University LECTURE Series Volume 52 Koszul Cohomology and Algebraic Geometry Marian Aprodu Jan Nagel American Mathematical Society http://dx.doi.org/10.1090/ulect/052 Koszul Cohomology and Algebraic Geometry University LECTURE Series Volume 52 Koszul Cohomology and Algebraic Geometry Marian Aprodu Jan Nagel M THE ATI A CA M L ΤΡΗΤΟΣ ΜΗ N ΕΙΣΙΤΩ S A O C C I I R E E T ΑΓΕΩΜΕ Y M A F O 8 U 88 NDED 1 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jerry L. Bona NigelD.Higson Eric M. Friedlander (Chair) J. T. Stafford 2000 Mathematics Subject Classification. Primary 14H51, 14C20, 14H60, 14J28, 13D02, 16E05. For additional information and updates on this book, visit www.ams.org/bookpages/ulect-52 Library of Congress Cataloging-in-Publication Data Aprodu, Marian. Koszul cohomology and algebraic geometry / Marian Aprodu, Jan Nagel. p. cm. — (University lecture series ; v. 52) Includes bibliographical references and index. ISBN 978-0-8218-4964-4 (alk. paper) 1. Koszul algebras. 2. Geometry, Algebraic. 3. Homology theory. I. Nagel, Jan. II. Title. QA564.A63 2010 512.46—dc22 2009042378 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2010 by the authors. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 151413121110 Contents Preface vii Chapter 1. Basic definitions 1 1.1. The Koszul complex 1 1.2. Definitions in the algebraic context 2 1.3. Minimal resolutions 4 1.4. Definitions in the geometric context 5 1.5. Functorial properties 7 1.6. Notes and comments 11 Chapter 2. Basic results 13 2.1. Kernel bundles 13 2.2. Projections and linear sections 15 2.3. Duality 20 2.4. Koszul cohomology versus usual cohomology 22 2.5. Sheaf regularity 24 24 2.6. Vanishing theorems 25 2.7. Notes and comments 27 Chapter 3. Syzygy schemes 29 3.1. Basic definitions 29 3.2. Koszul classes of low rank 37 3.3. The Kp,1 theorem 39 3.4. Rank-2 bundles and Koszul classes 43 3.5. The curve case 46 3.6. Notes and comments 50 Chapter 4. The conjectures of Green and Green–Lazarsfeld 53 4.1. Brill-Noether theory 53 4.2. Numerical invariants of curves 55 4.3. Statement of the conjectures 57 4.4. Generalizations of the Green conjecture 60 4.5. Notes and comments 63 Chapter 5. Koszul cohomology and the Hilbert scheme 65 5.1. Voisin’s description 65 5.2. Examples 68 5.3. Vanishing via base change 72 Chapter 6. Koszul cohomology of a K3surface 75 6.1. The Serre construction, and vector bundles on K3 surfaces 75 v vi CONTENTS 6.2. Brill-Noether theory of curves on K3 surfaces 76 6.3. Voisin’s proof of Green’s generic conjecture: even genus 79 6.4. The odd-genus case (outline) 88 6.5. Notes and comments 90 Chapter 7. Specific versions of the syzygy conjectures 91 7.1. Nodal curves on K3 surfaces 91 7.2. The specific Green conjecture 92 7.3. Stable curves with extra-syzygies 95 7.4. Curves with small Brill-Noether loci 97 7.5. Further evidence for the Green-Lazarsfeld conjecture 100 7.6. Exceptional curves 103 7.7. Notes and comments 103 Chapter 8. Applications 105 8.1. Koszul cohomology and Hodge theory 105 8.2. Koszul divisors of small slope on the moduli space. 111 8.3. Slopes of fibered surfaces 115 8.4. Notes and comments 116 Bibliography 119 Index 125 Preface The systematic use of Koszul cohomology computations in algebraic geometry can be traced back to the foundational paper [Gre84a]byM.Green.Inthis paper, Green introduced the Koszul cohomology groups Kp,q(X, L) associated to a line bundle L on a smooth projective variety X, and studied the basic properties of these groups. Green noted that a number of classical results concerning the generators and relations of the (saturated) ideal of a projective variety can be rephrased naturally in terms of vanishing theorems for Koszul cohomology, and extended these results using his newly developed techniques. In a remarkable series of papers, Green and Lazarsfeld further pursued this approach. Much of their work in the late 80’s centers around the shape of the minimal resolution of the ideal of a projective variety; see [Gre89], [La89] for an overview of the results obtained during this period. Green and Lazarsfeld also stated two conjectures that relate the Koszul coho- mology of algebraic curves to two numerical invariants of the curve, the Clifford index and the gonality. These conjectures became an important guideline for future research. They were solved in a number of special cases, but the solution of the general problem remained elusive. C. Voisin achieved a major breakthrough by proving the Green conjecture for general curves in [V02]and[V05]. This result soon led to a proof of the conjecture of Green–Lazarsfeld for general curves [AV03 ], [Ap04]. Since the appearance of Green’s paper there has been a growing interaction between Koszul cohomology and algebraic geometry. Green and Voisin applied Koszul cohomology to a number of Hodge–theoretic problems, with remarkable success. This work culminated in Nori’s proof of his connectivity theorem [No93]. In recent years, Koszul cohomology has been linked to the geometry of Hilbert schemes (via the geometric description of Koszul cohomology used by Voisin in her work on the Green conjecture) and moduli spaces of curves. Since there already exists an excellent introduction to the subject [Ei06], this book is devoted to more advanced results. Our main goal was to cover the re- cent developments in the subject (Voisin’s proof of the generic Green conjecture, and subsequent refinements) and to discuss the geometric aspects of the theory, including a number of concrete applications of Koszul cohomology to problems in algebraic geometry. The relationship between Koszul cohomology and minimal res- olutions will not be treated at length, although it is important for historical reasons and provides a way to compute Koszul cohomology by computer calculations. Outline of contents. The first two chapters contain a review of a number of basic definitions and results, which are mainly included to fix the notation and vii viii PREFACE to obtain a reasonably self–contained presentation. Chapter 3 is devoted to the theory of syzygy schemes. The aim of this theory is to study Koszul cohomology classes in the groups Kp,1(X, L) by associating a geometric object to them. This chapter contains a proof of one of the fundamental results in the subject, Green’s Kp,1–theorem. In Chapter 4 we recall a number of results from Brill–Noether theory that will be needed in the sequel, and state the conjectures of Green and Green–Lazarsfeld. Chapters 5–7 form the heart of the book. Chapter 5 is devoted to Voisin’s description of the Koszul cohomology groups Kp,q(X, L) in terms of the Hilbert scheme of zero–dimensional subschemes of X. This description yields a method to prove vanishing theorems for Koszul cohomology by base change, which is used in Voisin’s proof of the generic Green conjecture. In Chapter 6 we present Voisin’s proof for curves of even genus, and outline the main steps of the proof for curves of odd genus; the latter case is technically more complicated. Chapter 7 contains a number of refinements of Voisin’s result; in particular, we have included a proof of the conjectures of Green and Green–Lazarsfeld for the general curve in a given gonality stratum of the moduli space of curves. In the final chapter we discuss geometric applications of Koszul cohomology to Hodge theory and the geometry of the moduli space of curves. Acknowledgments. MA thanks the Fourier Institute Grenoble, the University of Bayreuth, Laboratoire Painlev´e Lille, and the Max Planck Institut f¨ur Mathematik Bonn, and JN thanks the Fourier Institute Grenoble, the University of Bayreuth, and the Simion Stoilow Institute Bucharest for hospitality during the preparation of this work. MA was partially supported by the grant PN-II-ID-PCE-2008-2, cod CNCSIS 1189. The authors are members of the L.E.A. “MathMode”. Marian Aprodu and Jan Nagel Bibliography [Ap02] Aprodu, M.: On the vanishing of higher syzygies of curves. Math. Z., 241, 1–15 (2002) [Ap04] Aprodu, M.: Green-Lazarsfeld gonality conjecture for a generic curve of odd genus. Int. Math. Res. Notices, 63, 3409–3414 (2004) [Ap05] Aprodu, M.: Remarks on syzygies of d-gonal curves. Math. Res. Lett., 12, 387–400 (2005) [AF08] Aprodu, M., Farkas, G.: Koszul cohomology and applications to moduli. arXiv : 0811.3117. To appear in Proc. Clay Math Inst. AMS [AN04] Aprodu, M., Nagel, J.: A Lefschetz type result for Koszul cohomology. Manuscr. Math., 114, 423–430 (2004) [AN07] Aprodu, M., Nagel, J.: Non-vanishing for Koszul cohomology of curves.