Moduli of Curves Joe Harris Ian Morrison Springer To Phil Griffiths and David Mumford Preface Aims Theaimofthisbookistoprovideaguidetoarichandfascinatingsub- ject: algebraic curves, and how they vary in families. The revolution thatthefieldofalgebraicgeometryhasundergonewiththeintroduc- tionofschemes,togetherwithnewideas,techniquesandviewpoints introduced by Mumford and others, have made it possible for us to understandthebehaviorofcurvesinwaysthatsimplywerenotpossi- bleahalf-centuryago.Thisinturnhasled,overthelastfewdecades, to a burst of activity in the area, resolving long-standing problems and generating new and unforeseen results and questions. We hope to acquaint you both with these results and with the ideas that have madethempossible. The book isn’t intended to be a definitive reference: the subject is developing too rapidly for that to be a feasible goal, even if we had the expertise necessary for the task. Our preference has been to fo- cus on examples and applications rather than on foundations. When discussingtechniqueswe’vechosentosacrificeproofsofsome,even basic,results—particularlywherewecanprovideagoodreference— inordertoshowhowthemethodsareusedtostudymoduliofcurves. Likewise, we often prove results in special cases which we feel bring outtheimportantideaswithaminimumoftechnicalcomplication. Chapters1and2provideasynopsisofbasictheoremsandconjec- tures about Hilbert schemes and moduli spaces of curves, with few or no details about techniques or proofs. Use them more as a guide totheliteraturethanasaworkingmanual.Chapters3through6are, by contrast, considerably more self-contained and approachable. Ul- timately,ifyouwanttoinvestigatefullyanyofthetopicswediscuss, you’llhavetogobeyondthematerialhere;butyouwill learnthetech- niquesfullyenough,andseeenoughcompleteproofs,thatwhenyou finishasectionhereyou’llbeequippedtogoexploringonyourown. Ifyourgoalistoworkwithfamiliesofcurves,we’dthereforesuggest thatyoubeginbyskimmingthefirsttwochaptersandthentacklethe laterchaptersindetail,referringbacktothefirsttwoasnecessary. viii Contents As for the contents of the book: Chapters 1 and 2 are largely exposi- tory:forthemostpart,wediscussingeneraltermstheproblemsas- sociated with moduli and parameter spaces of curves, what’s known about them, and what sort of behavior we’ve come to expect from them. In Chapters 3 through 5 we develop the techniques that have allowed us to analyze moduli spaces: deformations, specializations (ofcurves,ofmapsbetweenthemandoflinearseriesonthem),tools formakingavarietyofglobalenumerativecalculations,geometricin- variant theory, and so on. Finally, in Chapter 6, we use the ideas and techniques introduced in preceding chapters to prove a number of basic results about the geometry of the moduli space of curves and aboutvariousrelatedspaces. Prerequisites What sort of background do we expect you to have before you start reading?Thatdependsonwhatyouwanttogetoutofthebook.We’d hopethatevenifyouhaveonlyabasicgroundinginmodernalgebraic geometry and a slightly greater familiarity with the theory of a fixed algebraic curve, you could read through most of this book and get a senseofwhatthesubjectisabout:whatsortofquestionsweask,and some of the ways we go about answering them. If your ambition is to work in this area, of course, you’ll need to know more; a working knowledgewithmanyofthetopicscoveredinGeometryofalgebraic curves,I [7]firstandforemost.Wecouldcompilealengthylistofother subjectswithwhichsomeacquaintancewouldbehelpful.But,instead, we encourage you to just plunge ahead and fill in the background as needed;again,we’vetriedtowritethebookinastylethatmakessuch anapproachfeasible. Navigation In keeping with the informal aims of the book, we have used only two levels of numbering with arabic for chapters and capital letters for sections within each chapter. All labelled items in the book are numbered consecutively within each chapter: thus, the orderings of suchitemsbylabelandbypositioninthebookagree. There is a single index. However, its first page consists of a list of symbols, giving for each a single defining occurrence. These, and other,referencestosymbolsalsoappearinthemainbodyoftheindex wheretheyarealphabetized“asread”:forexample,referencestoM g willbefoundunderMgbar;toκiunderkappai.Boldfaceentriesinthe main body index point to the defining occurrence of the cited term. References to all the main results stated in the book can be found undertheheadingtheorems. ix Production acknowledgements This book was designed by the authors who provided Springer with the PostScript file from which the plates were produced. The type is averyslightlymodifiedversionoftheLucidafontfamilydesignedby ChuckBigelowandKristinHolmes.(Weaddedswashestoafewchar- acters in the \mathcal alphabet to make them easier to distinguish from the corresponding upper-case \mathit character. These alpha- betsareoftenpaired:a\mathcalcharacterisusedforthetotalspace ofafamilyandthe\mathitversionforanelement.)Itwascodedina customized version of the LATEX2e format and typeset using Blue Sky Research’s Textures TEX implementation with EPS figures created in Macromedia’sFreehand7illustrationprogram. A number of people helped us with the production of the book. Firstandforemost,wewanttothankGregLangmeadwhodidatruly wonderful job of producing an initial version of both the LATEX code andthefiguresfromourearlierWYSIWYGdrafts.DaveBayeroffered invaluable programming assistance in solving many problems. Most notably,hedevotedconsiderableefforttodevelopingasetofmacros foroverlayingtextgeneratedwithinTEXontofigures.Theseallowpre- cise one-time text placement independent of the scale of the figure and proved invaluable both in preparing the initial figures and in solving float placement problems. If you’re interested, you can ob- tain the macros, which work with all formats, by e-mailing Dave at [email protected]. Frank Ganz at Springer made a number of comments to improve the design and assisted in solving some of the formatting problems he raised. At various points, Donald Arseneau, Berthold Horn, Vin- centJalbyandSorinPopescuhelpedussolveorworkaroundvarious difficulties.Wearegratefultoallofthem. Lastly,wewishtothankourpatienteditor,InaLindemann,whowas neverinourwaybutalwaysreadytohelp. Mathematical acknowledgements YoushouldnothopetofindherethesequeltoGeometryofalgebraic curves, I [7] announced in the preface to that book. As we’ve already noted,ouraimisfarfromthe“comprehensiveandself-containedac- count” which was the goal of that book, and our text lacks its uni- formity.ThepromisedsecondvolumeisinpreparationbyEnricoAr- barello,MaurizioCornalbaandPhilGriffiths. A few years ago, these authors invited us to attempt to merge our then current manuscript into theirs. However, when the two sets of material were assembled, it became clear to everyone that ours was so far from meeting the standards set by the first volume that such amergermadelittlesense.Enrico,MaurizioandPhilthen,withtheir x usual generosity, agreed to allow us to withdraw from their project and to publish what we had written here. We cannot too strongly ac- knowledge our admiration for the kindness with which the partner- shipwasproposedandthegracewithwhichitwasdissolvednorour debttothemfortheinfluencetheirideashavehadonourunderstand- ingofcurvesandtheirmoduli. ThebookisbasedonnotesfromacoursetaughtatHarvardin1990, whenthesecondauthorwasvisiting,andwe’dliketothankHarvard Universityforprovidingthesupporttomakethispossible,andFord- hamUniversityforgrantingthesecondauthorboththeleaveforthis visit and a sabbatical leave in 1992-93. The comments of a number ofstudentswhoattendedtheHarvardcoursewereveryhelpfultous: in particular, we thank Dan Abramovich, Jean-Francois Burnol, Lucia Caporaso and James McKernan. We owe a particular debt to Angelo Vistoli, who also sat in on the course, and patiently answered many questionsaboutdeformationtheoryandalgebraicstacks. Therearemanyothersaswellwithwhomwe’vediscussedthevar- ious topics in this book, and whose insights are represented here. In additiontothosementionedalready,wethankespeciallyDavidEisen- bud,BillFultonandDavidGieseker. WetothankArmandBrumer,AntonDzhamay,CarelFaber,BillFul- ton, Rahul Pandharipande, Cris Poor, Sorin Popescu and Monserrat Teixidor i Bigas who volunteered to review parts of this book. Their comments enabled us to eliminate many errors and obscurities. For anythatremain,theresponsibilityisoursalone. Finally, we thank our respective teachers, Phil Griffiths and David Mumford. The beautiful results they proved and the encouragement they provided energized and transformed the study of algebraic curves—forusandformanyothers.Wegratefullydedicatethisbook tothem.