thatf[C]=andfD=P%ixiisarelativestablemap.Thespaceofsuchmorphismswas - PDF document

thatf[C]=andfD=P%ixiisarelativestablemap.Thespaceofsuchmorphismswascompactiedby[Ga]and[Li]andtheyusedthistodenerelativeGromov-Witteninvariants.Inthegenus0case,GathmanndenedthespaceofrelativestablemapsbytakingtheclosureinMg;n(X;)ofthemapsf:C!Xabove.Inthispaper,weshowthatthestackoftwistedstablemapsintoXD;rhasanopensubstackUg;n(XD;r;;~%)Kg;n(XD;r;)mappingisomorphicallyontothelocallyclosedsubstackofMg;n(X;)consistingofthosemaps.Hererisanyintegerlargerthanevery%i.Thisprovidesanalternatecompacticationofthespaceofrelativestablemaps.Oneadvantagetothisapproachisthatitworksingreatergenerality.WeonlyassumethatXisprojectiveoveraeldandthatDXisaneectiveCartierdivisor.Itisalsonecessarythatrisinvertibleinthebaseeld,butthisdoesnotconstraintheinitialdata.Sincerisnotxed,thismethoddoesnotproduceauniquecompactication.However,thecompacticationsformaninversesysteminlightof[AV,9.1.2].Thisrequiresfurtherinvestigation.Oneapplicationofthisideaistocountcurvesintheplanehavingcertaintangencyconditionswithrespecttoasmoothcubic.IfDP2Cisthecubic,thentheGromov-WitteninvariantsofP2D;2canbecomputedusingtheWDVVequations,asin[Ca1].Theinvariantscanthenbeusedtocountrationalcurveshavingacertainnumberoforder1and2contactswithDatbothspeciedandunspeciedpoints[Ca2].Insection2.2wedenetherthrootconstruction.Whilewearemostlyinterestedinapplyingittodivisors,itisjustaseasytoapplyittoapair(L;s)consistingofaninvertiblesheafLandaglobalsectionsofL.Inall,therearefournotationsusedforvariousrthrootconstructions:X(L;s;r),XD;~r,X(L;r),andXD;r.ThesearefoundinDenitions2.2.1,2.2.4,2.2.6,andjustbelowDenition2.2.4respectively.Insection2.3wegiveconditionsforX(L;s;r)tobeaDeligne-Mumfordstackanddiscusspresentationsandcoarsemodulispaces.Insection2.4,wegiveseveralexamples.ThemostimportanttheoremasidefromthemainresultisTheorem3.3.6.ForsmoothcurvesnotmappingintoD,itexhibitsabijectivecorrespondencebetweenordinarystablemapswithspeciedtangenciesatthemarkedpointsandtwistedstablemapshavingspeciedcontacttypesatthemarkedpoints.Thecontacttypeisafundamentalconceptforthiscorrespondence,andisintroducedaftertheproofofProposition3.3.3.Priortothis,weprovetwokeyresultsabouthowtherthrootconstructionaectsthePicardgroup,Corollaries3.1.2and3.1.3.Themainresult,whichprovestheisomorphismofstacksdescribedabove,isTheorem4.2.Thenalsectioncontainssomeimportantlemmaswhichareusedthroughout.NotationandconventionsWeworkoverSpecZ.Somebackgroundonstacksisrequired,andourreferenceis[LMB].Unfortunately,thisrequiresustoassumethatallschemesarequasi-separatedsothattherthrootconstructionproducesanalgebraicstackinthesenseof[LMB,4.1].Section3.1requiressomefamiliaritywithgroupschemes,notbeyondwhatisfoundinChapter2of[Ja].IfthereaderisonlyinterestedinapplicationsoverC,thenthisisunnecessary.Weusethefollowinggroupschemes.2 Forintuition,itmayhelptoviewamorphismX![A1=Gm]notasaprincipalGm-bundleoverXtogetherwithaGm-equivariantmaptoA1,butinsteadasaprincipalGm-bundleoverXtogetherwithasectionoftheassociatedA1-bundle.Thenthelemmafollowseasily.Notethatthesectionis0ifandonlyifthemorphismfactorsX!BGm[A1=Gm],whichrecoverstheequivalencebetweeninvertiblesheavesandmorphismsX!BGm.TheidentitymorphismBGm!BGmcorrespondstoaninvertiblesheafonBGmwhosepullbackunderanarbitrarymorphismX!BGmisisomorphictothecorrespondingin-vertiblesheafonX.WecallthisthetautologicalsheafonBGm.Notethatthetotalspaceofthecorrespondinggeometriclinebundleis[A1=Gm].2.2DenitionsLetXbeanalgebraicstack,letLbeaninvertiblesheafonX,lets2�(X;L),andletrbeapositiveinteger.Thepair(L;s)denesamorphismX![A1=Gm]asabove.Letr:[A1=Gm]![A1=Gm]bethemorphisminducedbyrthpowermapsonbothA1andGm.UndertheequivalenceofcategoriesinLemma2.1.1,rsendsapair(L;s)toitsrthtensorpower(Lr;sr).Recallthatthe2-categoryofalgebraicstacksisclosedunderberproducts[LMB,4.5].Denition2.2.1WedeneX(L;s;r)tobetheberproductX[A1=Gm];r[A1=Gm];andsaythatX(L;s;r)isobtainedfromXbytherthrootconstruction.Remark2.2.2Explicitly,ifXisascheme,thentheobjectsofX(L;s;r)overaschemeSarequadruples(f;M;t;');wheref:S!Xisamorphism,MisaninvertiblesheafonS,t2�(S;M),and':Mr!fLisanisomorphismsuchthat'(tr)=fs.Amorphismfrom(f;M;t;')(overS)to(g;N;u; )(overT)isapair(h;),whereh:S!Tisamorphismsuchthatgh=f,and:M!hNisanisomorphismsuchthat(t)=huandthefollowingdiagramcommutes.Mr' r// hNrh  fL=can:// hgLItiseasytogeneralizethiswhenXisastack.Remark2.2.3Fromthedenitionitisclearthatforanymorphismf:Y!X,wehaveanaturalisomorphismY(fL;fs;r)=X(L;s;r)XY.4 where':Lr!OSisanisomorphismofinvertiblesheaves,andwhosemorphisms(L;')!(M; )overSareisomorphismsL!Msothatthefollowingdiagramcommutes.Lr// ' Mr  OS OSOnBrwehaveaninvertiblesheafinducedbytheprojectionontothesecondfactorBr=SpecZBGmBGm!BGm:Thisisequivalenttothemorphisminducedbythenaturalinclusionr!Gm.AsthePicardgroupofBriscanonicallyidentiedwiththegroupofcharactersr!Gm[LMB,13.3.7],itfollowsthatthisinvertiblesheafisageneratorofthePicardgroupofBr.2.3BasicPropertiesWebeginourinvestigationoftherthrootconstructionwithaneasyspecialcase.Lemma2.3.1LetX=A1withcoordinatex.ThenX(OX;x;r)=[A1=r].Proof:ByRemark2.2.2,theobjectsofX(OX;x;r)onaschemeSareequivalenttotriples(M;t;'),whereMisaninvertiblesheafonS,tisaglobalsectionofM,and':Mr!OSisanisomorphism.Themorphismf:S!X(whichisthesameasasectionofOS)isdeterminedbythecondition'(tr)=f.Anobjectof[A1=r]overaschemeSisapairconsistingofaprincipalr-bundleandasectionoftheassociatedA1-bundle.FromExample2.2.7,thisisequivalenttosuchatriple(M;t;').Itiseasilyveriedthatmorphismsinthetwocategoriesarethesame.2ThefollowingtheoremshowsthatX(L;s;r)hasnitediagonaloverSpecZwheneverXdoes.Theorem2.3.2ThediagonalX(L;s;r)!X(L;s;r)XX(L;s;r)isnite.Proof:ThestatementislocalonX.Indeed,ifV!Xisasmooth,surjectivemorphismandifU=X(L;s;r)XV,thenweobtainthefollowingbersquarewhoseverticalarrowsaresmoothandsurjective.U // 2UVU // 2V X(L;s;r)// X(L;s;r)XX(L;s;r)// XSowemayassumethatXisaschemeandthatListrivial.ThenthemorphismX![A1=Gm]factorsthroughA1,soitsucestoassumeX=A1ands=x.ByLemma2.3.1,wehavereducedtoshowingthat[A1=r]!A1hasnitediagonal.Notethatthis6 Proof:Weusethefollowingdiagram,whichshouldbeviewedasacubewithonevertexmissing.X(L;s;r)//  2X [A1=Gm]r// 2[A1=Gm]PccHHHHHHHHHHH 2A1x7!xr// ::vvvvvvvvvA1ddIIIIIIIIIThemorphismsA1![A1=Gm]andP!XareprincipalGm-bundles.FillingintheremainingvertexwithA1[A1=Gm][A1=Gm][A1=Gm]Xmakeseveryfacea2-cartesiansquare,whichprovestherststatement.Thesecondstatementfollowssimilarly,replacing[A1=Gm]withBGmandA1withSpecZ.2RemarkItfollowsthatX(L;s;r)=[PA1A1=Gm]andX(L;r)=[P=Gm].ThisallowsonetoconstructsmoothpresentationsofX(L;s;r)andX(L;r)givenoneforX.Corollary2.3.6TheprojectionX(L;s;r)!Xisfaithfully atandquasi-compact.Proof:SincethisislocalonX,weassumeX=SpecAandListrivial,sos2A.WehaveapresentationSpecB!X(L;s;r),whereB=A[x;y;y�1]=(xr�sy),soitsucestoshowthatBisfaithfully atoverA.ClearlyA[y;y�1]isfaithfully atoverA,andBisafreeA[y;y�1]-moduleofniterank,hencefaithfully at.2RecallthatacoarsemodulispaceforanalgebraicstackXisanalgebraicspaceYtogetherwithamorphismX!YwhichisuniversalformorphismsfromXtoalgebraicspacesandinducesbijectionsX(k)=isom!Y(k)foranyalgebraicallyclosedeldk[AV,x2.2].Corollary2.3.7LetXbeaschemeandletL;s;rbeasusual.ThenXisacoarsemodulispaceforbothX(L;s;r)andX(L;r)undertheprojectionstoX.Proof:LetPbeasintheproposition.ItiseasytocheckthatXisageometricquotientofPA1A1byGm,andhenceacategoricalquotient.Likewise,XisageometricquotientofPbyGm.Apriori,thisonlymakesXacoarsemodulischeme.SinceX(L;s;r)!Xhasnitediagonal,itfollowsthatX(L;s;r)hasacoarsemodulispacewhichisseparatedoverX[LMB,19.1].By[LMB,A.2],itfollowsthatthecoarsemodulispaceisascheme,henceisomorphictoX.22.4ExamplesThroughoutthissubsectionweusethefollowingnotation.LetXbeascheme,letLbeaninvertiblesheafonX,letsbeaglobalsectionofLandletrbeapositiveinteger.8 Sincetherthrootconstructionbehaveswellunderbasechange,itfollowsthatXD;risthe atlimitofXC;rwhereCisasmoothconicapproachingD.SinceXC;randXD;(r;r)arebothsmoothstacks,theirGromov-Wittentheoriesarewell-dened,butoneshouldnotexpectthetheoriestoagree.Example2.4.6Any(smooth)complexorbicurvecanbeobtainedfromtherthrootcon-struction.LetXbeasmoothcurve,letD1;:::;DnbedistinctpointsofX,andletr1;:::;rnbeintegersgreaterthan1.Thenthecomplexorbicurve(X;D;~r)denedin[CR,2.2.2]isisomorphictoXD;~r.ThisfollowsfromthelocaldescriptionofExample2.4.1.Example2.4.7Thefollowingexamplewillappearthroughouttherestofthepaper.Let:C!SbeafamilyofcurvesoveraconnectedNoetherianschemeS(seetheintroductionforconventions)andsupposewehaveann-tupleDofdisjointCartierdivisorsDiCwhichmapisomorphicallytoS.Givenann-tupleofpositiveintegers~r=(r1;:::;rn),weobtainastackCD;~roverS.IftheintegersriareinvertibleonS,thisisafamilyoftwistedn-pointedcurvesoverSinthesenseof[AV,4.1.2].ThisiseasilyveriedusingCorollary2.3.4,Proposition3.0.1,andExamples2.4.1and2.4.3.Seealsothebeginningofsection4,whereitisshownthattheDiaredisjointfromthesingularlocusof.3MorphismsCD;~r!XD;rThegoalofthissectionistoclassifycertainmorphismsCD;~r!XD;r,whereCD;~risthetwistedcurveofExample2.4.7,Xisascheme,andDXisaneectiveCartierdivisor.Ingeneral,amorphismofstacksisdenedbyafunctorbetweentheunderlyingcategories.SuchafunctorwouldtakeanobjectofCD;~roveraschemeTtoaquadruple(f;M;t;')overT(seeRemark2.2.2).Itisequivalenttoclassifysuchquadruples(f;M;t;')overCD;~r,whichfollowsfrom[LMB,13.3.6].Itiseasytoclassifymorphismsf:CD;~r!X.Proposition3.0.1ThemorphismCD;~r!CmakesCthecoarsemodulispaceofCD;~r.Inparticular,anymorphismCD;~r!XcomesfromauniquemorphismC!X.Proof:Giventherststatement,thesecondfollowsfromtheuniversalpropertyofcoarsemodulispaces.Fortherst,sinceitislocalonCitsucestoconsiderthecaseofjustonedivisor.Indeed,sinceExample2.4.7assumesthatthedivisorsaredisjoint,onecancoverCbytheZariskiopensetsCn[j6=iDj;1in:ThecaseofasingledivisorwasdoneinCorollary2.3.7.210 3.ThecanonicalmorphismM!Misanisomorphism.Proof:WeclaimthatthesearelocalonX.Thesecondandthirdclearlyare.Fortherst,notethatgivenapresentationP!XandapresentationQ!PXX(L;s;r)oneobtainsapresentationQ!X(L;s;r),andapply[LMB,13.2.4].SowecanassumeX=SpecAandListrivialonX.ByExample2.4.1,X(L;s;r)=[SpecB=r],whereB=A[x]=(xr�s)andractsbyt
x=t�1xandt
a=afora2A.TheinvertiblesheafFon[SpecB=r]isequivalenttoaninvertibleB-moduleMtogetherwithanactionofronMwhichiscompatiblewiththeractiononB(e.g.,thisfollowsfromfpqcdescent[LMB,13.5.5]).AfterlocalizingonSpecA,wecanassumethatMisfreeasaB-module(thisfollowsfrom[Ei,Ex.4.13]).Sincerisdiagonalizable,MsplitsasadirectsumM=Mi2Z=(r)Mi;wheretheactionofronMiisgivenbyt
m=tim[Ja,x2.11].Bycompatibilityoftheraction,eachMiisanA-moduleandmultiplicationbyxsendsMitoMi�1.SinceMiisaprojectiveA-moduleandMisafreeB-module,itfollowsthatMiisinvertible(otherwisexr�1wouldannihilateM).BylocalizingagainonSpecA,wecanassumethateachMiisafreeA-module.Nowtherststatementreducestoshowingthatthenaturalmorphism :M0 AB!Misanisomorphism.By[Ei,4.4a],itsucestoshowthat issurjective.SincemultiplicationbyxonMisgraded,xMisagradedsubmoduleofM,soM=xMsplitsasadirectsumofMi=xMi+1.Thehypothesis(F)=0impliesthattheinducedr-actiononM=xMistrivial,soMi=xMi+1=0foralli6=0.Itfollowsthat issurjective,whichprovesstatement1.ThetautologicalsheafTisrepresentedbyBwithther-actiont
xi=t1�ix,andthetautologicalsectionisrepresentedbyx.Sostatement2reducestoshowingthatmul-tiplicationbyxkisanisomorphismfromM0toM�k.ThisfollowsfromthefactthatMi=xMi+1=0fori6=0.Forstatement3,notethatgivenanA-moduleNrepresentingM,NisrepresentedbyN AB,withNbeingther-invariantsubmodule.2Corollary3.1.2AssumethatZisconnectedandnonemptyandletFbeaninvertiblesheafonX(L;s;r).ThenthereisaninvertiblesheafMonXandanintegerksuchthat0krandF=M Tk:Moreover,kisuniqueandMisuniqueuptoisomorphism.Proof:Wedenektobetheintegerin[0;r�1]representedby(F).Byconstructionof,F T�ksatisesthehypothesisofTheorem3.1.1.LetM=(F T�k).Theorem12 Proof:Theconclusionisobviouswhenri=1,soassumeri�1.ByCorollary3.2.1,wehaveadecompositionGT0i= L QTkii,andbyCorollary3.2.2,thepullbackofthetautologicalsection0igoestoasectionoftheform sQkii.RaisingbothsidestothepowerriandusingtheuniquenessstatementofCorollary3.2.1andthefactthatg(Di)=Ei,itfollowsthatOC(Di)=Lri OC(Pj`jDj)byanisomorphismsendingsDitosriQjs`jDjwhere`j=rikj=rj.Bycomparingvanishingloci,itfollowsthatkj=0forj6=i.SincethegeometricbersofarenodalcurveswhoseintersectionwithDiisasinglesmoothpoint,itfollowsthatsdoesnotvanishatanypointofCandkiequals1.ThusListrivialandGT0i=Ti.2RemarkItfollowsthatanytwomorphismsCD;~r!BE;~rovergare2-isomorphic.3.3ClassicationFixaschemeX,aneectiveCartierdivisorDX,andapositiveintegerr.WearenowreadytoclassifycertainmorphismsCD;~r!XD;r(hopefullythetwousesoftheletterrwillnotcauseconfusion).Let :CD;~r!CbetheprojectionandxamorphismF:CD;~r!XD;r.ByProposition3.0.1,everymorphismCD;~r!Xisoftheformf forauniquemorphismf:C!X.ThusthemorphismFisgivenbyaquadruple(f;M;t;'),wheref:C!Xisamorphism,MisaninvertiblesheafonCD;~rwithglobalsectiont,and':Mr! fOX(D)isanisomorphismsuchthat'(tr)= f(sD).CD;~rF//  XD;r Cf// X(3.3.1)ByCorollary3.2.1,theinvertiblesheafMuniquelydeterminespositiveintegerskiwith0kiri�1andaninvertiblesheafLonCsothatM= L QTkii.Weneedacriterionforamorphismtoberepresentable.Thefollowinglemmaisaconsequenceof[Co,2.2.5-7],andforDeligne-Mumfordstacksaproofcanbefoundin[AV,4.4.3].Firstwexsomenotation.Ify:Speck!YisageometricpointofanalgebraicstackY,letGybethegroupschemeoverkgivenbyGy(R)=f2�automorphismsofSpecR!Speck!Ygforanyk-algebraR.Thiscanbethoughtofastheautomorphismgroupschemeofy,andforaDeligne-MumfordstackY,itissimplythe2-automorphismgroupofy.Lemma3.3.2AmorphismofalgebraicstacksY!Zisrepresentableifandonlyifforeveryalgebraicallyclosedeldkandeverymorphismy:Speck!Ywithinducedmorphismz:Speck!Z,theinducedmorphismGy!Gzisinjective.Proposition3.3.3ThemorphismFisrepresentableifandonlyifforeveryi,ridividesrandkiisrelativelyprimetori.14 Theorem3.3.6Let:C!SbeasmoothfamilyofcurvesoveraconnectedNoetherianbaseS,andletDiC,i=1;:::;n,beann-tupleofdisjointeectiveCartierdivisorswhichmapisomorphicallyontoS.Fixann-tupleofpositiveintegers~%,andlet~rbedeterminedfrom~%by(3.3.5).Givenamorphismf:C!Xsuchthat1.noberofmapsintoDand2.thereexistsaneectiveCartierdivisorZC(necessarilyuniquebyLemma5.3)suchthatfD=rZ+P%iDi,thereisaunique(upto2-isomorphism)representablemorphismCD;~r!XD;rofcontacttype~%whichmakesdiagram3.3.1commute.Conversely,anyrepresentablemorphismF:CD;~r!XD;rofcontacttype~%suchthatF�1Gdoesnotcontainanyberscomesfromsuchanf.HereGisthegerbeofXD;r(Denition2.4.4).Proof:Itremainstoproveonlytherstpart.ArepresentablemorphismF:CD;~r!XD;r,ifitexisted,wouldbegivenbyaquadruple(f;M;t;')asabove.Wewanttoshowthatthereisonlyonesuchquadrupleuptoisomorphism.Themorphismf:C!Xisalreadygiven.LetL=O(Z)andletsbethecanonicalsectionofLvanishingonZ.LetM= L QTkii,andlett= sQkii.Thencondition2impliesthatthereisanisomorphism':Mr! fO(D)sendingtrto fsD.ThisdenesamorphismFofcontacttype~%.Ifwechooseadierentquadruple(f;N;u; )givingrisetoamorphismofcontacttype~%,thenwehavealreadyshownthatthereisaninvertiblesheafL0onCandasections0sothatN= L0 QTkiiandu= s0Qkii.IfZ0Cisthevanishinglocusofs0,thenLemma5.3impliesthatZ=Z0,sothereisanisomorphismL!L0sendingstos0.ThisinducesanisomorphismM!Nsendingttou,anditnecessarilysends'to sincetanduarenonzeroonadenseopenset.24TwistedstablemapsInthissection,wecomparetwistedstablemapsintoXD;rwithordinarystablemapsintoX.WeassumethatXisprojectiveoveraeldkandthatrisinvertibleink.Webeginwithanobservationaboutnodaln-pointedcurvesoveraNoetherianschemeS.Inthestandarddenition,onehasa atmorphism:C!Swhosegeometricbersarenodalcurvestogetherwithnsectionsi:S!Cwhoseimagesaredisjointandwhichdon'tpassthroughanysingularpointsofthebers.ItisequivalenttoreplacethensectionsiwithndisjointeectiveCartierdivisorsDiCwhichmapisomorphicallytoS.Indeed,anysectionofaseparatedmorphismisaclosedembedding(thisfollowsfrom[Gr2,8.11.5]),andgivenasectionwhichdoesn'tpassthroughanynodes,itfollowsfromLemma5.1thatitsimageisaneectiveCartierdivisor.Conversely,ifitiseectiveCartierandmapsisomorphicallyontoS,thenitsrestrictiontoeachberisapointdenedbyasingleequation,soitcan'tbeanodeoftheber.Therefore,weconsideranodaln-pointedcurveoverStobegivenbythedata(:C!S;D1;:::;Dn).16 LetUg;n(XD;r;)Kg;n(XD;r;)betheopensubstackconsistingofstablemapsfromsmoothtwistedcurveswhichdonotmapintothegerbeGXD;rofDenition2.4.4.Toseethatthisisopen,itsucestoseethatforanyfamily(:C!S;1;:::;n;F:C!XD;r)ofstablemapsintoXD;r,thesetofs2SsuchthattheberCsissmoothanddoesnotmapintoGisopen.Thisisaneasyconsequenceofthefactthatisbothopenandclosed.ByTheorem4.1,anobjectofUg;n(XD;r;)overaconnectedschemeSisequivalenttothedata(:C!S;D;~r;F:CD;~r!XD;r),where(:C!S;D)isasmoothn-markedcurveoverS,~risann-tupleofpositiveintegers,andFisastablemorphismsuchthatnoberofmapsintoDundertheinducedmorphismf:C!X.ForFtobestablemeansthatitisrepresentableandfisstable(asamorphismfromann-markedcurveintoX).WewillthereforedenoteobjectsofUg;n(XD;r;)by(:C!S;D;~r;F).Equation3.3.4associatestoanysuchmorphismFann-tupleofintegerscalledthecontacttype.ItfollowsfromLemma3.2.3thatthecontacttypedenesnlocallyconstantfunctionsonthemodulistackofstablemaps.LetUg;n(XD;r;;~%)Ug;n(XD;r;)betheopenandclosedsubstackconsistingofstablemapswhichhavecontacttype~%.LetVg;n(X;)Kg;n(X;)betheopensubstackconsistingofstablemapsfromsmoothcurveswhichdonotmapintoD.BytheremarkfollowingLemma5.2,foranystablemap(:C!S;D;f)inVg;n(X;),f�1DisaneectiveCartierdivisor(whichwedenotefD).LetVg;n(X;;~%)bethestackwhoseobjectsoveraschemeSarequadruples(:C!S;D;f;Z),where(:C!S;D;f)isanobjectofVg;n(X;)andZCisaneectiveCartierdivisorsuchthatfD=rZ+P%iDi,andwhosemorphismsaremorphismsofstablemapswhichpreserveZ.WehaveamorphismVg;n(X;;~%)!Vg;n(X;)andweclaimthatthisisaclosedembedding.Let(:C!S;D;f)beanobjectofthesecondstack,andletTbetheberproduct,asinthediagrambelow.Vg;n(X;;~%)2// Vg;n(X;)TOO //  2SOO  Hilbd0C=Sh// HilbdC=SLetd=D
andletd0satisfyd=rd0+P%i,whichisanintegersinceweassumed~%tobeadmissible.ThestackTisisomorphictothestackwhoseobjectsoveraschemeUarepairs(g;Z),whereg:U!SisamorphismandZgCisaneectiveCartierdivisorsuchthat~gfD=rZ+P%i~gDi.Here~g:gC!Cistheprojection.SuchaZisnecessarily atoverSbyLemma5.2,soitdenesamorphismfromTtotheHilbertfunctorHilbd0C=Sparametrizinglengthd0subschemesofthebersofC!S.SinceXisprojectiveandf:C!Xisstable,CisprojectiveoverS.Inthissituation,theHilbertfunctorisrepresentablebyaschemewhichisprojectiveoverS[Gr1,3.2].Wehaveamorphismh:Hilbd0C=S!HilbdC=SsendingZtorZ+P%iDiwhichmakesabersquareasinthe18 Proof:WerstshowthattheidealsheafofDislocallygeneratedbyasingleelement.Forthis,itsucestoassumethatS=SpecAandX=SpecB,whereAandBarelocalringswithmaximalidealsmandM.LetIbetheidealofD.Wehaveanexactsequence.0// I// B// B=I// 0SincejDis at,Tor1A(B=I;A=m)=0,sothisremainsexactaftertensoringwithA=m.ThusI=mIistheidealoftherestrictionofDtoSpecB=mB.Thisisgeneratedbyasingleelementbyhypothesis,soI=MIisalso,andNakayama'sLemmaimpliesthatIisalsogeneratedbyasingleelement.WeclaimthatanylocalgeneratoroftheidealsheafofDisanonzerodivisor.Ifnot,thenthereisapointx2DwhichisanassociatedpointofX,meaningthatthemaximalidealofitslocalringconsistsofzerodivisors.Lets=(x).Sinceis at,itfollowsthatxisanassociatedpointofXs.Forexample,thisfollowsfromthefactthatdepthisadditivefor atmorphisms[Ma,23.3].ButthiscontradictsthehypothesisthatDsXsisaneectiveCartierdivisor.2Lemma5.2Let:X!Sbea atmorphismofNoetherianschemesallofwhosebersareintegral.LetDXbeaclosedsubschemewhoseidealsheafislocallygeneratedbyasingleelementandwhichdoesnotcontainanyberof.ThenDis atoverS,andinparticular,DXisaneectiveCartierdivisor.Proof:ThatDXiseectiveCartierfollowsfromLemma5.1giventhatDis atoverS.Toshow atness,wereducetothelocalcaseandusethenotationfromLemma5.1.Bythelocalcriterion[Ei,6.8]and atnessof,weneedonlyshowthatI=mI!B=mBisinjective.WearegiventhatIisgeneratedbyasingleelementf,soitsucestoshowthatfg2mBifandonlyifg2mB.ButthisfollowssincemBisaprimeidealwhichdoesnotcontainf.2RemarkOneapplicationofLemma5.2isthefollowing.LetSbeaNoetherianscheme,let:C!Sbeasmoothmorphismwithconnectedbers,letf:C!XbeamorphismtoaschemeX,andletDXbeaneectiveCartierdivisor.IfnoberofmapsintoD,thenf�1DisaneectiveCartierdivisor.Lemma5.3LetSbeaNoetherianscheme,let:C!Sbeasmoothmorphismofrelativedimension1withgeometricallyconnectedbers,letrbeapositiveintegerwhichisinvertibleonS,andletZCbeaneectiveCartierdivisorwhichdoesnotcontainanyberof.IfZ1andZ2aretwoeectiveCartierdivisorsonCsuchthatrZ1=rZ2=Z,thenZ1=Z2.Proof:FirstnotethatitsucestoprovethelemmaafterabasechangeS0!SwhereS0isthespectrumofanArtinianlocalring.Indeed,ifZ16=Z2,thenthereisapointp2CsuchthattheidealsofZ1andZ2dierinOp.ThisimpliesthattheydierinthecompletioncOp,sotheydierinsomequotientOp=mk,wheremOpisthemaximalideal.SoZ1andZ2woulddierafterthebasechangeSpecOp=mk!S.20 [Gr2]A.Grothendieck,ElementsdegeometriealgebriqueIV,PublicationsMathematiquesdel'I.H.E.S.24,28(1965-6).[Ja]J.C.Jantzen,RepresentationsofAlgebraicGroups,Amer.Math.Soc.,2003.[LMB]G.LaumonandL.Moret-Bailly,Champsalgebriques,Springer-Verlag,2000.[Li]J.Li,Stablemorphismstosingularschemesandrelativestablemorphisms,J.Dif-ferentialGeom.57(2001)509{578.[MO]K.MatsukiandM.Olsson,Kawamata-ViehwegvanishingasKodairavanishingforstacks,Math.Res.Lett.12(2005)207{218.[Ma]H.Matsumura,CommutativeRingTheory,CambridgeUniv.Press,1986.[Ol1]M.Olsson,Logarithmicgeometryandalgebraicstacks,Ann.Sci.EcoleNorm.Sup.36(2003)747{791.[Ol2]M.Olsson,On(log)twistedcurves,preprint.UniversityofMichigan2074EastHallAnnArbor,MI48109-1043cdcadman@umich.edu22