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1、ModuliofCurvesJoeHarrisIanMorrisonSpringerToPhilGriffithsandDavidMumfordPrefaceAimsTheaimofthisbookistoprovideaguidetoarichandfascinatingsub-ject:algebraiccurves,andhowtheyvaryinfamilies.Therevolutionthatthefieldofalgebraicgeometryhasundergonewiththeintroduc-tionofschemes,togetherwithnewi
2、deas,techniquesandviewpointsintroducedbyMumfordandothers,havemadeitpossibleforustounderstandthebehaviorofcurvesinwaysthatsimplywerenotpossi-bleahalf-centuryago.Thisinturnhasled,overthelastfewdecades,toaburstofactivityinthearea,resolvinglong-standingproblemsandgeneratingnewandunforeseenr
3、esultsandquestions.Wehopetoacquaintyoubothwiththeseresultsandwiththeideasthathavemadethempossible.Thebookisn’tintendedtobeadefinitivereference:thesubjectisdevelopingtoorapidlyforthattobeafeasiblegoal,evenifwehadtheexpertisenecessaryforthetask.Ourpreferencehasbeentofo-cusonexamplesandappl
4、icationsratherthanonfoundations.Whendiscussingtechniqueswe’vechosentosacrificeproofsofsome,evenbasic,results—particularlywherewecanprovideagoodreference—inordertoshowhowthemethodsareusedtostudymoduliofcurves.Likewise,weoftenproveresultsinspecialcaseswhichwefeelbringouttheimportantideaswi
5、thaminimumoftechnicalcomplication.Chapters1and2provideasynopsisofbasictheoremsandconjec-turesaboutHilbertschemesandmodulispacesofcurves,withfewornodetailsabouttechniquesorproofs.Usethemmoreasaguidetotheliteraturethanasaworkingmanual.Chapters3through6are,bycontrast,considerablymoreself-c
6、ontainedandapproachable.Ul-timately,ifyouwanttoinvestigatefullyanyofthetopicswediscuss,you’llhavetogobeyondthematerialhere;butyouwilllearnthetech-niquesfullyenough,andseeenoughcompleteproofs,thatwhenyoufinishasectionhereyou’llbeequippedtogoexploringonyourown.Ifyourgoalistoworkwithfamilie
7、sofcurves,we’dthereforesuggestthatyoubeginbyskimmingthefirsttwochaptersandthentacklethelaterchaptersindetail,referringbacktothefirsttwoasnecessary.viiiContentsAsforthecontentsofthebook:Chapters1and2arelargelyexposi-tory:forthemostpart,wediscussingeneraltermstheproblemsas-sociated