On Cohomology of the Square of an Ideal Sheaf

arXiv:alg-geom/9601027v221Mar1996ONCOHOMOLOGYOFTHESQUAREOFANIDEALSHEAFJonathanWahlUniversityofNorthCarolinaAbstract.ForasmoothsubvarietyX⊂PN,consider(analogouslytoprojectivenormality)thevanishingconditionH1(PN,I2X(k))=0,k≥3.ThisconditionisshowntobesatisfiedforallsufficientlylargeembeddingsofagivenX,andforaVeroneseembeddingofPn.ForC⊂Pg−1,thecanonicalembeddingofanon-hyperellipticcurve,thiscondi-tionguaranteesthevanishingofsomeobstructiongroupstodeformationsofthecone.RecallthatthetangentstodeformationsaredualtothecokerneloftheGaussian-Wahlmap.Theorem.SupposetheGaussian-WahlmapofCisnotsurjectiveandthevanishingconditionisfulfilled.ThenCisextendable:itisahyperplanesectionofasurfaceinPgnottheconeoverC.SuchasurfaceisaK3ifsmooth,butitcouldhaveserioussingularities.Theorem.Forageneralcurveofgenus≥3,thisvanishingholds.Conjecture.IftheCliffordindexis≥3,thisvanishingholds.0.IntroductionLetLbeaveryamplelinebundleonasmoothcomplexprojectivevarietyX,givinganembeddingX⊂PN.Itiswell-knownthatprojectivenormalityoftheembedding(ornormalgenerationofL)isequivalenttothevanishingH1(PN,IX(k))=0,allk,whereIXistheidealsheafdefiningX;further,allsufficientlyhighpowersofLarenormallygenerated.Inthispaper,weshallbeconcernedwiththeconditiononL(oritsembedding)(∗)H1(PN,I2X(k))=0,allk6=2.(Fork=2,Proposition1.8showsthisgroupisfrequentlythekerneloftheGaussianmapofL,henceisrarely0.)Thisquestionarisesnaturallybecausethesecohomol-ogygroupsgivethetorsionsubmoduleoftheK¨ahlerdifferentialsoftheaffineconeoverX(Proposition1.4).Ourfirstmainresultsare:12JONATHANWAHLTheorem2.1.(∗)holdsforX=PnandanyveryampleL.Corollary3.3.GivenanyXandveryampleL,allsufficientlyhighpowersofLsatisfy(∗).Corollary5.7.LetC⊂Pg−1bethecanonicalembeddingofageneralnon-hyperellipticcurveofgenusg≥3.Then(∗)holds.Verificationof(∗)forPnsurprisinglyturnsouttobenon-trivial;eventhen=1caserequiressomeseriousthought!OnemustdeduceviarepresentationtheoryofG=SL(n+1)andthemethodsof[W4]thesurjectivityofΓ(I(k))→Γ(I/I2(k)),k≥3.ThedifficultyisthatthesecondspaceisareducibleG-module,andonlyhalfofitsirreducibleconstituentsareobviouslyintheimage.Corollary3.3isdeducedfromTheorem2.1.Ingeneral,(∗)isdifficulttoverify;itholdsforthePl¨uckerembeddingofaGrassmannianG(2,n+1),andforanembeddingofacurveviaalinebundleofdegree≥2g+3(J.Rathmann[R],unpublished).ThemainpointofCorollary5.7istocalculateforCwhichispentagonal,i.e.,withabase-point-freeg15(Theorem5.3).SuchaCsitsnaturallyona4-dimensionalrationalscrollX,forwhich(∗)mustbeproved.Itisthenadelicatecalculationto“descend”thisresulttoC;thekeypointisthat(∗)holdsfor5pointsingeneralpositioninP3.As(∗)failsforthecanonicalembeddingofatetragonalcurveorplanesextic,onecanoptimisticallyhopetoprovethefollowingConjecture.LetCbeacurveofCliffordindex≥3.ThenthecanonicalembeddingofCsatisfies(∗).ThispaperismotivatedbythequestionofwhetheracanonicalcurveC⊂Pg−1isextendable,i.e.,isalinearsectionofanX⊂PgwhichisnottheconeoverC.SuchanXiscanonicallytrivial(c.t.):normal,ωX∼=OX,andh1(OX)=0([E],[W5]).Ifsmooth,XisaK-3;butanynormalquarticinP3isc.t.In[W2],extendabilityofCisstudiedviathedeformationtheoryoftheaffineconeAoverC.ThenT1A,themoduleoffirst-orderdeformations,isnon-trivialindegree−1;byduality,thismeansthattheGaussianmapΦK:∧2Γ(C,K)→Γ(C,K⊗3)isnotsurjective.Liftingafirst-orderdeformationofAtoahigherorderrequirescontroloftheobstructionspaceT2Awhich,bylocalduality,isdualtotorsionintheK¨ahlerdifferentials.Notingthegradingonthesemodules,andcombining(1.6)with(6.4)yieldstheTheorem.LetC⊂Pg−1beacanonicalcurvesatisfying(∗),andletAbetheaffinecone.(a)If3≤g≤10,g6=9,thenT2A=0.ONCOHOMOLOGYOFTHESQUAREOFANIDEALSHEAF3(b)Ifg=9org≥11,thenT2Aisconcentratedindegree−1.(Theg=9exceptionisthenon-injectivityofΦK,shownin[C-M]andreprovedin(6.5.2)usingworkofS.Mukai).Weusethistoproveapartialconversetothenon-surjectivityoftheGaussianforaK-3curve:Theorem7.1.LetC⊂Pg−1beacanonicalcurvesatisfying(∗).ThenCisextendableifftheGaussianΦKisnotsurjective.ThecanonicalembeddingofasmoothplanecurveCofdegree≥7satisfies(∗),withcorankΦK=10,henceisextendable;wegiveanexplicitdescriptionin(7.3),whereatypicalextensionXhasanon-smoothablesimpleellipticsingularity.SuchaCsitsonnoK-3surface,by[G-L]or[W5].Becauseofthespecialgeometryofc.t.surfaces,weoffertheConjecture.ABrill-Noether-Petrigeneralcurveofgenus≥8sitsonaK-3sur-faceifandonlyiftheGaussianisnotsurjective.By(1.13),condition(∗)isanaturalonetoconsiderforalinebundlealreadyknowntobenormallypresented(i.e.,satisfyingM.Green’scondition(N1)—pro-jectivelynormal,withhomogeneousidealgeneratedbyquadrics).CubicsvanishingtwiceonCvanishoneachsecantline,henceonthesecantvarietySec(C);so(∗)shouldberelatedtoSec(C)beingdefinedbycubics.FollowingworkofAaronBertram[B]andMichaelThaddeus[T],oneshouldstudymoregenerallyforpro-jectivelynormalC⊂PNthespacesH0(PN,InC(n+m)).Thepaperisorganizedasfollows:Section1introducesthegroupsH1(I2(k)),relatingthemtoconesandtoGaussianmappings;and(∗)iscomparedwithothernicepropertiesofhighpowersofamplelinebundles,specificallynormalpresenta-tion.InSection2,(∗)isprovedforallembeddingsofPnanddiscussedforcomplexhomogeneousspaces.Howto“