Graduate Texts in Mathematics 187Editorial Board

S. Axler F.W. Gehring K.A. Ribet

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1 TAKEUTI/ZARING. Introduction toAxiomatic Set Theory. 2nd ed.

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Homological Algebra. 2nd ed.5 MAC LANE. Categories for the Working

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Theory.11 CONWAY. Functions of One Complex

Variable I. 2nd ed.12 BEALS. Advanced Mathematical Analysis.13 ANDERSON/FULLER. Rings and Categories

of Modules. 2nd ed.14 GOLUBITSKY/GUTLLEMTN. Stable Mappings

and Their Singularities.15 BERBERIAN. Lectures in Functional

Analysis and Operator Theory.16 WINTER. The Structure of Fields.17 ROSENBLATT. Random Processes. 2nd ed.18 HALMOS. Measure Theory.19 HALMOS. A Hilbert Space Problem Book.

2nd ed.20 HUSEMOLLER. Fibre Bundles. 3rd ed.21 HUMPHREYS. Linear Algebraic Groups.22 BARNES/MACK. An Algebraic Introduction

to Mathematical Logic.23 GREUB. Linear Algebra. 4th ed.24 HOLMES. Geometric Functional Analysis

and Its Applications.25 HEWITT/STROMBERG. Real and Abstract

Analysis.26 MANES. Algebraic Theories.27 KELLEY. General Topology.28 ZARISKI/SAMUEL. Commutative Algebra.

Vol.1.29 ZARISKI/SAMUEL. Commutative Algebra.

Vol.11.30 JACOBSON. Lectures in Abstract Algebra I.

Basic Concepts.31 JACOBSON. Lectures in Abstract Algebra

0. Linear Algebra.32 JACOBSON. Lectures in Abstract Algebra

III. Theory of Fields and Galois Theory.

33 HIRSCH. Differential Topology.34 SPITZER. Principles of Random Walk.

2nd ed.35 ALEXANDER/WERMER. Several Complex

Variables and Banach Algebras. 3rd ed.36 KELLEY/NAMIOKA et al. Linear

Topological Spaces.37 MONK. Mathematical Logic.38 GRAUERT/FRITZSCHE. Several Complex

Variables.39 ARVESON. An Invitation to C*-Algebras.40 KEMENY/SNELL/KNAPP. Denumerable

Markov Chains. 2nd ed.41 APOSTOL. Modular Functions and

Dirichlet Series in Number Theory.2nd ed.

42 SERRE. Linear Representations of FiniteGroups.

43 GILLMAN/JERISON. Rings of ContinuousFunctions.

44 KENDIG. Elementary AlgebraicGeometry.

45 LOEVE. Probability Theory I. 4th ed.46 LOEVE. Probability Theory II. 4th ed.47 MOISE. Geometric Topology in

Dimensions 2 and 3.48 SACHS/WU. General Relativity for

Mathematicians.49 GRUENBERG/WEIR. Linear Geometry.

2nd ed.50 EDWARDS. Fermat's Last Theorem.51 KLINGENBERG. A Course in Differential

Geometry.52 HARTSHORNE. Algebraic Geometry.53 MANIN. A Course in Mathematical

Logic.54 GRAVER/WATKINS. Combinatorics with

Emphasis on the Theory of Graphs.55 BROWN/PEARCY. Introduction to

Operator Theory I: Elements ofFunctional Analysis.

56 MASSEY. Algebraic Topology: AnIntroduction.

57 CROWELL/FOX. Introduction to KnotTheory.

58 KOBLITZ. p-adic Numbers, p-adicAnalysis, and Zeta-Functions. 2nd ed.

59 LANG. Cyclotomic Fields.60 ARNOLD. Mathematical Methods in

Classical Mechanics. 2nd ed.

continued after index

Joe HarrisIan Morrison

Moduli of Curves

Springer

Joe Harris Ian MorrisonDepartment of Mathematics Department of MathematicsHarvard University Fordham UniversityCambridge, MA 02138 Bronx, NY 10458-5165USA USA

Editorial Board

S. Axler F.W. Gehring K.A. RibetMathematics Department Mathematics Department Department of MathematicsSan Francisco State East Hall University of California

University University of Michigan at BerkeleySan Francisco, CA 94132 Ann Arbor, MI48109 Berkeley, CA 94720-3840USA USA USA

Mathematics Subject Classification (1991): 14H10

library of Congress Cataloging-in-Publication DataHarris, Joe.

Moduli of curves / Joe Harris, Ian Morrison.p. cm. — (Graduate texts in Mathematics; 187)

Includes bibliographical references and index.ISBN 0-387-98438-0 (hardcover : alk. paper). — ISBN 0-387-98429-1

(pbk. : alk. paper)1. Moduli theory. 2. Curves, Algebraic. I. Morrison, Ian, 1950-

n. Title.QA564.H244 1998516.3'5—dc21 98-13036

© 1998 Springer-Verlag New York, Inc.All rights reserved. This work may not be translated or copied in whole or in part without thewritten permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York,NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Usein connection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use of general descriptive names, trade names, trademarks, etc., in this publication, even if theformer are not especially identified, is not to be taken as a sign that such names, as understood bythe Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

ISBN 0-387-98438-0 Springer-Verlag New York Berlin Heidelberg SPIN 10659835 (hardcover)ISBN 0-387-98429-1 Springer-Verlag New York Berlin Heidelberg SPIN 10659843 (softcover)

To Phil Griffiths

and David Mumford

Preface

AimsThe aim of this book is to provide a guide to a rich and fascinating sub-ject: algebraic curves, and how they vary in families. The revolutionthat the field of algebraic geometry has undergone with the introduc-tion of schemes, together with new ideas, techniques and viewpointsintroduced by Mumford and others, have made it possible for us tounderstand the behavior of curves in ways that simply were not possi-ble a half-century ago. This in turn has led, over the last few decades,to a burst of activity in the area, resolving long-standing problemsand generating new and unforeseen results and questions. We hopeto acquaint you both with these results and with the ideas that havemade them possible.

The book isn’t intended to be a definitive reference: the subject isdeveloping too rapidly for that to be a feasible goal, even if we hadthe expertise necessary for the task. Our preference has been to fo-cus on examples and applications rather than on foundations. Whendiscussing techniques we’ve chosen to sacrifice proofs of some, evenbasic, results — particularly where we can provide a good reference —in order to show how the methods are used to study moduli of curves.Likewise, we often prove results in special cases which we feel bringout the important ideas with a minimum of technical complication.

Chapters 1 and 2 provide a synopsis of basic theorems and conjec-tures about Hilbert schemes and moduli spaces of curves, with fewor no details about techniques or proofs. Use them more as a guideto the literature than as a working manual. Chapters 3 through 6 are,by contrast, considerably more self-contained and approachable. Ul-timately, if you want to investigate fully any of the topics we discuss,you’ll have to go beyond the material here; but you will learn the tech-niques fully enough, and see enough complete proofs, that when youfinish a section here you’ll be equipped to go exploring on your own.

If your goal is to work with families of curves, we’d therefore suggestthat you begin by skimming the first two chapters and then tackle thelater chapters in detail, referring back to the first two as necessary.

viii

ContentsAs for the contents of the book: Chapters 1 and 2 are largely exposi-tory: for the most part, we discuss in general terms the problems as-sociated with moduli and parameter spaces of curves, what’s knownabout them, and what sort of behavior we’ve come to expect fromthem. In Chapters 3 through 5 we develop the techniques that haveallowed us to analyze moduli spaces: deformations, specializations(of curves, of maps between them and of linear series on them), toolsfor making a variety of global enumerative calculations, geometric in-variant theory, and so on. Finally, in Chapter 6, we use the ideas andtechniques introduced in preceding chapters to prove a number ofbasic results about the geometry of the moduli space of curves andabout various related spaces.

PrerequisitesWhat sort of background do we expect you to have before you startreading? That depends on what you want to get out of the book. We’dhope that even if you have only a basic grounding in modern algebraicgeometry and a slightly greater familiarity with the theory of a fixedalgebraic curve, you could read through most of this book and get asense of what the subject is about: what sort of questions we ask, andsome of the ways we go about answering them. If your ambition isto work in this area, of course, you’ll need to know more; a workingknowledge with many of the topics covered in Geometry of algebraiccurves, I [7] first and foremost. We could compile a lengthy list of othersubjects with which some acquaintance would be helpful. But, instead,we encourage you to just plunge ahead and fill in the background asneeded; again, we’ve tried to write the book in a style that makes suchan approach feasible.

NavigationIn keeping with the informal aims of the book, we have used onlytwo levels of numbering with arabic for chapters and capital lettersfor sections within each chapter. All labelled items in the book arenumbered consecutively within each chapter: thus, the orderings ofsuch items by label and by position in the book agree.

There is a single index. However, its first page consists of a listof symbols, giving for each a single defining occurrence. These, andother, references to symbols also appear in the main body of the indexwhere they are alphabetized “as read”: for example, references toMgwill be found under Mgbar; to κi under kappai. Bold face entries in themain body index point to the defining occurrence of the cited term.References to all the main results stated in the book can be foundunder the heading theorems.

ix

Production acknowledgementsThis book was designed by the authors who provided Springer withthe PostScript file from which the plates were produced. The type isa very slightly modified version of the Lucida font family designed byChuck Bigelow and Kristin Holmes. (We added swashes to a few char-acters in the \mathcal alphabet to make them easier to distinguishfrom the corresponding upper-case \mathit character. These alpha-bets are often paired: a \mathcal character is used for the total spaceof a family and the \mathit version for an element.) It was coded in acustomized version of the LATEX2e format and typeset using Blue SkyResearch’s Textures TEX implementation with EPS figures created inMacromedia’s Freehand7 illustration program.

A number of people helped us with the production of the book.First and foremost, we want to thank Greg Langmead who did a trulywonderful job of producing an initial version of both the LATEX codeand the figures from our earlier WYSIWYG drafts. Dave Bayer offeredinvaluable programming assistance in solving many problems. Mostnotably, he devoted considerable effort to developing a set of macrosfor overlaying text generated within TEX onto figures. These allow pre-cise one-time text placement independent of the scale of the figureand proved invaluable both in preparing the initial figures and insolving float placement problems. If you’re interested, you can ob-tain the macros, which work with all formats, by e-mailing Dave atbayer@math.columbia.edu.

Frank Ganz at Springer made a number of comments to improvethe design and assisted in solving some of the formatting problemshe raised. At various points, Donald Arseneau, Berthold Horn, Vin-cent Jalby and Sorin Popescu helped us solve or work around variousdifficulties. We are grateful to all of them.

Lastly, we wish to thank our patient editor, Ina Lindemann, who wasnever in our way but always ready to help.

Mathematical acknowledgementsYou should not hope to find here the sequel to Geometry of algebraiccurves, I [7] announced in the preface to that book. As we’ve alreadynoted, our aim is far from the “comprehensive and self-contained ac-count” which was the goal of that book, and our text lacks its uni-formity. The promised second volume is in preparation by Enrico Ar-barello, Maurizio Cornalba and Phil Griffiths.

A few years ago, these authors invited us to attempt to merge ourthen current manuscript into theirs. However, when the two sets ofmaterial were assembled, it became clear to everyone that ours wasso far from meeting the standards set by the first volume that sucha merger made little sense. Enrico, Maurizio and Phil then, with their

x

usual generosity, agreed to allow us to withdraw from their projectand to publish what we had written here. We cannot too strongly ac-knowledge our admiration for the kindness with which the partner-ship was proposed and the grace with which it was dissolved nor ourdebt to them for the influence their ideas have had on our understand-ing of curves and their moduli.

The book is based on notes from a course taught at Harvard in 1990,when the second author was visiting, and we’d like to thank HarvardUniversity for providing the support to make this possible, and Ford-ham University for granting the second author both the leave for thisvisit and a sabbatical leave in 1992-93. The comments of a numberof students who attended the Harvard course were very helpful to us:in particular, we thank Dan Abramovich, Jean-Francois Burnol, LuciaCaporaso and James McKernan. We owe a particular debt to AngeloVistoli, who also sat in on the course, and patiently answered manyquestions about deformation theory and algebraic stacks.

There are many others as well with whom we’ve discussed the var-ious topics in this book, and whose insights are represented here. Inaddition to those mentioned already, we thank especially David Eisen-bud, Bill Fulton and David Gieseker.

We to thank Armand Brumer, Anton Dzhamay, Carel Faber, Bill Ful-ton, Rahul Pandharipande, Cris Poor, Sorin Popescu and MonserratTeixidor i Bigas who volunteered to review parts of this book. Theircomments enabled us to eliminate many errors and obscurities. Forany that remain, the responsibility is ours alone.

Finally, we thank our respective teachers, Phil Griffiths and DavidMumford. The beautiful results they proved and the encouragementthey provided energized and transformed the study of algebraiccurves — for us and for many others. We gratefully dedicate this bookto them.

Contents

Preface vii

1 Parameter spaces: constructions and examples 1A Parameters and moduli . . . . . . . . . . . . . . . . . . . . . 1B Construction of the Hilbert scheme . . . . . . . . . . . . . 5C Tangent space to the Hilbert scheme . . . . . . . . . . . . 12D Extrinsic pathologies . . . . . . . . . . . . . . . . . . . . . . 18

Mumford’s example . . . . . . . . . . . . . . . . . . . . . 19Other examples . . . . . . . . . . . . . . . . . . . . . . . . 24

E Dimension of the Hilbert scheme . . . . . . . . . . . . . . . 26F Severi varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 29G Hurwitz schemes . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 Basic facts about moduli spaces of curves 35A Why do fine moduli spaces of curves not exist? . . . . . . 35B Moduli spaces we’ll be concerned with . . . . . . . . . . . 40C Constructions of Mg . . . . . . . . . . . . . . . . . . . . . . 43

The Teichmuller approach . . . . . . . . . . . . . . . . . 43The Hodge theory approach . . . . . . . . . . . . . . . . 44The geometric invariant theory (G.I.T.) approach . . . 46

D Geometric and topological properties . . . . . . . . . . . . 52Basic properties . . . . . . . . . . . . . . . . . . . . . . . 52Local properties . . . . . . . . . . . . . . . . . . . . . . . 52Complete subvarieties of Mg . . . . . . . . . . . . . . . 55Cohomology of Mg : Harer’s theorems . . . . . . . . . . 58Cohomology of the universal curve . . . . . . . . . . . 62Cohomology of Hilbert schemes . . . . . . . . . . . . . 63Structure of the tautological ring . . . . . . . . . . . . . 67Witten’s conjectures and Kontsevich’s theorem . . . . 71

E Moduli spaces of stable maps . . . . . . . . . . . . . . . . . 76

3 Techniques 81A Basic facts about nodal and stable curves . . . . . . . . . 81

Dualizing sheaves . . . . . . . . . . . . . . . . . . . . . . 82

xii Contents

Automorphisms . . . . . . . . . . . . . . . . . . . . . . . 85B Deformation theory . . . . . . . . . . . . . . . . . . . . . . . 86

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Deformations of smooth curves . . . . . . . . . . . . . 89Variations on the basic deformation theory plan . . . 92Universal deformations of stable curves . . . . . . . . 102Deformations of maps . . . . . . . . . . . . . . . . . . . 105

C Stable reduction . . . . . . . . . . . . . . . . . . . . . . . . . 117Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

D Interlude: calculations on the moduli stack . . . . . . . . 139Divisor classes on the moduli stack . . . . . . . . . . . 140Existence of tautological families . . . . . . . . . . . . . 148

E Grothendieck-Riemann-Roch and Porteous . . . . . . . . . 150Grothendieck-Riemann-Roch . . . . . . . . . . . . . . . 150Chern classes of the Hodge bundle . . . . . . . . . . . . 154Chern class of the tangent bundle . . . . . . . . . . . . 159Porteous’ formula . . . . . . . . . . . . . . . . . . . . . . 161The hyperelliptic locus in M3 . . . . . . . . . . . . . . . 162Relations amongst standard cohomology classes . . . 165Divisor classes on Hilbert schemes . . . . . . . . . . . . 166

F Test curves: the hyperelliptic locus in M3 begun . . . . . 168G Admissible covers . . . . . . . . . . . . . . . . . . . . . . . . 175H The hyperelliptic locus in M3 completed . . . . . . . . . . 186

4 Construction of Mg 191A Background on geometric invariant theory . . . . . . . . . 192

The G.I.T. strategy . . . . . . . . . . . . . . . . . . . . . . 192Finite generation of and separation by invariants . . . 194The numerical criterion . . . . . . . . . . . . . . . . . . . 199Stability of plane curves . . . . . . . . . . . . . . . . . . 202

B Stability of Hilbert points of smooth curves . . . . . . . . 206The numerical criterion for Hilbert points . . . . . . . 206Gieseker’s criterion . . . . . . . . . . . . . . . . . . . . . 211Stability of smooth curves . . . . . . . . . . . . . . . . . 216

C Construction of Mg via the Potential Stability Theorem . 220The plan of the construction and a few corollaries . . 220The Potential Stability Theorem . . . . . . . . . . . . . . 224

5 Limit Linear Series and Brill-Noether theory 240A Introductory remarks on degenerations . . . . . . . . . . . 240B Limits of line bundles . . . . . . . . . . . . . . . . . . . . . . 247C Limits of linear series: motivation and examples . . . . . 253D Limit linear series: definitions and applications . . . . . . 263

Limit linear series . . . . . . . . . . . . . . . . . . . . . . 263

Contents xiii

Smoothing limit linear series . . . . . . . . . . . . . . . 266Limits of canonical series and Weierstrass points . . . 269

E Limit linear series on flag curves . . . . . . . . . . . . . . . 274Inequalities on vanishing sequences . . . . . . . . . . . 274The case ρ = 0 . . . . . . . . . . . . . . . . . . . . . . . . 276Proof of the Gieseker-Petri theorem . . . . . . . . . . . 280

6 Geometry of moduli spaces: selected results 286A Irreducibility of the moduli space of curves . . . . . . . . 286B Diaz’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

The idea: stratifying the moduli space . . . . . . . . . . 288The proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

C Moduli of hyperelliptic curves . . . . . . . . . . . . . . . . . 293Fiddling around . . . . . . . . . . . . . . . . . . . . . . . 293The calculation for an (almost) arbitrary family . . . . 295The Picard group of the hyperelliptic locus . . . . . . 301

D Ample divisors on Mg . . . . . . . . . . . . . . . . . . . . . 303An inequality for generically Hilbert stable families . 304Proof of the theorem . . . . . . . . . . . . . . . . . . . . 305An inequality for families of pointed curves . . . . . . 308Ample divisors on Mg . . . . . . . . . . . . . . . . . . . 310

E Irreducibility of the Severi varieties . . . . . . . . . . . . . 313Initial reductions . . . . . . . . . . . . . . . . . . . . . . . 314Analyzing a degeneration . . . . . . . . . . . . . . . . . 320An example . . . . . . . . . . . . . . . . . . . . . . . . . . 324Completing the argument . . . . . . . . . . . . . . . . . 326

F Kodaira dimension of Mg . . . . . . . . . . . . . . . . . . . 328Writing down general curves . . . . . . . . . . . . . . . 328Basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . 330Pulling back the divisors Drs . . . . . . . . . . . . . . . . 335Divisors on Mg that miss j(M2,1 \W) . . . . . . . . . 336Divisors on Mg that miss i(M0,g) . . . . . . . . . . . . 340Further divisor class calculations . . . . . . . . . . . . . 342Curves defined over Q . . . . . . . . . . . . . . . . . . . 342

Bibliography 345

Index 355