Arithmetically CohenMacaulay algebraic curves Marco Fr - PDF document

issurjective.Hissaidtobenormallygeneratedifthemapskaresurjectiveforallk2N.ThiscorrespondstosaythatthegradedringR(C;H)=Mk0H0(C;H k)isgeneratedindegree1.WeremarkthatifHisampleonCthenHturnsouttobenormallygeneratedifandonlyifHisveryampleandtheassociatedembeddedscheme'jHj(C)P(H0(C;H)_)isarithmeticallyCohen{Macaulay,i.e.,itshomogeneouscoordinateringisaCohen{Macaulyring.OurrstresultisthefollowingTheorem1.LetCbeanumericallyconnectedcurvecontainedinasmoothalgebraicsurfaceandletHnum!C A,withAanampleinvertiblesheafsuchthatdegHjB2pa(B)+18subcurveBCThenHisnormallygeneratedonCIn[5]itwasshownthattheabovetheoremistrueexceptinsomeexcep-tionalcases.InthepresentpaperweshowthatalsointheseparticularcasesHisnormallygenerated.TheproofofthetheoremrelyonthetheoryofKoszulcohomologydevelopedbyGreenin[6]andtheclassical\Mumford'sargu-ment"for1-dimensionCohen{Macaulayprojectivescheme(seeLemma5),viaadetailedanalysisofnumericallyconnectedcurveswithmultiplecomponents.ThestudyoftheringR(C;H)foranumericallyconnectedcurvehasmanyapplicationsinseveralaspectsofthetheoryofalgebraicsurfaces.Forinstanceifoneconsidertheresolutionofanormalsurfacesingularity:S!Xortherelativecanonicalalgebraofabrationf:S!B(cf.[5]).Anotherapplicationofthesekindofresultscanbefoundintheanalysisofpluricanonicalmapsofalgebraicsurfaceofgeneraltype.Indeed,simplybyrestrictiontodivisorsinanappropriatelinearsystemwecangetfurtherinformationonthedegreeofthegeneratorsofthecanonicalringsorontheprojectivenormalityofasurfaceembeddedbyapluricanonicalsystem.Inparticulartheabovetheorem1turnsouttobeparticularusefulinthespecialsituationswhereitisimpossibletoconsidertheoremsofBertinitype.OurresultonpluricanonicalembeddingsisthefollowingTheorem2.SupposethatSisasmoothsurfaceofgeneraltypewithKSample.Assume(KS)23,pg(S)=h0(S;KS)2andq=h1(S;OS)=0.ThenK 3SisnormallygeneratedonS.Weremarkthat,withmildassumptions,theveryamplenessofK 3Shasbeenprovedin[2](see[2,x5]fortheproof).Herethehypothesespg(S)22 Theorem4.AssumeC=Psi=1ni�itobeacurvecontainedinsmoothalgebraicsurface.Letd=(d1;:::;ds)2NsbesuchthatforeachinvertiblesheafF0ofmultidegreedwehavedegF0jBpa(B)+18subcurveBCThenfor[F]generalinPicd(C),jFjisabase-pointfreesystem.Foraproofoftheabovetheoremssee[5,Thm.3.1,3.2].2.3Mumford'sargumentToprovethesurjectionofacertainmultiplicationmapMumford'sargumentconsistsinndingausefulsubsheafandthenanalyzingthenaturaldecompo-sitionwhichcomeout.Proposition5(Mumford'sargument).LetL;HbeinvertiblesheavesonacurveCwithjHjandjLjbasepointfreesystems.LetF=H(�
)(
a0-dimensionalscheme)beaninvertiblesubsheafofHsothatjFjisabasepointfreesystemonC.Assumefurthermorethatthesequence0!F!H!O
!0isexactonglobalsectionsandthatthemultiplicationmapp1:H0(F) H0(L)!H0(F L)isonto.Thenwehaveasurjectionr1:H0(H) H0(L)H0(H L)Foraproofsee[9,Thm.6].ApplyingProp.8andThm.3{4asacorollaryofwegetthefollowing(see.[5,Thm.A])Theorem6.LetHbeaninvertiblesheafonCsuchthatHnumF G,whereF;GareinvertiblesheavessuchthatdegGjBpa(B)8subcurveBC(1)degFjBpa(B)+18subcurveBC(2)Thenthenaturalmultiplicationmap(H0(H)) 2!H0(H 2)isonto.Moreover,ifCisnumericallyconnectedthenHisnormallygeneratedonC.3Koszulcohomologygroupsofalgebraiccurves3.1DenitionandbasicresultsInthissectionwerecallthenotionofKoszulcohomologygroupsasintroducedanddevelopedbyGreenin[6],andwefocusonsomeapplicationsofKoszul4 Proposition7(Duality).LetWjFjbeabasepointfreesystemofdimensionr.ThenKp;q(C;W;H;F)d Kr�p�1;2�q(C;W;!C H�1;F)(whered meansdualityofvectorspace).Foraproofsee[5,Prop.1.4].Proposition8(H0-Lemma).LetF,HbeinvertiblesheavesonCandassumeWH0(C;F)tobeasubspaceofdim=r+1whichyieldsabasepointfreesystem.Ifeither(i)H1(C;H F�1)=0,or(ii)Cisnumericallyconnected,!C=H F�1andr2,or(iii)Cisnumericallyconnected,h0(C;!C H�1 F)r�1andthereexistsareducedsubcurveBCsuchthat:W,!WjB,H0(C;!C H�1 F),!H0(B;!C H�1 F),everynon{zerosectionofH0(C;!C H�1 F)doesnotvanishidenticallyonanycomponentofB;thenK0;1(C;W;H;F)=0,thatis,themultiplicationmapW H0(C;H)!H0(C;F H)issurjective.Proof.BydualityweneedtoprovethatKr�1;1(C;W;!C H�1;F)=0:Tothisaimletfs0;:::;srgbeabasisforWandlet=Psi1^si2^:::^sir�1 i1i2:::ir�12Vr�1W H0(C;!C H�1 F)beanelementintheKerneloftheKoszulmapdr�1;1.Incases(i)and(ii)obviously=0(see[5]fordetails).InthelattercasebyourassumptionswecanrestricttothecurveB.SinceBisreducedwecanchooser+1\sucientlygeneralpoints"onBsothatsj(Pi)=ij.Butthen2ker(dr�1;1)impliesforeverymultiindexI=fi1;:::ir�2gthefollowingequations(uptosign)j1i1:::ir�2
sj1+j2i1:::ir�2
sj2+j3i1:::ir�2sj3=0:(wherefi1;:::ir�2g[fj1;j2;j3g=f0;:::;r+1g).6 Denition11.LetC=Psi=1ni�ibeanumericallyconnectedcurvecontainedinasmoothalgebraicsurface.ThenCissaidtobeevenifdeg!Cj�iiseven8irreducible�iC(thisisequivalenttosay�i
(C��i)even8i.)Evencurvesandevendivisorsappearforinstanceifyouconsiderthecanoni-calsystemjKSjforasurfaceSofgeneraltypeandmayhaveusefulapplicationstotheanalysisofpluricanonicalmapsofalgebraicsurfaces.Indeed,byadjunc-tion,foreverycurveC2jKSjwehavej(2KS)jCj=jKCj,thatiseverycurveinthecanonicalsystemiseven.4.1CombinatorialpropertiesofevencurvesNowweanalyzesomeusefulcombinatorialpropertiesofnumericallyconnectedevencurves.FirstofallwehaveRemark12.LetC=Psi=1ni�ibeanumericallyconnectedevencurve.ThenthereexistsaninvertiblesheafGofmultidegree=(1;:::;s)suchthatG 2num!C:ThefollowingtechnicallemmaswillturnouttobeusefulintheanalysisoftheparticularcasesofTheorem1.Lemma13.LetC=Psi=1ni�ibeanumericallyconnectedcurvecon-tainedinasmoothalgebraicsurface.SupposethatthereexistsaninvertiblesheafGofmultidegree=(1;:::;s)suchthatG 2num!C:ThenforageneraleectivedivisorOC(
)inPiceverynon{zerosectioninH0(C;OC(
))doesnotvanishidenticallyonanypropersubcurveofC.Proof.ThelemmafollowssinceforeverydecompositionC=A+Bwehave0!H0(B;OB(
) OB(�A))!H0(C;OC(
))!H0(A;OA(
))!0andbyadegreeargumentitisH0(B;OB(
) OB(�A))=0.IndeedbydualityandadjunctionitisequivalenttoprovethatH1(B;!C OB(
)�1)=0.ThisfollowsbyThm.3sincebynumericallyconnectednessdeg(!C OC(
)�1)jB0=1 2deg!CjB0pa(B0)foreveryB0B. Finallyweconsidertheparticularsituationwhere�
(C��)=2foreveryirreduciblecomponent�8 Sinceforeveryiwehave�i(C��i)=2thisinequalitycanbereadaskXi=1�i
(C��i)�kXj=1j6=i�j=kXi=12�kXj=1j6=i�i
�j2whichisequivalenttokXi;j=1j6=i2�i
�j2k�2i.e.,8i=1;:::;k9uniquej�is.t.�i
�j=1andequalitiesholdthroughout.ThisexactlymeansthatBandthenCredarechainsofirreducibleandreducedcomponents,suchthatCred
B=2.InparticularBandCredarenumericallyconnectedsinceCis2-connected(cf.[4]).Thelastassertionfollowssinceforeveryiwehave�i(C��i)=2. 4.2NumericallyconnectedevenirreduciblecurvesNextwefocalizeonirreduciblebutnonreducedcurves.Lemma15.LetC=3�beanirreduciblebutreducedcurveofmultiplic-ity3containedinasmoothalgebraicsurface.SupposeC3-connected.Thenh1(C;!C(�
))=1forageneraleectivedivisor
of(multi)degree1 2(deg�!C).Proof.Firstofallnoticethat!Cisveryampleandyieldsanembedding'!C(C)PNsinceCis3-connectedandnothonestlyhyperelliptic(see[2,x3]),andthatforageneral
ofdegree1 2(deg�!C),by[5,Thm.3.1]itisH0(C;!C)H0(2�;!C)O
j2�Theproofwillbemadebyinductionon.For=1takeapointQ.
willbechosenamongalltheCartierdivisorsofdegree1withsupportQ.WerecallthatthesetofallCartierdivisorofmultidegree=1concentratedatQisisomorphictoBQ=f[g]jg2OQ;C;vQ((g))=1gwhere:C!�isthereductionmorphismandvQ(
)isthevaluationatQinOQ;�(see[10,x4]).FurthermoreifC=n�iscontainedinasmoothalgebraicsurfacethenBQ=An�1(seee.g.[10,4.3.1]).NowitisH0(C;!C)KQandH0(C;!C)
j2�foreverylocaltransversecut
supportedatQsince!Cisveryampleandlength(
j2�)=2.Finally,10 SetF:=H(�
)=H G�1.ThenwehavedegF 2j�idegHj�i8i,whencedegFjBpa(B)+1forallBC.Case1.Thereexistsanindexhs.t. h2hdh.InthiscaseFandGsatisfythehypothesesofTheorem6andthenwecanconclude.Case2.Foralli=1;:::;sdeg!Cj�iisevendegAj�i=1Herewewilltreatrstlythegeneralsituation,showingthatwemayndanewdecompositionHnumF0 G0,withF0andG0asinTheorem6,andthenwewilltreattheexceptionalcases.Tothisaimletusconsiderthefollowinglist(a)C=n�;n3;�2even;(b)C=�1+2�2;�1
�2=1;(c)Forallirreducible�C;�
(C��)=2.Claim16.Assumethatforalli=1;:::;sdeg!Cj�iisevenanddegAj�i=1.IfCdoesnotbelongtotheabovelistthenthereexistsadecom-positionHnumF0 G0,withF0andG0asinTheorem6.Proof.Byhypothesesthereexistsanirreduciblecomponent�hofmulti-plicitynhsuchthat�h
(C��h)4.TakingAhageneraltransversecutonthiscomponent,weletG0:=G Ah,F0:=F A�1h,andweinferthatG0andF0satisfyrespectivelycondition(3)andcondition(4)ofTheorem6.ForG0thisisobvious.AboutF0wehavedegF0jB=deg!CjB 2+degAjB�degAhjBThus,ifB=CtherequiredinequalityholdssincedegAjB�degAhjB2becausecase(a)and(b)donotoccur,whileifBCandB6=mh�hitholdsbecauseCis2-connectedanddegAjB�degAhjB1.Finally,ifB=mh�h(1mhnh)itholdsthankstoourchoiceof�h. Theexceptionalcongurations.Inthecongurations(a),(b),(c)listedabovewecannolongerndasuitabledecompositionwithF,!H,jFjb.p.f.andH1(F)=H1(H F�1)=0.IntheseparticularcongurationswehaveG=OC(
)isaninvertiblesheafsuchthat!CnumG 2(3)12 ToapplyProposition8weneedanestimateforh0(C;!C H�1 F).Thisfollowssincedeg(!C H�1 F)j�2=pa(�2),andfurthermore!C H�1 FnumG:thusbyLemma13,forageneral
,H0(C;!C H�1 F),!H0(�2;!C H�1 F)andwemayassumethisvectorspacebeingofdimension1.Conguration(c).Forallirreducible�C;�
(C��)=2.LetG=OC(
)andF=H(�
).Noticethatforallirreducible�C;wehavedegFj�=pa(�)+1anddegGj�=pa(�):IfCisreducedjFjisabasepointfreesystemofprojectivedimensionr2(sincebynumericallyconditionsdegA3)while!C H�1 FnumGsatisesLemma13.Toconcludeweonlyneedh0(C;!C H�1 F)h0(C;F)�2:Thiscanbeeasilyseensince,asabove,forageneral
wehaveH0(C;!C H�1 F),!H0(�;!C H�1 F)=C.ThusH0-lemmaholdsandwecanconclude.NowletusassumeCnonreduced.BytheaboveanalyseswemayassumethatCredhasatleast3component.LetusconsiderthesplittingofLemma14,C=Cred+BwhereB=Pki=1�iisachainofreducedcurvesuchthatCred
�1=Cred
�k=1;Cred
�i=08i=2;:::;k�1:Now,bydegreeconsideration(sincewehavedegFj�i=pa(�i)+1,)forFsucientlygeneral,wemayassumeh1(B;OB(F) OB(�Cred))=0i.e.,theexactsequence0!OB(F) OB(�Cred)!F!OCred(F)!0isexactonglobalsections.ThismeansthatwecanpickasubspaceWH0(C;F)suchthatWisisomorphictoH0(Cred;F)(henceofdimensionr+13)andWyieldsabasepointfreesubsystemofjFj(sinceitisbasepointfreeonCred).Moreover,(!C H�1 F) 2num!C,i.e.deg(!C H�1 F)j�i=pa(�i)foreveryi.ThusbytheparticularcongurationofCred(itisachainofcurves),applyinganinductionargumentonthenumberofcomponentswegeth0(Cred;!C H�1 F)=h0(C;!C H�1 F)=1.Finally,byLemma13everysectionofH0(C;!C H�1 F)doesnotvanishonanycomponentofCred.ThusH0-Lemmaappliesandthenthetheoremfollows.Q.E.D.forTheorem16ProjectivelynormaltricanonicalembeddingsofalgebraicsurfacesInthissectionweprovethattheimageofthetricanonicalembeddingofasurfaceofgeneraltypeisprojectivelynormal.14 References[1]G.Castelnuovo,Suimultiplidiunaserielinearedigruppidipuntiap-partenentiadunacurvaalgebrica,Rend.Circ.Mat.Palermo7(1893),89{110[2]F.Catanese,M.Franciosi,K.HulekandM.Reid,EmbeddingsofCurvesandSurfaces,NagoyaMath.J.154(1999),185{220[3]C.Ciliberto,Sulgradodeigeneratoridell'anellocanonicodiunasupercieditipogenerale,Rend.Sem.Mat.Univ.Pol.Torinovol.41,3(1983),83{111[4]C.Ciliberto,P.Francia,M.MendesLopes,Remarksonthebicanonicalmapsforsurfacesofgeneraltype,Math.Z.,224(1997),no.1,137{166[5]M.Franciosi,Adjointdivisorsonalgebraiccurves,AdvancesinMathemat-ics186(2004),317{333[6]M.Green,Koszulcohomologyandthegeometryofprojectivevarieties,J.Di.Geom.19(1984),125{171[7]K.Konno,1-2-3forcurvesonalgebraicsurface,J.reineangew.Math.533(2001),171{205[8]M.MendesLopes,Adjointsystemsonsurfaces,BollettinoU.M.I.(7)10-A(1996),169{179[9]D.Mumford,Varietiesdenedbyquadraticequations,in'Questionsonalgebraicvarieties',C.I.M.E.,IIICiclo,Varenna,1969,Ed.Cremonese,Rome,(1970),30{100[10]F.Oort,Reducibleandmultiplealgebraiccurves,VanGorcum&Comp.N.V.,H.J.Prakke&H.M.G.Prakke,Leiden,(1961)16