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1、ToPhilGriffithsandDavidMumfordPrefaceAimsTheaimofthisbookistoprovideaguidetoarichandfascinatingsub-ject:algebraiccurves,andhowtheyvaryinfamilies.Therevolutionthatthefieldofalgebraicgeometryhasundergonewiththeintroduc-tionofschemes,togetherwithnewideas,tech
2、niquesandviewpointsintroducedbyMumfordandothers,havemadeitpossibleforustounderstandthebehaviorofcurvesinwaysthatsimplywerenotpossi-bleahalf-centuryago.Thisinturnhasled,overthelastfewdecades,toaburstofactivityinthearea,resolvinglong-standingproblemsandgen
3、eratingnewandunforeseenresultsandquestions.Wehopetoacquaintyoubothwiththeseresultsandwiththeideasthathavemadethempossible.Thebookisn’tintendedtobeadefinitivereference:thesubjectisdevelopingtoorapidlyforthattobeafeasiblegoal,evenifwehadtheexpertisenecessar
4、yforthetask.Ourpreferencehasbeentofo-cusonexamplesandapplicationsratherthanonfoundations.Whendiscussingtechniqueswe’vechosentosacrificeproofsofsome,evenbasic,results—particularlywherewecanprovideagoodreference—inordertoshowhowthemethodsareusedtostudymodul
5、iofcurves.Likewise,weoftenproveresultsinspecialcaseswhichwefeelbringouttheimportantideaswithaminimumoftechnicalcomplication.Chapters1and2provideasynopsisofbasictheoremsandconjec-turesaboutHilbertschemesandmodulispacesofcurves,withfewornodetailsabouttechn
6、iquesorproofs.Usethemmoreasaguidetotheliteraturethanasaworkingmanual.Chapters3through6are,bycontrast,considerablymoreself-containedandapproachable.Ul-timately,ifyouwanttoinvestigatefullyanyofthetopicswediscuss,you’llhavetogobeyondthematerialhere;butyouwi
7、lllearnthetech-niquesfullyenough,andseeenoughcompleteproofs,thatwhenyoufinishasectionhereyou’llbeequippedtogoexploringonyourown.Ifyourgoalistoworkwithfamiliesofcurves,we’dthereforesuggestthatyoubeginbyskimmingthefirsttwochaptersandthentacklethelaterchapter
8、sindetail,referringbacktothefirsttwoasnecessary.viiiContentsAsforthecontentsofthebook:Chapters1and2arelargelyexposi-tory:forthemostpart,wediscussingeneraltermstheproblemsas-sociatedwithmoduliandparameterspacesofcurves,what’
