect to the flat topology.) ⊗(−m) For example, suppose S is of general type and L = ωS for some m ≥ 1. Then either H 1 (S, L) = 0 and one proceeds as in the classical case or there exists an inseparable cover Y → S, where ωY∨ is big and nef. The second alternative implies that S is inseparably dominated by a surface of special type, namely, Y , and a further analysis of the situation leads to either a contradiction (establishing the desired vanishing result) or an explicit counterexample to a vanishing result. For example, Theorem 8.3 is proved this way.

8.2 Castelnuovo’s Inequality Over the complex numbers, surfaces of general type satisfy Castelnuovo’s inequality c2 > 0. In [114], Raynaud constructs minimal surfaces of general type with c2 < 0 in every characteristic p ≥ 5, i.e., this inequality fails. On the other, we have the following structure result: Theorem 8.4 (Shepherd–Barron [130]). Let S be a minimal surface of general type: 1. If c2 (S) = 0, then S is inseparably dominated by a surface of special type. 2. If c2 (S) < 0, then the Albanese map S → Alb(S) has one-dimensional image, whose generic fiber is a singular rational curve. In particular, S is uniruled. In characteristic p ≥ 11, surfaces of general type satisfy χ (OS ) > 0. We refer to [130] for more detailed statements. There do exist surfaces of general type with c2 < 0, but in view of Noether’s formula 12χ = c21 + c2 , one might

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ask whether the stronger inequality χ > 0 still holds for surfaces of general type in positive characteristic. At least, we have the following analog of a theorem of Castelnuovo and DeFranchis: Proposition 8.5. Let S be a surface with χ (OS ) < 0. Then: 1. S is birationally ruled over a curve of genus 1 − χ (OS). Or 2. S is quasi-elliptic of Kodaira dimension κ = 1 and p ≤ 3. Or 3. S is a surface of general type and p ≤ 7. Proof. If κ = −∞, then S is birationally ruled over a curve of genus 1 − χ (OS ) and we get the first case. Also, by the explicit classification, there are no surfaces with κ = 0 and χ (OS ) < 0. For κ = 2, this is [130, Theorem 8]. If κ = 1, then S admits a (quasi-)elliptic fibration S → B, say, with generic fiber F. Also, we may assume that S is minimal. In case F is smooth then [33] yields c2 (S) ≥ e(F) · e(B) = 0, where e denotes the Euler number. Since c21 (S) = 0 for a relatively minimal (quasi-)elliptic fibration, Noether’s formula yields χ (OS ) = 0. Thus, if χ (OS ) < 0, then the fibration must be quasi-elliptic and such surfaces exist for p ≤ 3 only. Quasi-elliptic surfaces with χ (OS ) < 0 in characteristic p ≤ 3 can be found in [114], i.e., the first two cases of the previous proposition do exist. On the other hand, it is still unknown whether there do exist surfaces of general type with χ (OS ) ≤ 0.

8.3 Noether’s Inequality Every minimal surface of general type fulfills KS2 ≥ 2pg (S) − 4

(Noether’s inequality).

Moreover, if the canonical map is composed with a pencil, then KS2 ≥ 3pg (S) − 6

(Beauville’s inequality)

holds true. If the canonical map is birational onto its image, then KS2 ≥ 3pg (S) − 7

(Castelnuovo’s inequality)

holds true (see [79, 82]). In particular, this area of geography of surfaces of general type behaves as over the complex numbers. We recall that surfaces that are extremal with respect to Noether’s inequality are called Horikawa surfaces. More precisely, an even Horikawa surface is a minimal surface of general type with K 2 = 2pg − 4, whereas an odd Horikawa surface satisfies K 2 = 2pg − 3. These surfaces are classified in arbitrary characteristic in


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[79, 82]. Basically, the same structure results as over the complex numbers hold for them: most of them arise as double covers of rational surfaces via their canonical maps. In characteristic 2, the canonical map may become purely inseparable, and then, the corresponding Horikawa surfaces are unirational (see also Sect. 9). We refer to the aforementioned articles for the precise classification results, the description of the moduli spaces, as well as the description of the subsets in these moduli spaces corresponding to surfaces with inseparable canonical maps. Finally, unirational Horikawa surfaces in characteristic p ≥ 3 were systematically constructed in [85]. Also, Beauville’s result that minimal surfaces of general type with K 2 < 3pg − 7 are double covers of rational surfaces via their canonical maps still holds in positive characteristic [82].

8.4 Bogomolov–Miyaoka–Yau Inequality A minimal surface of general type over the complex numbers fulfills KS2 ≤ 9χ (OS ) or, equivalently, KS2 ≤ 3c2 (S). This is proved using analytic methods from differential geometry. Moreover, by a theorem of Yau, surfaces with c21 = 3c2 are uniformized by the complex 2-ball and thus, these surfaces are rigid by a theorem of Siu. Minimal surfaces of general type with c2 ≤ 0 (counterexamples to Castelnuovo’s inequality) provide counterexamples to the Bogomolov–Miyaoka–Yau inequality. But even if c2 is positive, it may fail, as shown by Parshin [110] and Szpiro [138, Sect. 3.4.1]. More precisely, they construct series of examples, where c2 is bounded and c21 tends to infinity. Let us also mention the counterexamples of [8, Kapitel 3.5.J], where covers of P2 ramified over special line configurations that only exist in positive characteristic are used. Similar constructions appeared in [34]. Since Parshin’s counterexamples have highly non-reduced Picard schemes, he asked in [111], whether surfaces of general type with reduced Picard schemes satisfy the Bogomolov–Miyaoka–Yau inequality. Also this turns out to be wrong by the examples of Jang [63]. In [37, Remark (i) to Proposition 2.14], a ten-dimensional family of surfaces with K 2 = 9 and χ (OS ) = 1 in characteristic 2 is constructed, i.e., rigidity on the Bogomolov–Miyaoka–Yau line fails. On the other hand, there is the following positive result [129]. Theorem 8.6 (Shepherd–Barron). If S is a minimal surface of general type in characteristic 2 that lifts over W2 (k) then c21 (S) ≤ 4c2 (S) holds true. We refer to [129] for results circling around Bogomolov’s inequality c21 (E) ≤ 4c2 (E) for stable rank 2 vector bundles.

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8.5 Global Vector Fields The tangent space to the automorphism group scheme of a smooth variety is isomorphic to the space of global vector fields. Since a surface of general type has only finitely many automorphisms, this implies that there are no global vector fields on a surface of general type in characteristic zero. However, in positive characteristic, the automorphism group scheme of a surface of general type has still finite length but may contain infinitesimal subgroup schemes, which have nontrivial tangent spaces. Thus, infinitesimal automorphism group schemes of surfaces of general type in positive characteristic give rise to nontrivial global vector fields. For examples, we refer to Lang’s article [71].

8.6 Inseparable Albanese Morphisms If X is smooth and projective, then the pullback of a nonzero global one-form from Alb(X) to X via the Albanese morphism is again nonzero by a result of Igusa (see Section 3.4). Now, for a smooth projective curve, the Albanese map is trivial (g = 0), an isomorphism (g = 1), or a closed immersion (g ≥ 2). Despite of Igusa’s result, it is still possible that the Albanese morphism becomes purely inseparable in positive characteristic: Theorem 8.7. Let k be an algebraically closed field of positive characteristic p. Then, for every Abelian surface A over k, there exist infinitely many surfaces of general type (actually, with arbitrary large KX2 , i.e., these examples do not lie in some bounded families), whose Albanese morphisms are generically finite and purely inseparable onto A. Proof. Let L be a very ample line bundle on A such that H 1 (A, L⊗i ) = 0 for all −p < i < 0. Then, for a generic section s ∈ H 0 (A, L⊗p ), we consider the purely inseparable cover π : X → A of degree p that is given by taking a p.th root of s. That " p−1 is, π∗ OX ∼ = i=0 L⊗(−i) and the OA -algebra structure is given by multiplication by s : L⊗(−p) → OA . Since s is generic, X has at worst canonical singularities of type A p−1, see [82, Theorem 3.4]. From ωX ∼ = π ∗ (ωA ⊗ L⊗(p−1) ), we find that ωX is ample, i.e., X is the canonical model of a surface of general type. Taking cohomology, and using the assumption h1 (L⊗i ) = 0 for −p < i < 0, we find h1 (OX ) = h1 (OA ) = 2. Since X has rational singularities, we find h1 (OX' ) = 2. This ' ≤ 4 for the first -adic Betti number (we don’t know yet gives the estimate b1 (X) ' is reduced). On the other hand, we have a surjective morphism whether Pic(X)

α : X' → X → A, ' ≥ 4. Thus, b1 (X) ' = 4 (and it onto an Abelian surface, giving the estimate b1 (X) ' is reduced, but we will not need this fact). In any case, the follows that Pic(X)


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' of X' is an Abelian surface. We claim that α is Albanese variety A := Alb(X) ' namely, let alb : X' → A be the Albanese map. By the Albanese morphism of X: universality, α factors over alb showing that alb is generically finite onto A . Also, the induced map A → A is finite. Since α is a generically finite of degree p, which is a prime, we get simply for degree reasons, that either X' is birational to an Abelian variety (which is not the case), or A and A are isomorphic. Thus, α is the Albanese morphism. Finally we note that every sufficiently ample line bundle on A satisfies the assumptions of our construction. In particular, for every A we obtain infinitely many such X with unbounded KX2 .

8.7 Nonclassical Godeaux Surfaces Since KS2 > 0 for a minimal surface of general type, it is natural to classify surfaces with KS2 = 1. It turns out that these fulfill 1 ≤ χ (OS ) ≤ 3 and thus, the lowest invariants possible are as follows: Definition 8.8. A numerical Godeaux surface is a minimal surface of general type with χ (OS ) = KS2 = 1. Such a surface is called classical if pg = h0,1 = 0 and otherwise nonclassical. In characteristic zero or in characteristic p ≥ 7, numerical Godeaux surfaces are classical [81]. Moreover, quotients of a quintic surface in P3 by a free Z/5Z-action (this construction is due to Godeaux) provide examples of classical Godeaux surfaces in characteristic p = 5. Classical and nonclassical Godeaux surfaces in characteristic p = 5 have been constructed by Lang [70] and Miranda [93]. Nonclassical Godeaux surfaces in characteristic p = 5 have been completely classified in [81]—it turns out that all of them arise as quotients of (possibly highly singular) quintic surfaces in P3 by Z/5Z or α5 . We finally note that nonclassical Godeaux surfaces are precisely those numerical Godeaux surfaces that have nonreduced Picard schemes. Quite generally, for every n there exists an integer P(n) such that minimal surfaces of general type with K 2 ≤ n in characteristic p ≥ P(n) have a reduced Picard scheme [80]. Thus, P(1) = 7, but P(n) is unbounded as a function of n.

8.8 Surfaces with pg = 0 For a minimal surface of general type with pg = 0 over the complex numbers, the inequality χ (OS ) > 0 forces h0,1 = 0, thus χ (OS ) = 1, and then, the Bogomolov– Miyaoka–Yau inequality implies 1 ≤ K 2 ≤ 9. Interestingly, these (in-)equalities hold over any field:

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Proposition 8.9. Let S be a minimal surface of general type with pg (S) = 0. Then, the equalities and inequalities h0,1 (S) = 0,

χ (OS ) = 1,

and

1 ≤ KS2 ≤ 9

hold true. Proof. From pg = 0 we get χ (OS ) ≤ 1. Suppose first that χ (OS ) = 1 holds. Then h0,1 = 0 and we find b1 = 0, which yields c2 = 2 − 2b1 + b2 ≥ 3. But then, Noether’s formula yields KS2 = 12χ (OS ) − c2 (S) ≤ 12 − 3 = 9, which gives the desired (in)equalities. If χ (OS ) ≤ 0, then Noether’s formula implies c2 (S) < 0. By Theorem 8.4, the Albanese morphism S → Alb(S) is a fibration over a curve. Thus, b2 ≥ ρ (S) ≥ 2 using Igusa’s inequality. Next, we have h0,1 = 1 − χ (OS ) and in particular, b1 ≤ 2(1 − χ (OS)). Thus, c2 (S) = 2 − 2b1 + b2 ≥ 2 − 4(1 − χ (OS)) + 2 = 4χ (OS ). But then, Noether’s formula implies 12 χ (OS ) = KS2 + c2 (S) ≥ 4χ (OS ) + KS2 and we obtain 0 ≥ 8 χ (OS ) ≥ KS2 , a contradiction. Thus, the χ (OS ) ≤ 0 case cannot happen and we are done.

The first examples of algebraically simply connected surfaces of general type with pg = 0 were constructed by Lee and Nakayama [75], by adapting Q-Gorenstein smoothing techniques to positive characteristic. Theorem 8.10 (Lee–Nakayama). There do exist algebraically simply connected surfaces of general type with pg = 0, all values 1 ≤ K 2 ≤ 4, and in all characteristics p ≥ 3.

9 Unirationality, Supersingularity, Finite Fields, and Arithmetic In this and the following section we discuss more specialized characteristic-ptopics. In this section, we circle around rationality, unirationality, their effect on Néron–Severi groups, and the formal Brauer group. We discuss these for K3 surfaces and surfaces over finite fields. Finally, we discuss zeta functions and the Tate conjecture.


265

9.1 An Instructive Computation To start with, let ϕ : X Y be a dominant and generically finite morphism in characteristic zero. Then, the pullback of a nonzero pluricanonical form is again a nonzero pluricanonical form. Thus, if κ (X) = −∞, also κ (Y ) = −∞ holds true. However, over a field of positive characteristic p, the example

ϕ : t → t p

and then

ϕ ∗ (dt) = dt p = pt p−1 dt = 0

shows that the pullback of a nonzero pluricanonical form may become zero after pullback. In particular, the previous characteristic zero argument, which shows that the Kodaira dimension cannot increase under generically finite morphisms, breaks down. However, if S is separably uniruled, i.e., if there exists a dominant rational map P1 ×C S such that the finite field extension k(S) ⊂ k(P1 ×C) is separable, then this phenomenon does not occur, we find κ (S) = −∞, and applying Theorem 4.4, we conclude: Theorem 9.1. A separably uniruled surface is birationally ruled. In particular, if a surface S is separably unirational, then p2 (S) = 0. But being dominated by a rational surface, its Albanese map is trivial and so b1 (S) = 0. Thus, S is rational by Theorem 4.6, and we have shown: Theorem 9.2. A separably unirational surface is rational.

9.2 Zariski Surfaces On the other hand, Zariski [152] gave the first examples of (inseparably) unirational surfaces in positive characteristic that are not rational: for a generic choice of a polynomial f (x, y) ∈ k[x, y] of sufficiently large degree, z p − f (x, y) = 0

(2)

extends to an inseparable cover X → P2 , where X has “mild” singularities and where ˜ ≥ 0 for some resolution of singularities X˜ → X. By construction, we usually κ (X) have an inclusion of function fields: √ p ˜ = k(x, y)[ p f (x, y)] ⊂ k( p x, √ k(x, y) ⊂ k(X) y), i.e., X˜ is unirational. Surfaces that arise as desingularizations of covers of the form (2) are called Zariski surfaces.

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Theorem 9.3 (Zariski). In every positive characteristic there do exist unirational surfaces that are not rational. However, we have seen in Theorem 4.6 that rational surfaces are still characterized as those surfaces that satisfy h0,1 = p2 = 0.

9.3 Quasi-Elliptic Surfaces If S → B is a quasi-elliptic fibration from a surface S with generic fiber F, then there exists a purely inseparable extension L/k(B) of degree p = char(k), such that FL := F ×Spec k(B) Spec L is not normal, i.e., the cusp “appears” over L (see [17]). Thus, the normalization of FL is isomorphic to P1L , and we get the following result: Theorem 9.4. Let S be a surface and S → B be a quasi-elliptic fibration. Then, there exists a purely inseparable and dominant rational map B × P1 S, i.e., S is (purely inseparably) uniruled. In particular, if S → P1 is a quasi-elliptic fibration, then S is a Zariski surface and, thus, unirational.

9.4 Fermat Surfaces If the characteristic p = char(k) does not divide n, then the Fermat surface Sn , i.e., the hypersurface Sn := { xn0 + xn1 + xn2 + xn3 = 0 } ⊂ P3k is smooth over k. For n ≤ 3 it is rational, for n = 4 it is K3, and for n ≥ 5 it is of general type. Shioda and Katsura have shown in [65, 131] that Theorem 9.5 (Katsura–Shioda). For n ≥ 4 and p n, the Fermat surface Sn in characteristic p is unirational if and only if there exists a ν ∈ N such that pν ≡ −1 mod n. Shioda [134] generalized this result to Delsarte surfaces. The example of Fermat surfaces shows that being unirational is quite subtle. Namely, one can show that the generic hypersurface of degree n ≥ 4 in P3k is not unirational, and thus, being unirational is not a deformation invariant. From the point of view of Mori theory it is interesting to note that unirational surfaces that are not rational are covered by singular rational curves. However, (unlike in characteristic zero) it is not possible to smoothen these families—after all, possessing a pencil of smooth rational curves implies that the surface in question is rational.


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9.5 Fundamental Group There do exist geometric obstructions to unirationality: being dominated by a rational surface, the Albanese morphism of a unirational surface is trivial, and we conclude b1 = 0. Moreover, Serre [126] showed that the fundamental group of a unirational surface is finite, and Crew [28] showed that it does not contain p-torsion in characteristic p. A subtle invariant is the formal Brauer group (see Sect. 9.8 below), whose height can prevent a surface from being unirational (and that may actually be the only obstruction to unirationality for K3 surfaces).

9.6 Horikawa Surfaces Let us recall from Sect. 8.3 that a minimal surface of general type is called an even Horikawa surface if it satisfies K 2 = 2pg − 4. This unbounded class is particularly easy to handle, since all such surfaces arise as double covers of rational surfaces. In view of the previous paragraph, let us also mention that they are algebraically simply connected. In [85], we constructed unirational Horikawa surfaces in arbitrarily large characteristics and for arbitrarily large pg . Thus, although the generic Horikawa surface is not unirational, being unirational is nevertheless a common phenomenon.

9.7 K3 Surfaces and Shioda-Supersingularity We recall that the Kummer surface Km(A) of an Abelian surface A is the minimal desingularization of the quotient of A by the sign involution. In characteristic p = 2 the Kummer surface is always a K3 surface. Shioda [132] determined when such surfaces are unirational—in particular, his result establishes the existence of unirational K3 surfaces in every characteristic p ≥ 3: Theorem 9.6 (Shioda). Let A be an Abelian surface in characteristic p ≥ 3. Then, the Kummer surface Km(A) is unirational if and only if A is a supersingular Abelian variety. We recall from Definition 7.2 that an Abelian variety is called supersingular if its p-torsion subgroup scheme A[p] is infinitesimal. To explain the notion of supersingularity introduced by Shioda [131] let us recall from Sect. 3.5 that Igusa’s inequality states ρ ≤ b2 , where ρ denotes the rank of the Néron–Severi group and b2 is the second Betti number. Definition 9.7. A surface S is called supersingular in the sense of Shioda if ρ (S) = b2 (S) holds true. This notion is motivated by the following result, also from [131].

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Theorem 9.8 (Shioda). Uniruled surfaces are Shioda-supersingular. The unirationality results on Kummer and Fermat surfaces show that these classes of surfaces are unirational if and only if they are supersingular in the sense of Shioda. This leads to the following: Conjecture 9.9 (Shioda). A K3 surface is unirational if and only if it is Shiodasupersingular. Apart from Kummer surfaces (basically Theorem 9.6), this conjecture is known to be true in characteristic 2: the Néron–Severi lattices of Shioda-supersingular K3 surfaces have been classified in [117] and using these results they show Theorem 9.10 (Rudakov–Shafarevich). Every Shioda-supersingular K3 surface in characteristic 2 possesses a quasi-elliptic fibration. In particular, these surfaces are Zariski surfaces and unirational. It is also known in the following cases: for supersingular K3 surfaces with Artin invariant σ0 ≤ 6 (see Definition 9.13) and p = 3 [117], for σ0 ≤ 3 and p = 5 [113], and for elliptic K3 surfaces with pn -torsion section [60].

9.8 K3 Surfaces and Artin-Supersingularity There exists yet another notion of supersingularity, apart from those of Definition 7.2 and Definition 9.7: for a K3 surface S over k, Artin [3] considers the functor that associates to every Artin-algebra A over k the Abelian group × ) → H 2 (S, OS× ) . Br : A → ker H 2 (S × A, OS×A This functor is pro-representable by a one-dimensional formal group law, the ( so-called formal Brauer group Br(S) of S. (Of course, this functor can be studied for arbitrary varieties, not just K3 surfaces. Under suitable conditions, which are satisfied for K3 surfaces, it is pro-representable by a formal group law of dimension h0,2 ; see [4].) Over a field of characteristic zero, there exists for every one-dimensional formal ˆ a. group law an isomorphism (the logarithm) to the additive formal group law G In positive characteristic, this need no longer be the case, and every formal group law has a discrete invariant, called the height. The height h is a strictly positive integer or infinity and measures the complexity of multiplication by p in the group law. For example, h = ∞ means that multiplication by p is equal to zero. By a result of Lazard, h determines the formal group law if the ground field is algebraically closed. Over an algebraically closed field, a one-dimensional group law of height h = 1 is ˆ m , whereas height h = ∞ corresponds isomorphic to the multiplicative group law G ˆ a . We refer to [49] for more on formal group laws and to to the additive group law G Artin’s and Mazur’s original article [4] for applications to geometry.


269

For K3 surfaces, the height h of the formal Brauer group satisfies 1 ≤ h ≤ 10 or h = ∞. This follows from the fact that b2 = 22 together with the formula

ρ (S) ≤ b2 (S) − 2h(S),

(3)

which holds if h = ∞ (see [3]). Moreover, the height h stratifies the moduli space of K3 surfaces: K3 surfaces with h = 1—these are called ordinary—are open in families, and surfaces with h ≥ h0 + 1 form a closed subset inside families of surfaces with h ≥ h0 . We refer to [3,42,107] for more on the geometry of the height stratification of the moduli space. Definition 9.11. A K3 surface is called supersingular in the sense of Artin, if its formal Brauer group has infinite height. Shioda-supersingular K3 surfaces are Artin-supersingular, which follows from formula (3). To prove the converse direction, one first reduces to the case of finite fields, where it follows from the Tate conjecture for K3 surfaces with h = ∞ (see the discussion below). For elliptic K3 surfaces with h = ∞, this latter conjecture was established by Artin [3], and for K3 surfaces possessing a degree 2 polarization, by Rudakov, Shafarevich, and Zink [119]. Finally, it was established by Charles [24] and Maulik [88] for every characteristic p ≥ 5. Theorem 9.12. Let X be a K3 surface in characteristic p > 0. Assume that p ≥ 5, or that X is elliptic, or that it possesses a degree 2 polarization. Then, X is Artinsupersingular if and only if it is Shioda-supersingular. To stratify the moduli space of Artin-supersingular K3 surfaces, we consider their Néron–Severi groups. The discriminant of the intersection form on NS(S) of an Artin-supersingular K3 surface S is equal to disc NS(S) = ±p2σ0 for some integer 1 ≤ σ0 ≤ 10 by [3]. Definition 9.13. The integer σ0 is called the Artin invariant of the Artinsupersingular K3 surface. In characteristic p ≥ 3, Shioda-supersingular K3 surfaces with σ0 ≤ 2 are Kummer surfaces of supersingular Abelian surfaces. Their moduli space is onedimensional but non-separated. Moreover, there is precisely one such surface with σ0 = 1, and it arises as Km(E × E), where E is a supersingular elliptic curve. We refer to [106, 133] for details and to [121] for the description in characteristic 2. From Theorem 9.6, it follows that Shioda-supersingular K3 surfaces with σ0 ≤ 2 are unirational (see also the discussion at the end of Sect. 9.7). Ogus [106] established a Torelli theorem for Shioda-supersingular K3 surfaces with marked Picard lattices in terms of crystalline cohomology. Finally, we refer to [118] for further results on K3 surfaces in positive characteristic.

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9.9 Zeta Functions and Weil Conjectures If X is a smooth and projective variety of dimension d over a finite field Fq , then we can count the number #X(Fqn ) of its Fqn -rational points and form its zeta function : ∞ tn . Z(X,t) := exp ∑ #X(Fqn ) n n=1 Weil conjectured many properties of Z(X,t), and it was Grothendieck’s insight that many of these properties would follow from the existence of a suitable cohomology theory, namely, -adic cohomology. These conjectures are now known to hold by work of Deligne, Dwork, Grothendieck, Weil, and others—we refer to [48, Appendix C] for an overview and to [92] for details. In particular, the zeta function is a rational function of the form Z(X,t) =

P1 (S,t) · P3 (S,t) · . . . · P2d−1 (X,t) , P0 (X,t) · P2 (X,t) · . . . · P2d (X,t)

where each Pi (X,t) is a polynomial with integral coefficients, with constant term 1, and of degree equal to the ith Betti number bi (X). In the extremal cases we have P0 (X,t) = 1 −t and P2d (X,t) = 1 − qd t. Moreover, over the complex numbers, these polynomials factor as Pi (X,t) =

bi (X)

∏ (1 − αi j t) , j=1

where the αi j are complex numbers (in fact, algebraic integers) of absolute value qi/2 . Finally, there is a functional equation Z(X,

1 ) = ± qdE/2 t E · Z(X,t), qd t

where E is the Euler number cd (X) = cd (ΘX ). The Frobenius morphism Fq : x → xq acts trivially on Fq and topologically generates the absolute Galois group Gal(Fq /Fq ). It also induces an Fq -linear morphism Fq : XFq → XFq . Now, an important characterization of Pi (X,t) is that it is equal to the characteristic polynomial det(1 − t · Fq∗ ) of the linear map Fq∗ induced by Fq on Heí t (XFq , Q ). In fact, taking this as definition for the Pi (X,t)’s, and noting that Fqn -rational points of X correspond to fixed points under Fqn , the rationality of the zeta function and the specific form of its factors as given above follow from Lefschetz fixed-point formulae for powers of Fq∗ on -adic cohomology [48, Appendix C.4]. There is an injective Chern map c1 : NS(XFq ) ⊗ Q → He´2t (XFq , Q ), which is equivariant with respect to the Galois actions of Gal(Fq /Fq ) on both sides. In particular, a non-torsion invertible sheaf L ∈ NS(X) is Galois-invariant, and thus,


271

c1 (L) is a nontrivial and Galois-invariant class in He´2t (XFq , Q ). Since Frobenius

topologically generates Gal(Fq /Fq ), we see that c1 (L) is an eigenvector of Fq∗ for the eigenvalue q. In particular, q−1 is a root of P2 (X,t) and thus, (1 − qt) divides P2 (X,t). Applying this argument to the whole of NS(X), we find that (1 − qt)ρ (X)

divides

P2 (X,t),

where ρ (X) denotes the rank of NS(X). In [140], Tate conjectured that the image c1 (NS(X) ⊗ Q ) is not only a subspace but is also in fact equal to the whole eigenspace of Fq∗ to the eigenvalue q. We shall now discuss this conjecture in greater detail.

9.10 Tate Conjecture Let us now specialize to the case where S := X is a smooth and projective surface over Fq . For the factorization of Z(S,t), we have P0 (S,t) = 1 − t and P4 (S,t) = 1−q2t. Since He´1t (XFq , Q ) is Galois-equivariantly isomorphic to He´1t (Alb(X)Fq , Q ) via the Albanese morphism, we conclude P1 (X,t) = P1 (Alb(X),t). And finally, by Poincaré duality, we have P3 (S,t) = P1 (S, qt). Thus, the “interesting” part of -adic cohomology and the zeta function is encoded in P2 (S,t). We have also just seen that (1 − qt)ρ (S) divides P2 (S,t). Now, suppose for a moment that SFq is Shioda-supersingular. After possibly replacing Fq by a finite extension, we may assume that all divisor classes of SFq

are defined over Fq and then, we have P2 (S,t) = (1 − qt)b2(S) . Moreover, if Alb(S) is trivial or a curve, then Z(S,t) is equal to the zeta function of a birationally ruled surface. This fits perfectly to Conjecture 9.9. Also, one might expect that if a surface over Fq satisfies P2 (S,t) = (1 − qt)b2(S) , then it is Shioda-supersingular. This expectation would follow from the following, more general conjecture of Tate [140]: Conjecture 9.14 (Tate conjecture). Let S be a smooth and projective surface over Fq and factor P2 (S,t) as P2 (S,t) =

b2 (S)

∏ (1 − α2, jt),

with α2, j ∈ Q ∩ Z.

j=1

Then, the Néron–Severi rank ρ (S) is equal to the number of times q occurs among the α2, j . For an overview, we refer to [144, Lecture 2]. For a relation of Tate’s conjecture with Igusa’s inequality and a conjecture of Artin and Mazur on Frobenius eigenvalues on crystalline cohomology, we refer to [55, Remarque II.5.13]. Also, Artin and

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C. Liedtke

Tate [142, (C)] refined Tate’s conjecture as follows: let D1 , . . . , Dρ be independent classes in NS(S) and set B := ∑i ZDi . Let #Br(S) be the order of the Brauer group, which is conjecturally finite. Then Conjecture 9.15 (Artin–Tate). We have P2 (S, q−s ) ∼ (−1)ρ (S)−1 ·

#Br(S) · det({Di · D j }i, j ) · (1 − q1−s)ρ (S) · (NS(S) : B)2

qχ (OS )−1+b1 (S)

as s tends to 1. In fact, Conjectures 9.14 and 9.15 are equivalent, as shown up to p-power by Artin and Tate [142] and the full equivalence was established by Milne [91]. For elliptic surfaces, the Artin–Tate conjecture is a function field analog of the Birch–Swinnerton–Dyer conjecture (see [5, 142]). For explicit examples, progress on this conjecture, and interrelations, we refer to [144]. The Tate conjecture is known in the following cases: Theorem 9.16 (Tate [141]). The conjectures of Tate and Artin–Tate hold for Abelian varieties and products of curves over finite fields. Let us discuss what is known for K3 surfaces: for elliptic K3 surfaces, it was established by Artin and Swinnerton–Dyer [5]. For ordinary K3 surfaces, it was established by Nygaard [103], and for K3’s with finite height of the formal Brauer group and p ≥ 5, by Nygaard and Ogus [104]. For K3 surfaces of infinite height (Artin-supersingular), equipped with a polarization of degree 2, it was established by Rudakov, Shafarevich, and Zink [119], if p is large with respect to a polarization degree by Maulik [88], and for p ≥ 5 by Charles [24]. Thus, we obtain Theorem 9.17. The conjectures of Tate and Artin–Tate hold for K3 surfaces in characteristic p ≥ 5. By [76], this implies that there exist only finitely many K3 surfaces defined over a fixed finite field of characteristic p ≥ 5. This is similar to the situation for Abelian varieties: by [150], there exist only finitely many Abelian varieties of a fixed dimension over a fixed finite field. Coming back to P2 (S,t), we note that Poincaré duality implies that if β is among the α2, j , then so is q/β . For K3 surfaces, using the fact that degP2 (S,t) = b2 (S) = 22 is even, this has the following surprising consequence (see [14, Theorem 13] for a proof): Theorem 9.18 (Swinnerton–Dyer). Let S be a K3 surface over Fq , and assume that the Tate conjecture holds for S. Then, the geometric Néron–Severi rank ρ (SFq ) is even. Interestingly, there are more restrictions on P2 (S,t) if S is a K3 surface, than those coming from the Weil conjectures (see [39, 151]). Let us finally note that if we have an α2, j in the factorization of P2 (S,t) of some surface S over Fq that is not of the form μ · q, where μ is a root of unity, then SFq is


273

not Shioda-supersingular and thus, not unirational. For example, the zeta function of a Fermat surface Sn ⊂ P3 over F p can be computed explicitly using Gauß- and Jacobi-sums. From this, one concludes that if (Sn )F p is Shioda-supersingular, then there must exist a ν such that pν ≡ −1 mod n (see [65] or Theorem 9.5).

10 Inseparable Morphisms and Foliations In this section we study inseparable morphisms of height one in greater detail. On the level of function fields this is Jacobson’s correspondence, a kind of Galois correspondence for purely inseparable field extensions. However, this correspondence is not via automorphisms but via derivations. On the level of geometry, this translates into p-closed foliations. For surfaces, it simplifies to p-closed vector fields. For other overviews, we refer to [36] and [94, Lecture III].

10.1 Jacobson’s Correspondence Let us recall the classical Galois correspondence: given a field K and a finite and separable extension L, there exists a minimal Galois extension of K containing L, the Galois closure Kgal of L. By definition, the Galois group G = Gal(Kgal /K) of this extension is the group of automorphism of Kgal over K, which is finite of degree equal to [Kgal : K]. Finally, there is a bijective correspondence between subgroups of G and intermediate fields K ⊆ M ⊆ Kgal . In particular, there are only finitely many fields between K and Kgal . In Sect. 2.1 we encountered extensions of height one of a field K. It turns out that automorphism of purely inseparable extensions is trivial and thus gives no insight into these extensions. However, there does exist a Galois-type correspondence for such extensions, Jacobson’s correspondence [62, Chap. IV]. Instead of automorphisms, one studies derivations over K: Namely, let L be a purely inseparable extension of height one of K, i.e., K ⊆ −1 −1 L ⊆ K p , or, equivalently, L p ⊆ K. We remark that K p plays the role of a Galois closure of L. Next, we consider the Abelian group Der(L) := {δ : K p

−1

−1

→ K p , δ is a derivation and δ (L) = 0}.

Since δ (x p ) = p · x p−1 · δ (x) = 0, these derivations are automatically K-linear and thus, Der(L) is a K-vector space. Also, Der(L) is a subvector space of Der(K). In case K is of finite transcendence degree n over some perfect field k, then Der(K) is n-dimensional. Now, these vector spaces carry more structure: if δ and η are derivations, then in general their composition δ ◦ η is no derivation, which is why one studies their Lie bracket, i.e., the commutator [δ , η ] = δ ◦ η − η ◦ δ , which is again a derivation.

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Now, over fields of positive characteristic p, it turns out that the p-fold composite δ ◦ . . . ◦ δ is again a derivation. The reason is that expanding this composition the binomial coefficients occurring that usually prevent this composition from being a derivation are all divisible by p, i.e., vanish. This p-power operation is denoted by δ → δ [p] . It turns out that the K-vector spaces Der(K) and Der(L) are closed under the Lie bracket, as well as the p-power operation. Definition 10.1. A p-Lie algebra or restricted Lie algebra is a Lie algebra over a field of characteristic p together with a p-power map δ → δ [p] satisfying the axioms in [61, Definition 4 of Chap. V.7]. We refer to [61, Chap. V.7] for general results on p-Lie algebras. So far, we have associated to every finite and purely inseparable extension L/K of height one a sub-p-Lie algebra of Der(K). Conversely, given such a Lie algebra (V, −[p] ), we may form the fixed set −1

[p] )

(K p )(V,−

−1

:= {x ∈ K p | δ (x) = 0 ∀δ ∈ V } ,

which is easily seen to be a field. Since elements of V are K-linear derivations, this −1 field contains K. Moreover, by construction, it is contained in K p , i.e., of height one. Theorem 10.2 (Jacobson). There is a bijective correspondence { height one extensions of K } ↔ { sub-p-Lie algebras of Der(K) } . Let us mention one important difference to Galois theory: suppose K is of transcendence degree n over an algebraically closed field k, e.g., the function field of −1 an n-dimensional variety over k. Then, the extension K p /K is finite of degree pn . For n ≥ 2 there are infinitely many sub-p-Lie algebras of Der(K) and, thus, infinitely −1 many fields between K and K p .

10.2 Curves Let C be a smooth projective curve over a perfect field k with function field K = k(C). Then, the purely inseparable field extension K p ⊂ K is of degree p and corresponds to the k-linear Frobenius morphism F : C → C(p) . Since every purely inseparable extension L/K of degree p is of the form L = √ −1 K[ p x] for some x ∈ L, such extensions are of height one, i.e., K ⊆ L ⊆ K p . Simply for degree reasons, we see that the k-linear Frobenius morphism is the only purely inseparable morphism of degree p between normal curves. Since every finite purely inseparable field extension can be factored successively into extensions of degree p, we conclude


275

Proposition 10.3. Let C and D be normal curves over a perfect field k and let ϕ : C → D be a purely inseparable morphism of degree pn . Then, ϕ is the n-fold composite of the k-linear Frobenius morphism.

10.3 Foliations From dimension two onwards, there are many more purely inseparable morphisms than just compositions of Frobenius. In fact, if X is an n-dimensional variety with n ≥ 2 over an algebraically closed field k, then the k-linear Frobenius morphism has degree pn and it factors over infinitely many height one morphisms. To classify height one morphisms ϕ : X → Y from a fixed smooth variety X over a perfect field k, we geometrize Jacobson’s correspondence as follows: Definition 10.4. A (p-closed) foliation on a smooth variety X is a saturated subsheaf E of the tangent sheaf ΘX that is closed under the Lie bracket (E is involutive) and the p-power operation. Then, Jacobson’s correspondence translates into Theorem 10.5. There is a bijective correspondence

finite morphisms ϕ : X → Y ↔ { foliations in ΘX }. of height one with Y normal The saturation assumption is needed because an involutive and p-closed subsheaf and its saturation (which will also be involutive and p-closed) define the same extension of function fields and, thus, correspond to the same normal variety. We refer to [36] or [94, Lecture III] for details. Let us also mention [94, Lecture III.2], where a connection between p-closed foliations and non-stability of tangent bundles and uniruledness of varieties (not only in positive characteristic, but also in characteristic zero!) is discussed.

10.4 Surfaces In order to describe finite morphisms of height one ϕ : X → Y from a smooth surface onto a normal surface, we have to consider foliations inside ΘX . The sheaf ΘX and its zero subsheaf correspond to the k-linear Frobenius morphism and the identity, respectively. Thus, height one morphisms of degree p correspond to foliations of rank one inside ΘX . To simplify our exposition, let us only consider A2k , i.e., X = Spec R with R = k[x, y] and assume that k is perfect. Then, ΘX corresponds to the R-module generated by ∂ /∂ x and ∂ /∂ y. Now, a finite morphism of height one ϕ : X → Y with Y normal corresponds to a ring extension:

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C. Liedtke

R p = k[x p , y p ] ⊆ S ⊆ R = k[x, y] , where S is normal. By Jacobson’s correspondence, giving S is equivalent to giving a foliation inside ΘX , which will be of rank one if S = R, R p . This amounts to giving a regular vector field:

δ = f (x, y)

∂ ∂ + g(x, y) ∂x ∂y

for some f , g ∈ R. Since the Lie bracket of a one-dimensional Lie algebra is zero, every rank one subsheaf of ΘX is involutive. Thus, we only have to check closedness under the p-power operation, which translates into

δ [p] = h(x, y) · δ

for some

h(x, y) ∈ R,

i.e., δ is a p-closed vector field. We may assume that f and g are coprime. Then, the zero set of the ideal ( f , g) is of codimension two and is called the singular locus of the vector field. It is not difficult to see that S is smooth over k outside the singular locus of δ (cf. [116]). Finally, a purely inseparable morphism ϕ : X → Y is everywhere ramified, i.e., ΩX/Y has support on the whole of X. Nevertheless, the canonical divisor classes of X and Y are related by a kind of Riemann–Hurwitz formula and the role of the ramification divisor is played by a divisor class that can be read off from the foliation (see [116]). As an application, let us give the main result of [116]: let S be a K3 surface over k, and suppose that we had H 0 (S, ΘS ) = 0. Then, there exists in fact a 0 = δ ∈ H 0 (S, ΘS ) that is p-closed. As explained above, this δ gives rise to an inseparable morphism S → S/δ . A careful analysis of the hypothetical quotient S/δ and its geometry finally leads to a contradiction, and we conclude H 0 (S, ΘS ) = 0, which proves Theorem 7.3.

10.5 Quotients by Group Schemes Let X be a smooth but not necessarily proper variety of any dimension over a perfect field k. We have seen that a global section 0 = δ ∈ H 0 (X, ΘX ) gives rise to an inseparable morphism of degree p and height one if and only if δ is pclosed, i.e., δ [p] = c · δ for some c ∈ H 0 (X, OX ). Now, if X is proper over k, then c ∈ H 0 (X, OX ) = k, and after rescaling δ , we may in fact assume c = 1 or c = 0. Definition 10.6. A vector field δ is called multiplicative if δ [p] = δ and it is called additive if δ [p] = 0. Let δ be additive or multiplicative. Applying a (truncated) exponential series to δ , one obtains on X an action of some finite and flat group scheme G, which is infinitesimal of length p (see [121, Sect. 1]). Then, the inseparable morphism


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ϕ : X → Y corresponding to δ is the quotient morphism X → X/G. Moreover, the one-dimensional p-Lie algebra generated by δ is the p-Lie algebra of G, i.e., the Zariski tangent space of G with p-power map coming from Frobenius. We recall from Theorem 2.8 that the only infinitesimal group schemes of length p are α p and μ p . Putting these observations together, we obtain Proposition 10.7. Additive (resp., multiplicative) vector fields correspond to purely inseparable morphisms of degree p that are quotients by α p - (resp., μ p -) actions. This also explains the terminology for these vector fields: α p (resp., μ p ) is a subgroup scheme of the additive group Ga (resp., multiplicative group Gm ).

10.6 Singularities Let us finally assume that X is a smooth surface and let δ be a multiplicative vector field. By [116], such a vector field can be written near a singularity in local coordinates x, y as

δ =x

∂ ∂ + a·y ∂x ∂y

for some

a ∈ F× p .

Let ϕ : X → Y be the inseparable morphism corresponding to δ . In [51] it is shown that Y has toric singularities of type 1p (1, a). Thus, quotients by μ p behave very much like cyclic quotient singularities in characteristic zero. On the other hand, quotients by α p are much more complicated—the singularities need not even be rational and we refer to [78] for examples.

11 Witt Vectors and Lifting This section deals with lifting to characteristic zero. There are various notions of lifting, and the nicest ones are projective lifts over the Witt ring. For example, in the latter case, Kodaira vanishing and degeneracy of the Frölicher spectral sequence hold true. Unfortunately, although such lifts exist for curves, they do not exist in general in dimension at least two.

11.1 Witt Vectors Let k be a field of positive characteristic p. Moreover, assume that k is perfect, e.g., algebraically closed or a finite field.

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Then, one can ask whether there exist rings of characteristic zero having k as residue field. It turns out that there exists a particularly nice ring W (k), the so-called Witt ring, or ring of Witt vectors, which has the following properties: 1. W (k) is a discrete valuation ring of characteristic zero. 2. The unique maximal ideal m of W (k) is generated by p and the residue field R/m is isomorphic to k. 3. W (k) is complete with respect to the m-adic topology. 4. The Frobenius map x → x p on k lifts to a ring homomorphism of W (k). 5. There exists an additive map V : W (k) → W (k), called Verschiebung (German for “shift”), which is zero on the residue field k and such that multiplication by p on W (k) factors as p = F ◦ V = V ◦ F. 6. Every complete discrete valuation ring with quotient field of characteristic zero and residue field k contains W (k) as subring. We remark that the last property characterizes W (k) up to isomorphism. To obtain W (k), one constructs successively rings Wn (k), which are local Artin rings of length n with residue field k. One has W1 (k) = k and surjective projection maps Wn+1 (k) → Wn (k). By definition, W (k) is the projective limit over the Wn (k) (cf. [128, Chap. II.6]). The main example to bear in mind is the following: Example 11.1. For the finite field F p we have Wn (F p ) ∼ = Z/pn Z and thus W (F p ) = lim Z/pn Z ←− is isomorphic to Z p , the ring of p-adic integers. The maximal ideal of W (F p ) is generated by p and W (F p ) is complete with respect to the p-adic topology. In this special case, F is the identity on W (F p ) and V is multiplication by p. Witt’s construction W (−) makes sense for every commutative ring R. However, already W (k) for a non-perfect field k is not Noetherian, and its maximal ideal is not generated by p. This is why we will assume k to be perfect for the rest of this section. We refer to [128, Chap. II.6] and [49] for more on Witt vectors.

11.2 Lifting over the Witt Ring Let X be a scheme of finite type over some perfect field k of positive characteristic p. Then, there are different notions of what it means to lift X to characteristic zero. To make it precise, let R be a ring of characteristic zero with maximal ideal m and residue field R/m ∼ = k. For example, we could have R = W (k) and m = (p). Definition 11.2. A lift (resp. formal lift) of X over R is a scheme (resp. formal scheme) X of finite type and flat over Spec R (resp. Spf R) with special fiber X.


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In case R = W (k), i.e., if X admits a (formal) lift over the Witt ring, many “characteristic-p-pathologies” cannot happen. We have already encountered the following results in Sect. 3: 1. If X is of dimension d ≤ p and lifts over W2 (k), then its Frölicher spectral sequence from Hodge to deRham cohomology degenerates at E1 by a result of Deligne and Illusie (see [32] and [57, Corollary 5.6]). 2. If X is of dimension d ≤ p and lifts over W2 (k), then ample line bundles satisfy Kodaira vanishing (see [32] and [57, Theorem 5.8]). 3. If X lifts over W (k), then crystalline cohomology coincides with deRham cohomology of X /W (k). Actually, the last property is the starting point of crystalline cohomology (see the discussion in Sect. 3.8). Example 11.3. Smooth curves and birationally ruled surfaces lift over the Witt ring by Grothendieck’s existence theorem [58, Theorem 5.19].

11.3 Lifting over More General Rings Let R be an integral ring with maximal ideal m, residue field R/m ∼ = k, and quotient field K of characteristic zero. Let X be a smooth projective variety over k, let X be a lift of X over Spec R, and denote its generic fiber by XK → Spec K. After choosing a DVR dominating (R, m) and after passing to the m-adic completion, we may assume that (R, m) is a local and m-adically complete DVR. By the universal property of the Witt ring, R contains W (k) and m lies above (p) ⊂ W (k). Thus, it makes sense to talk about the ramification index, usually denoted by e, of R over W (k). This ramification index is an absolute invariant of R. To give a flavor of the subtleties that occur when dealing with lifting problems, let us mention the following examples: 1. Abelian varieties admit formal lifts over the Witt ring by an unpublished result of Grothendieck [58, Theorem 5.23]. However, to obtain algebraic lifts, one would like to have an ample line bundle on a formal lift in order to apply Grothendiecks’ existence theorem (see [58, Theorem 4.10]). However, even if one succeeds in doing so, this is usually at the prize that this new formal lift (which then is algebraic) may exist over a ramified extension of the Witt ring only. For Abelian varieties, this was established by Mumford [100] and Norman and Oort [102]. 2. K3 surfaces have unobstructed deformations by Theorem 7.3 and, thus, admit formal lifts over the Witt ring. Deligne [30] has shown that one can lift with every K3 surface also an ample line bundle, which gives an algebraic lifting–again at the prize that this lift may exist over ramified extensions of the Witt ring only.

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3. By results of Lang [72], Illusie [55], Ekedahl and Shepherd–Barron [38], and [83], Enriques surfaces–even in characteristic 2–lift to characteristic zero. However, the Frölicher spectral sequence of a supersingular Enriques surface in characteristic 2 does not degenerate at E1 by [55, Proposition II.7.3.8]. Thus, these latter surfaces only lift over ramified extensions of the Witt ring, but not over the Witt ring itself. 4. Lang [74] gave examples of hyperelliptic surfaces that lift to a ramified extension of W (k) of ramification index e = 2, but whose Frölicher spectral sequences do not degenerate at E1 . Thus, these surfaces do not lift over W (k). Rather subtle examples of non-liftable smooth elliptic fibrations were given by Partsch [112]. However, even if X lifts “only” over a ramified extension of the Witt ring, this does imply something: flatness of X over Spec R implies that χ (O) of special and generic fiber coincide, and smoothness of X over Spec R implies that the -adic Betti numbers of special and generic fiber coincide. For surfaces, we have additional results from [66, Sect. 9]: Theorem 11.4 (Katsura–Ueno). Let S be a lift of the smooth projective surface S over Spec R with generic fiber SK . Then, bi (S) = bi (SK ) χ (OS ) = χ (OSK ) κ (S) = κ (SK )

c2 (S) = c2 (SK ) KS2 = KS2 K

Moreover, S is minimal if and only SK is minimal. If S is of general type then Pn (S) = Pn (SK ) for n ≥ 3 since these numbers depend only on χ and K 2 by Riemann–Roch and [37, Theorem II.1.7]. However, in general, pg (S) may differ from pg (SK ), as the examples in [124, 137] show. More precisely, Hodge invariants are semi-continuous, i.e., in general we have hi, j (S) ≥ hi, j (SK )

for all

i, j ≥ 0 .

In case of equality for all i, j, the Frölicher spectral sequence of S degenerates at E1 . Theorem 11.4 implies that from dimension two on, there exist smooth projective varieties that do not admit any sort of lifting, namely, Example 11.5. Let S be: 1. A minimal surface of general type with KS2 > 9χ (OS ), i.e., violating the Bogomolov–Miyaoka–Yau inequality (see Sect. 8.4), or 2. A quasi-elliptic surface with κ (S) = 1 and χ (OS ) < 0 (see Sect. 8.2) Then, S does not admit an algebraic lifting whatsoever, i.e., not even over a ramified extension of the Witt ring. The first example of such a smooth and projective variety that does not admit an algebraic lifting is due to Serre [127].


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For this and related questions (see also [58, Sect. 5F]). Moreover, we have the following highly non-explicit result: namely, “Murphy’s law” holds for moduli spaces of surfaces of general type with very ample canonical sheaves [145]. Thus, we can find any kind of obstructed lifting behavior already on surfaces, for example, Theorem 11.6 (Vakil). For every integer n > 0 and every prime p > 0, there exists a smooth and projective surface over F p that lifts over Wn (F p ) but not over Wn+1 (F p ).

11.4 Birational Nature One can also ask to what extent liftability is a birational invariant. If X and Y are smooth, proper, and birational varieties of dimension at most 2, then their lifting behavior is the same. However, in dimension ≥ 3, or when allowing canonical singularities in dimension two, this is no longer the case. We refer to [84] for details, some positive results, and (counter)examples.

11.5 Canonical Lifts For an ordinary Abelian variety or K3 surface, there even exists a distinguished formal lift over the Witt ring, the canonical lift, or Serre–Tate lift. Quite generally, ordinary means that Newton and Hodge polygons on crystalline cohomology coincide, and we note that this property is open in equi-characteristic families. For a g-dimensional Abelian variety A over a field k of characteristic p, being ordinary is equivalent to A[p](k) ∼ = (Z/pZ)g , which is the maximum possible (see also ( Definition 7.2). For a K3 surface S, being ordinary is equivalent to h(Br(S)) =1 (see Sect. 9.8). We refer to [90] for details on canonical lifts of ordinary Abelian varieties. For ordinary Abelian varieties, this canonical lift is characterized by the property that the Frobenius morphism lifts. For the general case, we refer to [31,67]. ( Finally, K3 surfaces with h(Br(S)) < ∞ still possess quasi-canonical lifts, which has been used to prove the Tate conjecture for them (see [104] and Sect. 9.10).

12 Rational Curves on K3 Surfaces In the final section we give an application of characteristic p and lifting techniques to a characteristic zero conjecture. Namely, we show how infinitely many rational curves on complex projective K3 surfaces of odd Picard rank can be established by reduction modulo p, then finding the desired rational curves over finite fields, and eventually lifting cycles of them to characteristic zero.

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12.1 Rational Curves Let C be a smooth projective curve of genus g over an algebraically closed field k. Then, the Riemann–Hurwitz formula implies that if there exists a nonconstant map P1 → C then g = 0, i.e., C ∼ = P1 . Similarly, one can ask about nonconstant maps 1 from P to higher-dimensional varieties, i.e., whether they exist, and if so, how many and whether they move in families. First of all, let us introduce the following notion: Definition 12.1. A rational curve on a variety X is a reduced and irreducible curve C ⊂ X whose normalization is isomorphic to P1 . Let us study rational curves on surfaces in detail: clearly, if S is a non-minimal surface, then every exceptional (−1)-curve is a rational curve. Also, since surfaces with κ (S) = −∞ are birationally ruled by Theorem 4.4, they contain moving families of rational curves. On the other extreme, Serge Lang [68] conjectured that complex surfaces of general type contain only finitely many rational curves. However, we note that uniruled surfaces of general type in positive characteristic (see Sect. 9.6) contain infinitely many rational curves.

12.2 K3 Surfaces In between these extremes lie surfaces of Kodaira dimension zero. If S is an Abelian variety, then every map P1 → S factors over the Albanese variety of P1 , which is a point. Thus, Abelian varieties contain no rational curves at all. On the other hand, there is the well-known Conjecture 12.2 (Bogomolov). A projective K3 surface contains infinitely many rational curves. In characteristic zero, rational curves cannot move inside their linear systems, for otherwise the K3 surface in question would have to be uniruled, which is impossible. But even in positive characteristic, where uniruled K3 surfaces do exist, they are rather special, namely, supersingular by Theorem 9.8. The first important step towards Bogomolov’s conjecture is to establish the existence of at least one rational curve, and we refer to [95] for the following result: Theorem 12.3 (Bogomolov–Mumford). Let S be a projective K3 surface over an algebraically closed field, and let L be a nontrivial and effective invertible sheaf. Then, there exists a divisor ∑i niCi inside |L|, where ni ≥ 1 and the Ci are rational curves on S. For polarized K3 surfaces (S, H), say, of degree H 2 = 2d, there exists a moduli space M2d , which is smooth and irreducible over the complex numbers (see, e.g.,


283

[7, Chap. VIII]). Using degenerations of K3 surfaces to unions of rational surfaces, Chen [25] showed, among other things, Theorem 12.4 (Chen). A very general complex projective K3 surface in M2d contains infinitely many rational curves. Here, very general is meant in the sense that there exists a countable union of analytic divisors inside M2d , outside of which the statement is true. Although this result strongly supports Conjecture 12.2, it does not give even a single example of a K3 surface containing infinitely many rational curves!

12.3 Explicit Results It is shown in [12, Sect. 4] or [14, Example 5] that complex projective Kummer K3 surfaces contain infinitely many rational curves. In particular, since every complex K3 surface of Picard rank ρ ≥ 19 is rationally dominated by a Kummer surface, these surfaces contain infinitely many rational curves. In [11], elliptic K3 surfaces S → P1 are studied. There, the authors define an ntmultisection to be a multisection M of the fibration such that for a general point b ∈ P1 there exist two points in the fiber pb , pb ∈ Sb ∩ M such that the divisor pb − pb , considered as a point of the Jacobian of Sb , is non-torsion. Establishing infinitely many nt-multisections that are rational curves, we find infinitely many rational curves on elliptic K3 surfaces of Picard rank ρ ≤ 19 (see [11, Corollary 3.28]). We note that K3 surfaces of Picard rank ρ ≥ 5 are automatically elliptic: namely, in this case, by the theory of integral quadratic forms, there exists an isotropic vector in Pic(S), which gives rise to an elliptic fibration. Combining these results, we obtain the following Theorem 12.5 (Bogomolov–Tschinkel). Let S be a complex projective K3 surface that: 1. Carries an elliptic fibration, or 2. Is a Kummer surface, or 3. Has Picard rank ρ ≥ 5 Then, S contains infinitely many rational curves. Moreover, in case the effective cone of a K3 surface is not finitely generated, we find infinitely many rational curves using Theorem 12.3. Also, if the automorphism group is infinite, there are infinitely many rational curves. Combining these observations with the previous results, one can show that there are infinitely many rational curves for K3 surfaces of Picard rank ρ ≥ 4, possibly with the exception of two Picard lattices of rank 4. We refer to [11, Sect. 4] and [14, Sect. 2] for details, as well as to [11, Example 4.8] for an example of a K3 surface with ρ = 4, where infinity of rational curves is currently still unknown.

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On the other hand, a very general K3 surface in M2d has Picard rank ρ = 1, does not carry an elliptic fibration, and has a finite automorphism group. Thus, these are hard to come by, as they do not possess much geometric structure to work with.

12.4 Reduction Modulo p In [14], Bogomolov, Hassett, and Tschinkel gave an approach to the case of Picard rank ρ = 1, which uses reduction modulo finite characteristic. First, using degeneration techniques, they reduced to the number field case: Proposition 12.6 (Bogomolov–Hassett–Tschinkel). Bogomolov’s conjecture 12.2 holds for complex projective K3 surfaces if and only if it holds for K3 surfaces that are defined over number fields. Now, let S be a K3 surface over some number field K. Replacing K by a finite extension, we may assume that all divisor classes of SC are already defined over K. Embedding S into some projective space PNK and taking the closure of its image inside PNOK , we get a model of S over OK . After localizing at a finite set of places P depending on S and this embedding, we obtain a smooth projective model S → Spec OK,P , i.e., a smooth projective scheme over OK,P with generic fiber SK ∼ = S. In particular, for every prime ideal p of OK,P , the reduction Sp of S modulo p is a K3 surface over the finite field OK,P /p. The crucial observations and strategy of [14] are as follows: let (S, H) be a polarized K3 surface over K with geometric Picard rank ρ = 1, or, more generally, ρ odd. If char(OK /p) ≥ 5, then the Tate conjecture holds for Sp by Theorem 9.17. In particular, if we denote by Sp the base change of Sp to the algebraic closure of OK /p, then the Picard rank of Sp is even by Theorem 9.18. On the other hand, the specialization map sp

p Pic(S) ∼ = Pic(S) −→ Pic(Sp )

is injective. Since ρ is odd, there exists for every prime p not lying over 2 or 3 an invertible sheaf Lp on Sp that does not lift to S. We may assume Lp to be effective, and then, by Theorem 12.3, we find an effective divisor in |Lp | that is a sum of rational curves. Since Lp does not lift, there is at least one rational curve Cp in this sum that does not lift to S either. However, if Np is a sufficiently large integer, then |Np H −Cp | is effective, and by Theorem 12.3, there exist rational curves Rp,i on Sp and positive integers ni such that Cp + ∑ ni Rp,i ∈ |Np H| . i

(4)


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This sum of rational curves can be represented by a stable map of genus zero and, so, defines a point of the moduli space of stable maps M0 (Sp , Np H). Next, we want this stable map to be rigid, i.e., the stable map allows at most infinitesimal deformations, i.e., the moduli space is zero-dimensional at this point. The first problem is that rational curves can move on K3 surfaces in positive characteristic (in which case we might not be able to find a rigid representation). But then, the K3 surface is uniruled and, in particular, Artin-supersingular (see Sect. 9). By results of Bogomolov and Zarhin [13] (independently also obtained by Joshi and Rajan, but unpublished), we can always find infinitely many places p such that Sp is not Artin-supersingular, which is sufficient for our application. Now, take of these infinitely many primes of non-supersingular reduction and suppose (we comment on that below) that we can find a rigid stable map representing Eq. (4). We denote by k the algebraic closure of the finite field OK /p, let W (k) be the Witt ring of k, and base change the family S → Spec OK,P to W (k). Then, dimension estimates of the relative formal moduli space M0 (S, Np H) → Spf W (k) imply that our stable map to Sp extends to a stable map to the family S (here, rigidity is crucial). Thus, the stable map lifts over a possibly ramified extension of W (k), and in particular, there exists a rational curve on SC , whose reduction modulo p contains Cp . Thus, for infinitely many p we get rational curves on SC and eventually obtain the following result [14]: Theorem 12.7 (Bogomolov–Hassett–Tschinkel). Let S be a complex projective K3 surface with Picard group Pic(S) = Z · H such that H 2 = 2. Then, S contains infinitely many rational curves. The main issue is the representation of Eq. (4) by a rigid stable map, for otherwise it is not clear whether one can lift this sum of rational curves to characteristic zero. For degree 2 and ρ = 1, such a rigid representation exists by exploiting the involution on K3 surfaces of degree 2 (see [14]). In general, this difficulty was overcome in [77] by introducing rigidifiers: by definition, these are ample and irreducible rational curves with at worst nodal singularities. Then, every sum of rational curves can be represented by a rigid stable map after adding sufficiently many rigidifiers to them. Unfortunately, the surface Sp may not contain rigidifiers. However, surfaces containing rigidifiers are dense in the moduli space of polarized K3 surfaces. Using deformation techniques and rigidifiers, we obtained in [77] Theorem 12.8 (Li–Liedtke). Let S be a complex projective K3 surface, whose Picard rank is odd. Then, S contains infinitely many rational curves. More generally, the method of proof works whenever a K3 surface S is defined over some field K, and we can find a DVR R with quotient field K, as well as infinitely many primes p of R such that the geometric Picard rank of the reduction Sp is strictly larger than that of S. For example, if S is a complex projective K3 surface that cannot be defined over a number field, then S can be realized as generic fiber of a non-isotrivial family S → B over some positive dimensional base of characteristic zero. Using results on the jumping of Picard ranks of K3 surfaces in families from [18] or [105], we obtain

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Theorem 12.9. Let S be a complex projective K3 surface that cannot be defined over a number field. Then, S contains infinitely many rational curves. In view of these results and Theorem 12.5, it remains to deal with K3 surfaces of Picard rank ρ = 2 and ρ = 4 that are defined over number fields, in order to establish Conjecture 12.2 for all complex projective K3 surfaces. To apply the techniques of [14, 77], we need jumping of Picard ranks for infinitely places of non-supersingular reduction. For example, such jumping results for certain classes of K3 surfaces with ρ = 2 and ρ = 4 over number fields were established in [23]. We end by giving a heuristic reason why we always expect to find infinitely many places with non-supersingular reduction and jumping Picard rank (as in the case of odd rank), which would imply Conjecture 12.2. However, in view of the results in [89] and [23, Theorem 1], the situation may be more subtle than expected. In any case, here is our heuristic: The universal polarized K3 surface has Picard rank ρ = 1. All its (nonsupersingular) specializations to surfaces over finite fields have a larger geometric Picard rank, and the extra invertible sheaves extend (at least, formally) along divisors inside the moduli space. Also, these invertible sheaves must have unbounded intersection number with the polarization (otherwise some of them would lift to the universal K3 surface, which was excluded). Thus, the moduli space of polarized K3 surfaces over the integers is “flooded” by infinitely many divisors on which Picard ranks jump. It is likely that given a K3 surface over a number field, infinitely many of its non-supersingular reductions hit these divisors, establishing the desired jumping behavior of Picard ranks. Acknowledgements These lecture notes grew out of a lecture series given at Sogang University, Seoul, in October 19–22, 2009. I thank Yongnam Lee for the invitation to Sogang University and hospitality. It was a pleasure visiting him and giving these lectures. Also, I thank Fabrizio Catanese, Hisanori Ohashi, Holger Partsch, Sönke Rollenske, Nguyen Le Dang Thi, Yuri Tschinkel, Tong Zhang, and the referee for suggestions, remarks, and pointing out mistakes in earlier versions. I thank the referee especially for clarifications and providing me with more references. I wrote up a first version of these notes at Stanford University and I thank the department for hospitality. I gratefully acknowledge funding from DFG under research grants LI 1906/1-1 and LI 1906/1-2.

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Arithmetic of Del Pezzo surfaces Anthony Várilly-Alvarado

Mathematics Subject Classification codes (2010): 14 G05, 14 J26, 11 G35.

Introduction These notes were written to accompany a minicourse delivered during the conference “Arithmetic of Surfaces,” which was held at the Lorentz Center in Leiden, during October, 2010. The minicourse was aimed at graduate students and consisted of three lectures, each corresponding to one section of these notes. These notes are a faithful transcript of the material we went over, with some added details and references. I have resisted the temptation to add any more material. In particular, many aspects of the arithmetic of del Pezzo surfaces are not treated here (most notably perhaps the uniqueness of Brauer-Manin obstructions). In preparing this document, I used the “background material” chapters of my doctoral dissertation [51] quite freely (especially for the first and third sections); these chapters were not meant for publication.

1 Geometry of Del Pezzo Surfaces 1.1 Guiding Questions in Diophantine Geometry Let k be a global field, i.e., a finite extension of Q or F p (t) for some prime p, let Ak denote its ring of adèles, and let X be a smooth projective geometrically integral A. Várilly-Alvarado () Department of Mathematics MS-136, Rice University, Houston, TX 77005, USA e-mail: [email protected] F. Bogomolov et al. (eds.), Birational Geometry, Rational Curves, and Arithmetic, DOI 10.1007/978-1-4614-6482-2__12, © Springer Science+Business Media New York 2013

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variety over k. Generally speaking, diophantine geometers seek to “describe” the set X(k) of k-rational points of X. For example, we are interested in determining whether X(k) is empty or not. If X(k) = 0, / then we may further want to know something about the qualitative nature of X(k): is it dense for the Zariski topology of X? Is the image of the natural embedding X(k) → X(Ak ) dense for the adèlic topology? If not, can we account for the paucity of k-rational points? We may also pursue a more quantitative study of X(k). For instance, we might try to prove asymptotic formulas for the number of k-points of bounded height on some special Zariski-open subset of X. On the other hand, if X(k) = 0, / then we might try to account for the absence of k-rational points. For example, the existence of embeddings X(k) → X(kv ) for every completion kv of k shows that a necessary condition for X to have a k-rational point is X(kv ) = 0/ for all completions kv of k.

(1)

We say that X is locally soluble whenever Eq. (1) is satisfied. Whenever checking (1) suffices to show that X(k) = 0, / we say that X satisfies the Hasse principle.1 Many classes of varieties, such as quadrics, satisfy the Hasse principle. Perhaps the first known counterexample to the Hasse principle is due to Lind and Reichardt, who show that the genus 1 plane curve over Q with affine model given by 2y2 = x4 − 17 is locally soluble, but lacks Q-rational points; see [32,39]. Failures of the Hasse principle are often explained by the presence of cohomologically flavored obstructions, such as the Brauer-Manin obstruction. These kinds of obstructions may also produce examples of varieties X as above, with X(k) = 0, / for which the embedding X(k) → X(Ak ) is not dense. Notation. The following notation will remain in force throughout. First, k denotes a field, k¯ is a fixed algebraic closure of k, and ks ⊆ k¯ is the separable closure of k ¯ If k is a global field then we write Ak for the adèle ring of k, Ωk for the set in k. of places of k, and kv for the completion of k at v ∈ Ωk . By a k-variety X we mean a separated scheme of finite type over k (we will omit the reference to k when it can cause no confusion). If X and Y are S-schemes then we write XY := X ×S Y . However, if Y = Spec A then we write XA instead of XSpec A . A k-variety X is said to be nice if it is smooth, projective and geometrically integral. If T is a k-scheme, then we write X(T ) for the set of T -valued points of X. If, however, T = SpecA is affine, then we write X(A) instead of X(Spec A).

authors refer only to the Hasse principle in the context of a class S of varieties and say that S satisfies the Hasse principle if for every X ∈ S , the implication X(kv ) = 0/ for all v ∈ Ωk =⇒ 1 Many

X(k) = 0/ holds.

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1.2 Birational Invariance and a Theorem of Iskovskikh Let X be a nice k-variety. Many properties of X(k), such as “being nonempty,” depend only on X up to birational equivalence as follows: Existence of a Smooth k-point. The Lang-Nishimura lemma guarantees that if X X is a birational map between proper integral k-varieties then X has a smooth k-point if and only if X has a smooth k-point; see [31,38]. We give a short proof here due to Kollár and Szabó [40, Proposition A.6]. Lemma 1.1 (Lang-Nishimura). Let k be a field, and let f : X X be a rational map of k-schemes. Assume that X has a smooth k-point and that X is proper. Then X(k) = 0. / Proof. We use induction on n := dim X . The case n = 0 is clear. Let x be a smooth kpoint of X . Consider the blow-up Blx X of X at x with exceptional divisor E ∼ = Pn−1 k , as well as the composition Blx X → X X. By the valuative criterion of properness, this composition is defined outside a set of codimension at least 2, so the restricting to E we obtain a rational map E X. Now X(k) = 0/ by induction. (Note that induction is only necessary in the case where k is finite.) Zariski Density of k-rational Points. If X, X are two nice birationally equivalent k-varieties, then X(k) is Zariski dense in X if and only if X (k) is Zariski dense in X : the key point to keep in mind is that any two nonempty open sets in the Zariski topology have nonempty intersection. Weak Approximation. Let X be a geometrically integral variety over a global field k. We say that X satisfies weak approximation if the diagonal embedding X(k) →

∏ X(kv )

v∈Ωk

is dense for the product of the v-adic topologies. If X is a nice k-variety then X(Ak ) = ∏v X(kv ), the latter considered with the product topology of the v-adic topologies; see [45, pp. 98–99]. In this case X satisfies weak approximation if the image of the natural map X(k) → X(Ak ) is dense for the adèlic topology. Note also that if X does not satisfy the Hasse principle, then automatically X does not satisfy weak approximation. Lemma 1.2. If X and X are smooth, geometrically integral, and birationally equivalent varieties over a global field k, then X satisfies weak approximation if and only if X satisfies weak approximation.

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Proof (Sketch of proof). It is enough to prove the lemma in the case X = X \ W , where W is a proper closed subvariety of X, i.e., X is a dense open subset of X. Then, if X satisfies weak approximation, then clearly so does X . On the other hand, by the v-adic implicit function theorem, the set X (kv ) is dense in X(kv ); see [9, Lemma 3.1.2]. Suppose that X satisfies weak approximation and let (xv ) ∈ ∏v X(kv ) be given. Choose (yv ) ∈ ∏v X (kv ) ⊆ ∏v X(kv ) as close as desired to (xv ) for the product topology. By hypothesis, there is a rational point y ∈ X (k) whose image in ∏v X (kv ) is arbitrarily close to (yv ); then y is also close to (xv ), and X satisfies weak approximation. Hasse Principle. If k is a global field, and if X and X are two nice birationally equivalent k-varieties, then X satisfies the Hasse principle if and only if X satisfies the Hasse principle: this follows from two applications of the Lang-Nishimura lemma. It is thus natural to ask the qualitative questions of Sect. 1.1 in the context of a fixed birational class for X. In particular, we will fix the dimension of X. We will consider these questions only for nice surfaces. In addition, we require that X be geometrically rational, i.e., X ×k k¯ is birational to P2k¯ . The reason for this last restriction is the existence of the following beautiful classification theorem due to Iskovskikh, which describes the possible birational classes for X. Theorem 1.3 ([26, Theorem 1]). Let k be a field, and let X be a smooth projective geometrically rational surface over k. Then X is k-birational to either a del Pezzo surface of degree 1 ≤ d ≤ 9 or a rational conic bundle. Remark 1.4. It is possible for X as in Theorem 1.3 to be k-birational to both a del Pezzo surface and a rational conic bundle. More precisely, a rational conic bundle is birational to a minimal del Pezzo surface if and only if d = 1, 2, or 4, and there are two distinct representations of X as a rational conic bundle; see [26, Theorems 4 and 5].

1.3 Del Pezzo Surfaces In light of Theorem 1.3, we take a moment to review the definition and some basic properties of del Pezzo surfaces. In this section, we work over an arbitrary field k. We begin by recalling some basic facts and setting some notation. If X is a nice surface, then there is an intersection pairing on the Picard group ( · , · )X : Pic X × Pic X → Z; see [27, Appendix B]. We omit the subscript on the pairing if no confusion can arise. For such an X, we identify Pic(X) with the Weil divisor class group (see [24, Corollary II.6.16]); in particular, we will use additive notation for the group law on Pic X. If X is a nice k-variety, then we write KX for the class of the canonical sheaf ωX in Pic X; the anticanonical sheaf of X is ωX⊗−1 . An exceptional curve on a smooth projective k-surface X is an irreducible curve


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C ⊆ Xk¯ such that (C,C) = (KX ,C) = −1. By the adjunction formula (see [41, IV.8, Proposition 5]), an exceptional curve on X has arithmetic genus 0, and hence it is ¯ k-isomorphic to P1k¯ . Definition 1.5. A del Pezzo surface X is a nice k-surface with ample anticanonical sheaf. The degree of X is the intersection number d := (KX , KX ). If X is a del Pezzo surface then the Riemann-Roch theorem for surfaces and Castelnuovo’s rationality criterion show that X is geometrically rational. Moreover, Xks is isomorphic to either P1ks × P1ks (in which case d = 8) or the blowup of P2ks at r ≤ 8 distinct closed points (in which case d = 9 − r); this is the content of Theorem 1.6 below. In the latter case, the points must be in general position: this condition is equivalent to ampleness of the anticanonical class on the blown-up surface; see [19, Théorème 1, p. 27].

1.4 Del Pezzo Surfaces are Separably Split Throughout this section, k denotes a separably closed field and k¯ a fixed algebraic closure of k. A collection of closed points in P2 (k) is said to be in general position if no 3 points lie on a line, no 6 points lie on a conic, and no 8 points lie on a singular cubic, with one of the points at the singularity. Our goal is to prove the following strengthening of [35, Theorem 24.4]. Theorem 1.6. Let X be a del Pezzo surface of degree d over k. Then either X is isomorphic to the blowup of P2k at 9 − d points in general position in P2 (k), or d = 8 and X is isomorphic to P1k × P1k . We need two results of Coombes as follows: Proposition 1.7 ([15, Proposition 5]). Let f : X → Y be a birational morphism of smooth projective surfaces over k. Then f factors as X = X0 → X1 → · · · → Xr = Y, where each map Xi → Xi+1 is a blowup at a closed k-point of Xi+1 . ¯ The main step in The above proposition is well known if we replace k with k. the proof of Proposition 1.7 is to show that the blowup at a closed point whose residue field is a nontrivial purely inseparable extension of k cannot give rise to a smooth surface. Using Iskovskikh’s classification theorem (Theorem 1.3), Coombes deduces the following proposition. Proposition 1.8 ([15, Proposition 7]). The minimal smooth projective rational surfaces over k are P2k and the Hirzebruch surfaces Fn := P OP1 ⊕ OP1 (n) , where k k either n = 0 or n ≥ 2. Finally, we need the following lemma.

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Lemma 1.9 ([35, Theorem 24.3(ii)]). Let X be a del Pezzo surface over an algebraically closed field. Then every irreducible curve with negative self-intersection is exceptional. Proof. Let C ⊂ X be an irreducible curve with (C,C) < 0, and let pa (C) denote its arithmetic genus. Since −K X is ample, we have (C, −K X ) > 0. On the other hand, by the adjunction formula, we know that 2pa (C) − 2 = (C,C) − (C, −K X ). Since C is irreducible, we have pa (C) ≥ 0. All this forces pa (C) = 0 and (C,C) = (C, KX ) = −1, and thus C is exceptional. Proof (of Theorem 1.6). Let f : X → Y be a birational k-morphism with Y minimal, and write (2) X = X0 → X1 → · · · → Xr = Y for a factorization of f as in Proposition 1.7. By Proposition 1.8 we need only consider the following cases: 1. Y = P2k . We claim that no point that is blown up in one step of the factorization Eq. (2) may lie on the exceptional divisor of a previous blowup: otherwise Xk¯ would contain a curve with self-intersection less than −1, contradicting Lemma 1.9. Hence X is the blowup of P2k at r distinct closed k-points. We conclude that d = KX2 = 9 − r, as claimed; note that d = (KX , KX ) ≥ 1 since −KX is ample, and so 0 ≤ r ≤ 8. Suppose that three of these points lie on a line L. Let fk¯−1 Lk¯ denote the strict transform of Lk¯ for the base-extension fk¯ : Xk¯ → Yk¯ . Then ( fk¯−1 Lk¯ , fk¯−1 Lk¯ ) < −1, but this is impossible by Lemma 1.9. Similarly, if 6 of the blown-up points lie on a conic Q or if 8 points lie on a singular cubic C with one of the points at the singularity, then ( fk¯−1 Qk¯ , fk¯−1 Qk¯ ) < −1 or ( fk¯−1Ck¯ , fk¯−1Ck¯ ) < −1, respectively, which is not possible. Hence the blown-up points are in general position. 2. Y = P1k × P1k . If X = Y then X is a del Pezzo surface of degree 8. Otherwise, we may contract the two nonintersecting (−1)-curves of Xr−1 and obtain a birational morphism φ : Xr−1 → P2k . We may use the map φ to construct a new birational morphism X → P2k , given by φ

→ P2k , X = X0 → X1 → · · · → Xr−1 − and thus we may reduce this case to the previous case. 3. Y = Fn , n ≥ 2. There is a curve C ⊆ (Fn )k¯ whose divisor class satisfies (C,C) < −1. Let fk¯−1 (C) denote the strict transform of C in Xk¯ for the base-extension fk¯ : Xk¯ → (Fn )k¯ . Then ( fk¯−1C, fk¯−1C) < −1, but this is impossible by Lemma 1.9.


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1.5 Further Properties of Del Pezzo Surfaces The basic references on the subject are [35], [19] and [28, III.3].

The Picard Group Let X be a del Pezzo surface over a field k of degree d. Recall that an exceptional curve on X is an irreducible curve C on Xk¯ such that (C,C) = (C, KX ) = −1. Theorem 1.6 shows that exceptional curves on X are already defined over ks . We have seen that if Xks P1ks × P1ks then Xks is isomorphic to a blowup of P2ks at r := 9 − d closed points {P1 . . . , Pr } in general position. It follows that the group Pic Xks is isomorphic to Z10−d (see [24, Proposition V.3.2]); if d ≤ 7 then it is generated by the classes of exceptional curves. Let ei be the class of an exceptional curve corresponding to Pi under the blow-up map, and let be the class of the pullback of a line in P2ks not passing through any of the Pi . Then {e1 , . . . , er , } is a basis for Pic Xks . Note that (ei , e j ) = −δi j ,

(ei , ) = 0,

(, ) = 1,

where δi j is the usual Kronecker delta function. With respect to this basis, the anticanonical class is given by −KX = 3 − ∑ ei . This basis also allows us to interpret the exceptional curves in terms of strict transforms of our blowup: for example, if C = − e1 − e2 as classes in Pic Xks , then C is the strict transform of the line in P2 through the points P1 and P2 . The number of exceptional curves on X is finite and is computed as follows: if C = a − ∑ri=1 bi ei is an exceptional curve, then r

a2 − ∑ b2i = −1. i=1

r

3a − ∑ bi = 1. i=1

These equations are easily solved (keep in mind that a and bi are integers). The number of exceptional curves on X as r (and hence d) varies is shown in Table 1.

Root Systems Suppose that r = 9 − d ≥ 3. The orthogonal complement KX⊥ of KX in Pic Xks ⊗ R, equipped with the negative of the intersection form on X, is a Euclidean space.

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A. Várilly-Alvarado Table 1 Number of exceptional curves on X d(X) # of exceptional curves

7 3

6 6

5 10

4 16

3 27

2 56

1 240

Table 2 Root systems on Pic Xks r Rr

3 A1 × A2

4 A4

5 D5

6 E6

7 E7

8 E8

Its group of orthogonal transformations O(KX⊥ ), being an intersection of a compact group and a discrete group, is finite. The set of vectors Rr := {v ∈ Pic Xks : (v, KX ) = 0, (v, v) = −2} is a root system of rank r. Using standard facts about root systems, it is not too hard to identify Rr as r varies; see [35, Theorem 25.4]. An important fact for our purposes is that the group of automorphisms of Pic Xks that preserve the intersection form and KX coincides with the Weyl group W (Rr ) of the root system Rr [35, Theorem 23.9], i.e., O(KX )⊥ = W (Rr ) (Table 2).

Galois Action on the Picard Group The Galois group Gal(ks /k) acts on Pic Xks as follows. For σ ∈ Gal(ks /k), let σ˜ : Spec ks → Spec ks be the corresponding morphism. Then idX ×σ˜ : Xks → Xks induces an automorphism (idX ×σ˜ )∗ of Pic Xks . This gives a group homomorphism Gal(ks /k) → Aut(Pic Xks )

σ → (idX ×σ˜ )∗ .

The action of Gal(ks /k) on Pic(Xks ) fixes the canonical class KX and preserves the intersection pairing; in particular, the action of Gal(ks /k) takes exceptional curves to exceptional curves (see [35, Theorem 23.8]). By our discussion in Sect. 1.5, it follows that Gal(ks /k) acts as a subgroup of W (Rr ) on Pic Xks . Let K be the smallest extension of k in ks over which all exceptional curves of X are defined. We say that K is the splitting field of X. The natural action of Gal(ks /k) on Pic Xks ∼ = Pic XK factors through the quotient Gal(K/k), giving a homomorphism

φX : Gal(K/k) → Aut(Pic XK ).

(3)

If we have equations with coefficients in K for an exceptional curve C of X, then an element σ ∈ Gal(K/k) acts on C by applying σ to each coefficient. The map (3) conjecturally determines much of the arithmetic of X: there is a conjecture of Colliot-Thélène and Sansuc (first asked as a question in [12], based on


301

evidence published in [9, 13]) that says that the Brauer-Manin obstruction explains all violations of the Hasse principle for del Pezzo surfaces over a number field k. One can use φX to compute the Brauer-Manin obstruction, so if the conjecture is true, then one can decide whether X has a k-point.

Anticanonical Models For any scheme X and line sheaf L on X, we may construct the graded ring R(X, L ) :=

H 0 (X, L ⊗m ).

m≥0

When L = ωX⊗−1 , we call R(X, ωX⊗−1 ) the anticanonical ring of X. If X is a del Pezzo surface then X is isomorphic to the scheme ProjR(X, ωX⊗−1 ), because ωX⊗−1 is ample. This scheme is known as the anticanonical model of the del Pezzo surface. The construction of anticanonical models is reminiscent of the procedure that yields a Weierstrass model of an elliptic curve. In fact, we can use the RiemannRoch theorem for surfaces (and Kodaira vanishing—which is valid even in positive characteristic for rational surfaces) to prove the following dimension formula for a del Pezzo surface X over k of degree d: m(m + 1) d + 1; h0 X, −mKX = 2 see [28, Corollary III.3.2.5] or [17]. For example, if X has degree 1, then the anticanonical model for X is a smooth sextic hypersurface in Pk (1, 1, 2, 3), and we may compute such a model, up to isomorphism, as follows: 1. Choose a basis {x, y} for the 2-dimensional k-vector space H0 X, −KX . 2. The elements x2 , xy, y2of H0 X, −2KX are linearly independent. However, h0 X, −2KX = 4; choose an element z to get a basis {x2 , xy, y2 , z} for this k-vector space. 3. The elements x3 , x2 y, xy2 , y3 , xz, yz of H0 X, −3KX are linearly independent, but h0 X, −3KX = 7. Choose an element w to get a basis {x3 , x2 y, xy2 , y3 , xz, yz, w} for this k-vector space. 4. The vector space H0 X, −6KX is 22-dimensional, so the 23 elements {x6 , x5 y, x4 y2 , x3 y3 , x2 y4 , xy5 , y6 , x4 z, x3 yz, x2 y2 z, xy3 z, y4 z, x2 z2 , xyz2 , y2 z2 , z3 , x3 w, x2 yw, xy2 w, y3 w, xzw, yzw, w2 } must be k-linearly dependent. Let f (x, y, z, w) = 0 be a linear dependence relation among these elements. Then an anticanonical model of X is Projk[x, y, z, w]/( f ), where x, y, z, w are variables with weights 1, 1, 2, and 3, respectively. This way X may be described as the (smooth) sextic hypersurface V ( f ) in Pk (1, 1, 2, 3).

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For more details on this construction, see [17, pp.1199–1201]. Remark 1.10. If k is a field of characteristic not equal to 2 or 3, then in step (4) above we may complete the square with respect to the variable w and the cube with respect to the variable z to obtain an equation f (x, y, z, w) = 0 involving only the monomials {x6 , x5 y, x4 y2 , x3 y3 , x2 y4 , xy5 , y6 , x4 z, x3 yz, x2 y2 z, xy3 z, y4 z, z3 , w2 }. Moreover, we may also rescale the variables so that the coefficients of w2 and z3 are ±1. Remark 1.11. If X has degree d ≥ 3, then the anticanonical model recovers the usual description of X as a smooth degree d surface in Pdk . In particular, when d = 3, we get a smooth cubic surface in P3k . If X has degree 2 then the anticanonical model is a smooth quartic hypersurface in the weighted projective space Pk (1, 1, 1, 2); such a surface can then be thought of as a double cover of a P2k ramified along a quartic curve. Remark 1.12. If we write a del Pezzo surface X of degree 1 over a field k as the smooth sextic hypersurface V ( f (x, y, z, w)) in Pk (1, 1, 2, 3), then {x, y} is a basis for H0 X, −KX . In particular, |−KX | has a unique base point: [0 : 0 : 1 : 1].

2 Arithmetic of Del Pezzo Surfaces of Degree at Least 5 The goal for this section is to prove the following theorem. Theorem 2.1. Let X be a del Pezzo surface of degree d ≥ 5. If X(k) = 0, / then X is k-birational to P2k . This hypothesis is automatically satisfied if d = 5 or 7. In particular, if k is a global field, then X satisfies weak approximation. In addition, X(k) = 0/ provided X(kv ) = 0/ for all v ∈ Ωk (i.e., del Pezzo surfaces of degree at least 5 satisfy the Hasse principle). We will prove this theorem by a case-by-case analysis on the degree of X, making attributions as we go along. Our exposition is influenced by [8, Sect. 4].

2.1 Case 1: d(X) = 9 By Theorem 1.6, Xk¯ is isomorphic to P2k¯ , i.e., Xk¯ is a form of the projective plane. It is a classical theorem of Châtelet that such a surface is k-isomorphic to P2k if and only if X(k) = 0. / We will prove this result more generally for Severi-Brauer varieties.


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Definition 2.2. A Severi-Brauer variety X is a projective scheme over a field k that becomes isomorphic to some n-dimensional projective space upon passage to an algebraic closure k¯ of k, i.e., Xk¯ ∼ = Pnk¯ for some n. Theorem 2.3 (Châtelet). Let X be a Severi-Brauer variety of dimension n over a field k. The following are equivalent: 1. X is k-isomorphic to Pnk . 2. X(k) = 0. / Proof. The proof we give here is due to Endre Szabó; we follow the exposition in [21]. The implication (1) =⇒ (2) is clear. We claim that if X(k) = 0/ then X contains a twisted linear subvariety D of codimension 1. Let π : Y → X be the blowup of X at a k-rational point P. The variety Yk¯ is isomorphic to the blowup of Pnk¯ at a closed point and we can think of it as a subvariety of Pnk¯ × Pn−1 . Let k¯ n−1 ψk¯ : Yk¯ → Pk¯ be the projection onto the second factor. Choose a hyperplane L on the exceptional divisor E ∼ of the blowup. The subvariety Dk¯ := = Pn−1 k −1 πk¯ ψk¯ (ψk¯ (Lk¯ )) of Xk¯ is a hyperplane in Pnk¯ . ¯ The linear system Choose an ample divisor A of X, of degree d over k. |π ∗ A − dE| defines a k-rational map φ : Y PNk . Since (π ∗ A − dE)k¯ has degree 0 on the fibers of ψk¯ and degree d on Ek¯ , the map φk¯ factors as ψk¯ followed by the d-uple embedding. In particular, φ is defined everywhere. The subvariety D := π φ −1 (φ (L)) of X is defined over k and becomes Dk¯ after base extension ¯ to k. ¯ D becomes a The linear system |D| gives a rational map φD : X Pnk 1 . Over k, hyperplane and thus (φD )k¯ is an isomorphism with Pnk¯ . This shows that n1 = n and φ is an everywhere-defined isomorphism. Over a global field, forms of Pnk satisfy the Hasse principle. This also follows from work of Châtelet [6]. Theorem 2.4 (Châtelet). Severi-Brauer varieties over a global field satisfy the Hasse principle. Proof (Idea of the proof). Let SBn (k) be the pointed set of isomorphism classes of Severi-Brauer varieties of dimension n over k (the base point is the class of Pnk ). There is a base-point preserving bijection SBn(k) ←→ H1 Gal(ks /k), PGLn+1 (ks ) ; see [21, Theorem 5.2.1]. If K is a finite Galois extension of k, then there is a basepoint preserving bijection between k-isomorphism classes of central simple algebras of degree n split by K and H1 Gal(K/k), PGLn+1 (K) ; see [21, Theorem 2.4.3]. Thus, to a Severi-Brauer variety X over k, we may naturally associate a class of Br(k) in a unique way, and this class is trivial if and only if X ∼ = Pnk , i.e., if and only if X(k) = 0, / by Theorem 2.3. The theorem now follows from the fact that the map

304


Br(k) →

Br(kv ),

A → (A ⊗k kv )

v∈Ωk

is an injection [37, Theorem 8.1.17].

2.2 Case 2: d(X) = 8 By Theorem 1.6, Xks is isomorphic to either a blowup of P2ks at a point or to P1ks × P1ks . We deal with these cases separately. In the former case, there is exactly one exceptional curve on X, and Gal(ks /k) must fix it (see 1.5). Blowing down this curve we obtain a del Pezzo surface of degree 9 with a k-point, so by Theorem 2.3 we conclude that X is k-birational to P2k . In particular, X(k) = 0, / so these surfaces trivially satisfy the Hasse principle. Suppose now that Xks is isomorphic to P1ks × P1ks . Then Pic Xks ∼ = ZL1 ⊕ ZL2 , where L1 and L2 give the two rulings of P1ks × P1ks . Note that KX = −2(L1 + L2 ), and hence L1 + L2 is stable under the action of Gal(ks /k). If X(k) = 0, / or if k is a global field and X(kv ) = 0/ for all v ∈ Ωk , then the inclusion Gal(ks /k) Pic X → Pic Xks is an isomorphism (see Sect. 3.4). In either case, the class of L1 + L2 may be represented over k by some divisor D. The linear system |D| gives an embedding φD : X → P3k of X as a quadric surface, defined over k. If P ∈ X(k), then composing φD with projection away from φD (P), we obtain a k-birational map X P2k . If k is a global field and X(kv ) = 0/ for all v ∈ Ωk , then (φD )kv (Xkv ) is a quadric in P3kv and φD (X)(kv ) = 0/ for all v ∈ Ωk . Quadrics over global fields satisfy the Hasse principle, so φD (X)(k) = 0/ and hence X(k) = 0. /

2.3 Case 3: d(X) = 7 By Theorem 1.6, Xks is isomorphic to a blowup of P2ks at two points. There are only three exceptional curves on X, and their configuration is quite simple. Label these curves e1 , e2 , and e3 ; without loss of generality we have (e1 , e2 ) = (e2 , e3 ) = 1

and (e1 , e3 ) = 0.

Since the action of Gal(ks /k) on Pic Xks respects the intersection pairing, it follows that e2 is defined over k. Contracting it we obtain a del Pezzo surface of degree 8 with a point, and hence X(k) = 0/ by the Lang-Nishimura lemma. In particular, del Pezzo surfaces of degree 7 trivially satisfy the Hasse principle. It follows from our work above that X is k-birational to P2k .


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2.4 Case 4: d(X) = 6 By Theorem 1.6, Xks is isomorphic to a blowup of P2ks at three noncollinear points. If we have a point P ∈ X(k) that lies on at least one exceptional curve then one can show that X is not minimal and conclude using our work above. If P is not on any exceptional curve, then we may blowup P to obtain a del Pezzo surface of degree 5. On this surface, there are three exceptional curves that meet the exceptional divisor of the blowup, are pairwise skew, and form a Gal(ks /k)-set. Contracting them we obtain a del Pezzo surface of degree 8 that contains a rational point. Hence X is k-birational to P2k . The exceptional curves of Xks form a “hexagon.” Let e1 , . . . , e6 be the exceptional curves of X, numbered to correspond clockwise to the vertices of the hexagon. Thus {e1 , e3 , e5 } and {e2 , e4 , e6 } are triplets of curves that do not pairwise intersect, and {e1 , e4 }, {e2 , e5 } and {e3 , e6 } are couples of curves whose vertices in the hexagon are “as far apart as possible.” Since the action of Gal(ks /k) respects the intersection form, it induces an action on the sets T := {e1 , e3 , e5 }, {e2 , e4 , e6 } and D := {e1 , e4 }, {e2 , e5 }, {e3 , e6 } . If any element of T or D is fixed by the action of Gal(ks /k), then it is defined over k. Hence there exist: • An element of T defined over a field extension K of k with [K : k] | 2 • An element of D defined over a field extension L of k with [L : k] | 3 On XK we may contract the curves in the fixed element of T to obtain a del Pezzo surface of degree 9. On XL we may contract the curves in the fixed element of D to obtain a del Pezzo surface of degree 8 that is a form of P1L × P1L . If k is a global field and X(kv ) = 0/ for all v ∈ Ωv , then both XL and XK also have points everywhere locally. Since del Pezzo surfaces of degree 8 and 9 satisfy the Hasse principle, it follows that XK is K-birational to P2K and XL is L-birational to P2L . This means that there is a closed point P1 of X consisting of a pair of K-points that are conjugate, as well as a closed point P2 of X consisting of three L-points that are conjugate. Choose a four-dimensional linear subspace L of P4k that passes through P1 and P2 . By Bezout’s theorem, Xk¯ ∩ Lk¯ consists of 6 points, counted with multiplicity (because X has degree 6—this assumes the intersection is the right dimension; when the intersection contains a curve one has to do a little more work). The points P1 and P2 account for 5 of the geometric points in Xk¯ ∩ Lk¯ . The remaining point must be defined over k (how else would Galois act on it?). Thus del Pezzo surfaces of degree 6 satisfy the Hasse principle (this proof is essentially due to Colliot-Thélène [7]).

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2.5 Case 5: d(X) = 5 We shall assume first that X(k) = 0. / If X contains a k-point P that does not lie on any exceptional curve, then the blowup of X at P is a del Pezzo surface Y of degree 4. Let e be the exceptional curve of Y corresponding to the blown-up point P. Examining the graph of intersections of exceptional curves on Y , we see that there are five curves that do not intersect e and that are pairwise skew to each other. These five curves form an invariant set under the action of Gal(ks /k) because e is defined over k. Contracting this set we obtain a del Pezzo surface of degree 9 with k-point (since X(k) = 0), / so by Theorem 2.3 we conclude that X is k-birational to P2k . An easy case-by-case analysis shows that if P lies on at least one exceptional curve, then it is possible to contract at least one exceptional curve over k and thus reduce to previous cases to conclude that X is k-birational to P2k . Theorem 2.5 (Enriques, Swinnerton-Dyer, Skorobogatov, Shepherd-Barron, Kollár, Hassett). Let X be a del Pezzo surface of degree 5 over a field k. Then X(k) = 0. / In particular, if k is a global field, then del Pezzo surfaces of degree 5 trivially satisfy the Hasse principle. Proof (Sketch of proof). We follow Swinnerton-Dyer’s approach [47]; other proofs can be found in [20, 25, 28, 42, 44]. Recall that the anticanonical model of X is a quintic surface in P5k (i.e., the linear system |−K X | embeds X as a quintic in P5k ; see Sect. 1.5). Under this embedding, X is cut out by five quadrics, defined over k. It suffices to verify this claim after a base extension of the field. Over ks , the surface Xks is isomorphic to the blowup of P2ks at four points, no three of which are collinear. By a projective transformation (defined over ks ), we may assume that these points are [1 : 0 : 0], [0 : 1 : 0], [0 : 0 : 1] and [1 : 1 : 1]. It is now not difficult to check that the ideal I(Xks ) of Xks ⊆ P5ks is generated by five quadrics. Let Q be a general element of I(Xks )2 , let Π1 be a general plane lying in Q, and let L be a general threefold through Π1 of X. Then Q · L = Π1 + Π2 , where Π2 is another plane residual to Π1 . One can show that X · L = (X · Π1 )Q + (X · Π1 )Q and that, after possibly interchanging Π1 and Π2 , deg(X · Π1 )Q = 2

and

deg(X · Π2 )Q = 3.

These two equalities can now be used to show that Πi may be chosen in such a way that it is defined over an extension K of k with [K : k] = 2n with 0 ≤ n ≤ 3. Suppose for simplicity that n = 1. Let P be a K-point of Π1 and let Pc be its conjugate over k. Let H ⊂ P5k be any hyperplane defined over K. Then Hks · Xks is a curve C of genus


307

one defined over K containing the zero-cycle P + Pc and a zero-cycle of degree 5 obtained by intersecting X with any 3-dimensional linear subspace of H. Hence C contains a zero-cycle of degree 1 and thus a k-rational point. If n > 1, we just repeat this procedure a total of n times. Remark 2.6. Here is an alternative strategy [25, Exercise 3.1.4]: show that if Q1 , Q2 , Q3 are general elements of I(X)2 then V (Q1 , Q2 , Q3 ) = X ∪W, where Wks is isomorphic to the blowup of P2ks at a point. By our work in Sect. 2.2, the exceptional divisor of Wks is defined over k. Now show that this exceptional divisor intersects X in one point. Proof (Proof of Theorem 2.1). At this point, it only remains to note that since X is k-birational to P2k , it must satisfy weak approximation, by Lemma 1.2.

3 Counterexamples in Small Degree We have seen that del Pezzo surfaces of degree at least 5 over global fields satisfy both the Hasse principle and weak approximation. This is no longer the case for surfaces of lower degree. Of course, a counterexample to the Hasse principle immediately gives a counterexample to weak approximation. However, in degrees 2, 3, or 4, there exist examples of surfaces with a Zariski dense set of points for which weak approximation fails. Curiously, del Pezzo surfaces of degree 1 always have a k-rational point: the unique base point of the anticanonical linear system (see Remark 1.12). These surfaces, however, need not satisfy weak approximation. In Table 3 we have compiled a list of references for the earliest recorded counterexamples to the Hasse principle and weak approximation in low degrees. A little care must be taken with counterexamples to weak approximation: for example, by Lemma 1.2, if X is a counterexample to weak approximation and X(k) = 0, / then the blowup of X at a k-point is also a counterexample to weak approximation. Thus, when dealing with counterexamples to weak approximation on del Pezzo surfaces, it is important to make sure that the surfaces are k-minimal: the surface should not have a Galois-stable set of pairwise skew exceptional curves.

3.1 The Brauer-Manin Set I Let X be a nice variety over a global field k. Since X is proper, we have ∏v X(kv ) = X(Ak ). In [34], Manin used the Brauer group of the variety to construct an intermediate “obstruction set” between X(k) and X(Ak ): X(k) ⊆ X(Ak )Br ⊆ X(Ak ).

308

A. Várilly-Alvarado Table 3 Arithmetic phenomena on del Pezzo surfaces over global fields. A check mark ( ) indicates that the relevant arithmetic phenomenon holds for the indicated class of surfaces Phenomenon Hasse principle Weak approximation

d ≥5

d=4 [1] [11]

d =3 [46] [46]

d=2 [29] [30]

d =1

[50]

An entry with a reference indicates the existence of a counterexample to the arithmetic phenomenon which can be found in the paper cited

In fact, the set X(Ak )Br already contains the closure of X(k) for the adelic topology and thus may be used to explain the failure of both the Hasse principle and weak approximation on many classes of varieties. Definition 3.1. Let X be a nice variety over a global field k, and assume that X(Ak ) = 0. / We say that X is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction if X(Ak )Br = 0. / We say that X is a counterexample to the weak approximation explained by the Brauer-Manin obstruction if X(Ak ) \ X(Ak )Br = 0. /

3.2 Brauer Groups of Schemes Recall that the Brauer group of a field k can be defined in two different ways: as the set of similarity classes of central simple algebras over k or as the Galois cohomology group H2 Gal(ks /k), ks∗ . These two points of view can be naturally identified, and each has its own advantages, depending on context. Both definitions have natural generalizations to schemes, but these generalizations need no longer be naturally isomorphic. Definition 3.2. An Azumaya algebra on a scheme X is an OX -algebra A that is coherent and locally free as an OX -module, such that the fiber A(x) := A ⊗OX,x k(x) is a central simple algebra over the residue field k(x) for each x ∈ X. Two Azumaya algebras A and B on X are similar if there exist nonzero locally free coherent OX -modules E and F such that A ⊗OX EndOX (E) ∼ = B ⊗OX EndOX (F ). Definition 3.3. The Azumaya Brauer group of a scheme X is the set of similarity classes of Azumaya algebras on X, with multiplication induced by tensor product of sheaves. We denote this group by BrAz X. The inverse of [A] ∈ BrAz X is the class [Aop ] of the opposite algebra of A; the identity element is [OX ] (see [22, p. 47]). Definition 3.4. The Brauer group of a scheme X is Br X := H2ét X, Gm .


309

Remark 3.5. Note that if k is a field, then BrAz Spec k = Br Speck = Br k, the usual Brauer group of a field. For any scheme X there is a natural inclusion BrAz X → Br X; see [36, Theorem IV.2.5]. The following result of Gabber, a proof of which can be found in [18], determines the image of this injection for a scheme with some kind of polarization. Theorem 3.6 (Gabber, de Jong). If X is a scheme quasiprojective over a noetherian ring, then the natural map BrAz X → Br X induces an isomorphism ∼

BrAz X − → (Br X)tors . If X is an integral scheme with function field k(X), then the inclusion Speck(X) → X gives rise to a map Br X → Br k(X) via functoriality of étale cohomology. If further X is regular and quasi-compact then this induced map is injective; see [36, Example III.2.22]. On the other hand, the group Br k(X) is torsion, because it is a Galois cohomology group. These two facts imply the following corollary of Theorem 3.6. Corollary 3.7. Let X be a regular quasiprojective variety over a field. Then BrAz X ∼ = Br X. This corollary allows us to think of elements in “the” Brauer group of a nice k-variety either as Azumaya algebras or as étale cohomology classes. Each point of view has its advantages: the former is useful for computations, while the latter allows us to use theorems from étale cohomology to deduce things about the structure of the Brauer group.

3.3 The Brauer-Manin Set II Let X be a nice variety over a global field k. For each A ∈ Br X and each field extension K/k there is a specialization map evA : X(K) → Br K,

x → Ax ⊗OX,x K.

In words, evA (x) is the central simple algebra over the residue field of x encoded in A. These specialization maps may be put together to construct a pairing

φ : Br X × X(Ak ) → Q/Z,

(A, (xv )) →

∑

v∈Ωk

invv (evA (xv )),

(4)

310


where invv : Br kv → Q/Z is the usual invariant map from local class field theory. The sum in Eq. (4) is in fact finite because for (xv ) ∈ X(Ak ) we have evA (xv ) = 0 ∈ Br kv for all but finitely many v; see [45, p. 101]. For A ∈ Br X we obtain a commutative diagram X(k)

X(Ak )

(5) φ (A,−)

evA

0

Br k

evA ∑v invv

"

v Br kv

Q/Z

0

where the bottom row is the usual exact sequence from class field theory. Manin’s observation is that the kernel of the map φ (A, −) contains X(k) (because the bottom row of the above commutative diagram is a complex), and thus an element A ∈ Br X can be used to “carve out” a subset of X(Ak ) that contains X(k): X(Ak )A := (xv ) ∈ X(Ak ) : φ (A, (xv )) = 0 . Moreover, if Q/Z is given the discrete topology, then the map

φ (A, −) : X(Ak ) → Q/Z is continuous, so X(Ak )A is a closed subset of X(Ak ); see [23, Sect. 3.1]. This shows that X(k) ⊆ X(Ak )A . Definition 3.8. Let X be a nice variety over a global field k. We call X(Ak )Br :=

)

X(A)A

A∈Br X

the Brauer-Manin set of X. The structure map X → Spec k gives rise to a map Br k → Br X by functoriality. The group Br0 X := im (Br k → Br X) is known as the subgroup of constant algebras. The exactness of the bottom row of Eq.*(5) implies that if A ∈ Br0 X, then X(Ak )A = X(Ak ). This shows that to compute A∈Br X X(Ak )A it is enough to calculate the intersection over a set of representatives for the group Br X/ Br0 X.

3.4 The Hochschild-Serre Spectral Sequence in Étale Cohomology Let X be a nice locally soluble variety over a global field k. If Br Xks = 0, then the Hochschild-Serre spectral sequence in étale cohomology provides a tool for computing the group Br X/ Br k.


311

Let K be a finite Galois extension of k, with Galois group G. The HochschildSerre spectral sequence p,q q p+q E2 := H p G, Hét XK , Gm =⇒ Hét X, Gm =: L p+q gives rise to the usual “low-degree” long exact sequence 0 → E21,0 → L1 → E20,1 → E22,0 → ker L2 → E20,2 → E21,1 → E23,0 which in our case is 0 → Pic X → (Pic XK )G → H2 G, K ∗ → ker(Br X → Br XK ) → H1 G, Pic XK → H3 G, K ∗ .

(6)

Taking the direct limit over all finite Galois extensions of k gives the exact sequence 0 → Pic X → (Pic Xks )Gal(k /k) → Br k → ker(Br X → Br Xks ) (7) → H1 Gal(ks /k), Pic Xks → H3 Gal(ks /k), ks∗ . Furthermore, if k is a global field, then H3 Gal(ks /k), ks∗ = 0; this fact is due to Tate—see [37, 8.3.11(iv), 8.3.17]. For each v ∈ Ωk , local solubility of X gives a morphism Spec kv → X that splits the base extension πv : Xkv → Spec kv of the structure map of X. Thus, by functoriality of the Brauer group, the natural maps πv∗ : Br kv → Br Xkv split for every v ∈ Ωk . The exactness of the bottom row of Eq. (5) then shows that the natural map Br k → Br X coming from the structure morphism of X is injective. Moreover, if X is a del Pezzo surface, then Br Xks = 0 and thus Eq. (7) gives rise to the short exact sequence 0 → Br k → Br X → H1 Gal(ks /k), Pic Xks → 0 s

and hence to an isomorphism ∼ Br X/ Br k − → H1 Gal(ks /k), Pic Xks .

(8)

If K is a splitting field for X, i.e., a field extension K of k where a set of generators for Pic Xks have representatives defined over K, then the inflation map H1 Gal(K/k), Pic XK → H1 Gal(ks /k), Pic Xks is an isomorphism, because the cokernel maps into the first cohomology group of a free Z-module with trivial action by a profinite group, which is trivial. Hence Br X/ Br k ∼ (9) = H1 Gal(K/k), Pic XK . Finally, we note that since X(Ak ) = 0, / if H is a subgroup of G, then by Eq. (6) and the injectivity of the map Br k → Br X, we know that ∼

→ (Pic XK )H , Pic XK H − where K H is the fixed field of K by H.

312


In summary, one way of constructing Brauer-Manin obstructions on del Pezzo surfaces of small degree is to compute the group H1 Gal(K/k), Pic XK on “reasonable” surfaces and then try to invert the isomorphism (9). Many authors have pursued this set of ideas and not just for del Pezzo surfaces (see, for example, [3– 5, 9, 10, 14, 16, 29, 30, 33, 35, 48–50], to name but a few references).

3.5 A Counterexample to Weak Approximation in Degree 1 We will use the remainder of this section to go through the details of a counterexample to weak approximation. We will deal with del Pezzo surfaces of degree 1 for two reasons: (1) we were explicitly asked to talk a little bit about our research in these lectures and (2) the example shows that del Pezzo surfaces of degree 1 are not as scary as they may appear at first, if one is willing to use a computer to work out a little bit of algebra that is beyond what is reasonable to do with pencil and paper. Let us quickly review what we know about del Pezzo surfaces of degree 1. First, recall their anticanonical model is a smooth sextic hypersurface in Pk (1, 1, 2, 3) := Proj(k[x, y, z, w]), e.g., w2 = z3 + Ax6 + By6 ,

A, B ∈ k∗ .

Conversely, any smooth sextic X in Pk (1, 1, 2, 3) is a del Pezzo surface of degree 1. The surface Xks is isomorphic to the blowup of P2ks at 8 points in general position, so in particular, Pic Xks ∼ = Z9 . Fix a primitive sixth root of unity ζ in Q. Our goal is to prove the following theorem. Theorem 3.9 ([50]). Let X be the del Pezzo surface of degree 1 over k = Q(ζ ) given by w2 = z3 + 16x6 + 16y6 in Pk (1, 1, 2, 3). Then X is k-minimal and there is a Brauer-Manin obstruction to weak approximation on X. Moreover, the obstruction arises from a cyclic algebra class in Br X/ Br k. In order to compute the Galois cohomology group H1 Gal(ks /k), Pic Xks , we need an explicit description of the action of Gal(ks /k) on Pic Xks . Recall that Pic Xks is generated by the exceptional curves of X. The following theorem, which can be deduced from work of Shioda on Mordell-Weil lattices (see [43, Theorem 10.10]— we also give a different proof in [50, Theorem 1.2]), helps us compute the exceptional curves and a splitting field for a del Pezzo surface of degree 1. Theorem 3.10. Let X be a del Pezzo surface of degree 1 over a field k, given as a smooth sextic hypersurface V ( f (x, y, z, w)) in Pk (1, 1, 2, 3). Let


313

Γ = V (z − Q(x, y), w − C(x, y)) ⊆ Pks (1, 1, 2, 3), where Q(x, y) and C(x, y) are homogenous forms of degrees 2 and 3, respectively, in ks [x, y]. If Γ is a divisor on Xks , then it is an exceptional curve of X. Conversely, every exceptional curve on X is a divisor of this form. We explain how to use this theorem to compute the exceptional curves on the del Pezzo surface X of Theorem 3.9. Let a, b, c, r, s, t, and u be indeterminates, and let Q(x, y) = ax2 + bxy + cy2, C(x, y) = rx3 + sx2 y + txy2 + uy3 . The identity C(x, y)2 = Q(x, y)3 + 16x6 + 16y6 gives a3 − r2 + 16 = 0, 3a2 b − 2rs = 0, 3a2 c + 3ab2 − 2rt − s2 = 0, 6abc + b3 − 2ru − 2st = 0, 3ac2 + 3b2c − 2su − t 2 = 0, 3bc2 − 2tu = 0, c3 − u2 + 16 = 0. We can use Gröbner bases to solve this system of equations. We get 240 solutions, be read one for each exceptional curve of the surface. The action of Gal(ks /k) can√ off from the coefficients of the equations of the exceptional curves. Let s = 3 2, and consider the exceptional curves on X given by E1 = V (z + 2sx2 , w − 4y3), E2 = V (z − (−ζ + 1)2sx2 , w + 4y3), E3 = V (z − 2ζ sx2 + 4y2, w − 4s(ζ − 2)x2 y − 4(−2ζ + 1)y3), E4 = V (z + 4ζ sx2 − 2s2(2ζ − 1)xy − 4(−ζ + 1)y2, w − 12x3 − 8s(−ζ − 1)x2 y − 12ζ s2xy2 − 4(−2ζ + 1)y3), E5 = V (z + 4ζ sx2 − 2s2(ζ − 2)xy − 4ζ y2 w + 12x3 − 8s(2ζ − 1)x2 y − 12s2xy2 − 4(−2ζ + 1)y3 ), E6 = V (z − 2s(−s2ζ + s2 − 2s + 2ζ )x2 − 2s(2s2 ζ − 2s2 + 3s − 4ζ )xy − 2s(−s2ζ + s2 − 2s + 2ζ )y2,

314


w − 4(2s2ζ − 4s2 + 2sζ + 2s − 6ζ + 3)x3 − 4(−5s2ζ + 10s2 − 6sζ − 6s + 16ζ − 8)x2y − 4(5s2ζ − 10s2 + 6sζ + 6s − 16ζ + 8)xy2 − 4(−2s2ζ + 4s2 − 2sζ − 2s + 6ζ − 3)y3), E7 = V (z − 2s(−s2 − 2sζ + 2s + 2ζ )x2 − 2s(−2s2ζ + 3s + 4ζ − 4)xy − 2s(−s2ζ + s2 + 2sζ − 2)y2 , w − 4(2s2ζ + 2s2 + 2sζ − 4s − 6ζ + 3)x3 − 4(10s2ζ − 5s2 − 6sζ − 6s − 8ζ + 16)x2y − 4(5s2ζ − 10s2 − 12sζ + 6s + 8ζ + 8)xy2 − 4(−2s2ζ − 2s2 − 2sζ + 4s + 6ζ − 3)y3), E8 = V (z − 2s(s2 ζ + 2sζ + 2ζ )x2 − 2s(2s2 + 3s + 4)xy − 2s(−s2ζ + s2 − 2sζ + 2s − 2ζ + 2)y2, w − 4(−4s2ζ + 2s2 − 4sζ + 2s − 6ζ + 3)x3 − 4(−5s2ζ − 5s2 − 6sζ − 6s − 8ζ − 8)x2y − 4(5s2ζ − 10s2 + 6sζ − 12s + 8ζ − 16)xy2 − 4(4s2ζ − 2s2 + 4sζ − 2s + 6ζ − 3)y3 ), as well as the exceptional curve E9 = V (z − 2ζ s2 xy, w − 4x3 + 4y3 ). Then

Pic Xks = Pic XK ∼ =

8 i=1

Z[Ei ] ⊕ Z[H] = Z9 ,

√ where H = E1 + E2 + E9 . The exceptional curves of X are defined over K := k( 3 2). Let ρ be a generator for the cyclic group Gal(K/k). To invert the isomorphism Eq. (8), we will use the following diagram, whose individual maps we now explain:


Br X/ Br k

315 ∼

H1 Gal(ks /k), Pic Xks inf

(10)

∼

H1 Gal(K/k), Pic XK

Br k(X)/ Br k

∼

Brcyc (X, K)

ψ ∼

ker N¯ K/k / im Δ

First, the map Br X → Br k(X) is obtained by functoriality from the inclusion of the generic point Spec k(X) → X (see Sect. 3.2). Since X is nice and X(Ak ) = 0/ (because X(k) = 0), / this map induces an injection Br X/ Br k → Br k(X)/ Br k. Second, we have maps

Δ : Pic XK → Pic X

N K/k : Pic XK → Pic X ρ2

[D] → [D + ρD + D]

[D] → [D − ρD]

and the isomorphism H1 Gal(K/k), Pic XK ∼ = ker N¯ K/k / im Δ comes from Tate cohomology, since K is a cyclic extension of k. To define the group Brcyc (X, K), we first need to recall the notion of a cyclic algebra. If L/k is a finite cyclic extension of fields of degree n, and if ρ is a generator for Gal(L/k), then we let L[x]ρ be the “twisted” polynomial ring, where x = xρ for all ∈ L. Given b ∈ k∗ , we write (L/k, b) for the central simple k-algebra L[x]σ /(xn − b). Let f ∈ k(X)∗ ; since X is geometrically integral, we have Gal(k(XL )/k(X)) ∼ = Gal(L/k), and hence we write (L/k, f ) instead of (k(XL )/k(X), f ) for the cyclic algebra in Br k(X) given by f . We now define the group

classes [(K/k, f )] in the image of the Brcyc (X, K) := . map Br X/ Br k → Br k(X)/ Br k The map ψ in Eq. (10) is defined by

ψ : ker N¯ K/k / im Δ → Brcyc (X, K) [D] → [(K/k, f )], where f ∈ k(X)∗ is any function such that NK/k (D) = ( f ). This map is a group isomorphism; see [50, Theorem 3.3].

316


With our explicit generators for Pic KK in hand, we use Magma [2] to compute ker N¯ K/k / im Δ ∼ = (Z/3Z)4 . The classes h1 = [E2 + 2E8 − H],

h2 = [E5 + 2E8 − H],

h3 = [E7 + 2E8 − H],

h4 = [3E8 − H]

of Pic XK give a set of generators for this group. Proof (of Theorem 3.9). Consider the divisor class h1 − h2 = [E2 − E5 ] ∈ Pic XK . It gives rise to a cyclic algebra A := (K/k, f ) ∈ Brcyc (X, K), where f ∈ k(X)∗ is any function such that NK/k (E2 − E5) = ( f ). To wit, f is a function with zeroes along E2 + ρ E2 + ρ E2 2

and poles along E5 + ρ E5 + ρ E5 . 2

Using the explicit equations for E2 and E5 we find that f :=

w + 4y3 w + (2ζ + 2)zy + (−8ζ + 4)y3 + 12x3

does the job. Recall that X is given by w2 = z3 + 16x6 + 16y6. Note that P1 = [1 : 0 : 0 : 4] and P2 = [0 : 1 : 0 : 4] are in X(k). Let p be the unique prime above 3 in k. We compute invp (A (P1 )) = 0 and

invp (A (P2 )) = 1/3.

Let P ∈ X(Ak ) be the point that is equal to P1 at all places except p and is P2 at p. Then

∑

v∈Ωk

invv (A (Pv )) = 1/3,

so P ∈ X(Ak ) \ X(Ak )Br and X is a counterexample to weak approximation.

Acknowledgments I thank the conference organizers (Hendrik Lenstra, Cecilia Salgado, Lenny Taelman and Ronald van Luijk) for inviting me to give this minicourse and for their hospitality in Leiden. I also thank the staff at the Lorentz Center for all their help and professionalism. Finally, I thank Jean-Louis Colliot-Thélène for the comments on these notes.


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