Universal relations on stable map spaces in genus

arXiv:math/0607431v2[math.AG]12Aug2006UNIVERSALRELATIONSONSTABLEMAPSPACESINGENUSZEROANCAM.MUSTAT¸ˇAANDANDREIMUSTAT¸ˇAAbstract.Weintroduceafactorizationforthemapbetweenmodulispacesofstablemapswhichforgetsonemarkedpoint.Thisleadstoastudyofuniversalrelationsinthecohomologyofstablemapspacesingenuszero.IntroductionThemodulispacesofstablemapsM0,m(X,d)provideexamplesofDeligne-Mumfordstackswhoseintersectiontheoryisbothaccessibleandinteresting,asindicatedbytheconsiderablesuccessofGromov-Wittentheoryingenus0.Whenthetargetisapoint,M0,misasmoothprojectivevarietyanditscohomologyringhasbeencomputedbyKeel([Ke]).RecentstudiesleadtoamorecomprehensiveviewofthecohomologyandChowgroupsofthesemodulispacesforothertargets:[BO],[GP],[O1],[C1],[C2],[MM1],[MM2].Letd∈H2(X)beacurveclassonasmoothprojectivevarietyX.ThespaceM0,0(X,d)parametrizesmapsfromrationalsmoothornodalcurvesintoXwithimageclassd,suchthatanycontractedcomponentcontainsatleast3nodes.OverM0,0(X,d)thereexistsatowerofmodulispacesofstablemapswithmarkedpointsandmorphismsf:M0,m+1(X,d)→M0,m(X,d)forgettingonemarkedpoint,suchthatM0,m+1(X,d)istheuniversalfamilyoverM0,m(X,d).Inthispaperweintroduceafactorizationoftheforgetfulmapf,whichgraduallycontractspartoftheboundary.ThisallowsadetailedstudyofthecohomologyandChowringsofM0,m+1(X,d)asalgebrasovertheringsofM0,m(X,d).Wefindaseriesofuniversalrelationsoverfamiliesofstablemaps.(Theorems3.3and3.5intext).Thispaperisthethirdinaseriesdedicatedtotheintersectionringsofthesemodulispaces.PreviouslywehavefoundpresentationsfortheChowringsM0,m(Pn,d)forallm0([MM1],[MM2]).Thecasem=0seemedlessaccessiblefromourpointofview,asinasenseM0,m(Pn,d)hasmorestructurewhenm0.AgoodparallelisinthestudyoftheintersectionringforcompleteflagvarietiesversusthatoftheGrassmannian.Infact,whenDate:February2,2008.12ANCAM.MUSTAT¸ˇAANDANDREIMUSTAT¸ˇAd=1,M0,0(Pn,1)=Grass(P1,Pn),whileM0,1(Pn,1)isaflagvariety.Thissuggestedanindirectapproach,understandingH∗(M0,0(Pn,d))bystudyingtheextensionofalgebrasH∗(M0,0(Pn,d))→H∗(M0,1(Pn,d)).WeviewthisstepasprototypicalforstudiesofChowquotientsbySL2–actionfromanintersection–theoreticalpointofview.ThedescriptionofH∗(M0,0(Pn,d))iscompletedin[MM3].TherelationtoChowquotientsinthesenseof[Ka]isasfollows.WemayregardthecoarseschemeM0,0(X,d)astheSL2–ChowquotientofasimplercompactificationforthespaceofmapsP1→X×P1withimageclass(d,1):thespaceΣdXofquasi-mapsofX,alsoknownasthe”linearsigmamodel”.ThisistrueatleastforXconvex.LetαbetheclassofthegenericSL2–orbitinΣdX.TheChowquotientofΣdXisnaturallyasubvarietyoftheChowvarietyChow(ΣdX,α).ThegraphspaceU:=M0,0(X×P1,(d,1))isthenthe”universalfamily”overM0,0(X,d)inthesenseoftheabove,admittingacanonicalmapintoΣdX.Moreover,intermediatespacesbetweenUandΣdX,asdescribedin[MM1],appearnaturallywhenoneconsidersthegeometryoftheSL2–action.Finally,ifBbetheBorelsubgroupofSL2,thenM0,1(X,d)isthequotientU/B.OurstudymaythusbeviewedasaprocedureforunderstandingtheintersectiontheoryofSL2–ChowquotientsY/SL2foraprojectivevarietyY,undersuitableassumptionsonthestabilizers.ThefirststepregardstherelationbetweenthecohomologyofYandthatofthequotientU/B,asin[MM1].ThesecondstepconcernsthemorphismU/B→Y/SL2,asinthispaper.[MM4]considersotherapplicationsofthisviewpoint.Anothermotivationforourstudycomesfromthemorecomplexsituationofhighergenuscurves.Letg1.Thenbymethodssimilartothosepresentedinsection1ofthispaper,themorphismMg,1(Pn,d)→Mg,0(Pn,d)×Mg,0Mg,1factorsinaseriesofbirationalmorphismswithrationalexceptionalfibers.Onemayextract,forexample,thePicardgroupforthebasisintermsofthatofthetarget.Thelineofinquiryopenedinthispaperpartiallyappliestothemapsabove.Wenotethatsomeoftheexceptionallocihavecodimensiongreaterthan1,however,thissituationisremediedwhenrestrictingovereachofthestratainthenaturalstratificationMg,0=SMγ.HereγdenotestablegraphsandMγareopensetsofclosedsubstrataMγ,representingspecificsplittingtypesofagenusgcurve.Thecaseg=1isspecialandrequiresseparatetreatment.Thecomputationsinthispaperrelyontwotechnicaltoolsintroducedin[MM1],[MM2].OnetoolistheintersectionringB∗associatedtoanetworkoflocalregularembeddings,motivatedbythestructureofan´etalecoverofthetargetstack.Inthecaseofthemodulispaceofstablemaps,themainadvantageofB∗overtheusualintersectionringliesintheoverallsimplificationofintersectionontheboundary.Indeed,thereisanaturalUNIVERSALRELATIONSONSTABLEMAPSPACESINGENUSZERO3stratificationofM0,m(X,d),indexedbystabletreeswithdegreeandmarkingdecorations.IncontrasttoH∗(M0,m(X,d)),B∗allowstheclassesofeachofthesestratatobedecomposedasapolynomialofdivisorclassesinanaturalway.Thesecondtoolisasetofstabilityconditionsformapstoprojectivevari-etieswhichengendervariousspacesbirationaltoMg,m(X,d),viamorphismsthatcontracttheboundaryofMg,m(X,d).WecallthesetheintermediatespacesofMg,m(X,d).ThefiberedproductoftheuniversalfamiliesoverthesenewspaceswithMg,m(X,d)interpolatesbetweenMg,m+1(X,d)andMg,m(X,d).ThesequenceofspacesbirationaltoM0,0(Pn,d)asaboveisalsocon-structedbyParker([Par])viaGITquotientsundertheSL2–action.Indeed,heregardsM0,0(Pn,d)asaGITquotientofthegraphspaceM0,0(Pn×P1,(