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1、ModuliofCurvesJoeHarrisIanMorrisonSpringerToPhilGriffithsandDavidMumfordPrefaceAimsTheaimofthisbookistoprovideaguidetoarichandfascinatingsub-ject:algebraiccurves,andhowtheyvaryinfamilies.Therevolutionthatthefieldofalgebraicgeometryhasundergonewiththeintroduc-tionofschemes,t
2、ogetherwithnewideas,techniquesandviewpointsintroducedbyMumfordandothers,havemadeitpossibleforustounderstandthebehaviorofcurvesinwaysthatsimplywerenotpossi-bleahalf-centuryago.Thisinturnhasled,overthelastfewdecades,toaburstofactivityinthearea,resolvinglong-standingproblems
3、andgeneratingnewandunforeseenresultsandquestions.Wehopetoacquaintyoubothwiththeseresultsandwiththeideasthathavemadethempossible.Thebookisn’tintendedtobeadefinitivereference:thesubjectisdevelopingtoorapidlyforthattobeafeasiblegoal,evenifwehadtheexpertisenecessaryforthetask.
4、Ourpreferencehasbeentofo-cusonexamplesandapplicationsratherthanonfoundations.Whendiscussingtechniqueswe’vechosentosacrificeproofsofsome,evenbasic,results—particularlywherewecanprovideagoodreference—inordertoshowhowthemethodsareusedtostudymoduliofcurves.Likewise,weoftenprov
5、eresultsinspecialcaseswhichwefeelbringouttheimportantideaswithaminimumoftechnicalcomplication.Chapters1and2provideasynopsisofbasictheoremsandconjec-turesaboutHilbertschemesandmodulispacesofcurves,withfewornodetailsabouttechniquesorproofs.Usethemmoreasaguidetotheliterature
6、thanasaworkingmanual.Chapters3through6are,bycontrast,considerablymoreself-containedandapproachable.Ul-timately,ifyouwanttoinvestigatefullyanyofthetopicswediscuss,you’llhavetogobeyondthematerialhere;butyouwilllearnthetech-niquesfullyenough,andseeenoughcompleteproofs,thatwh
7、enyoufinishasectionhereyou’llbeequippedtogoexploringonyourown.Ifyourgoalistoworkwithfamiliesofcurves,we’dthereforesuggestthatyoubeginbyskimmingthefirsttwochaptersandthentacklethelaterchaptersindetail,referringbacktothefirsttwoasnecessary.viiiContentsAsforthecontentsofthebook
8、:Chapters1and2arelargelyexposi-tory:forthemostpart,wediscussingeneraltermstheproblemsas-sociated