Linkage on singular rational normal surfaces and t

arXiv:math/0105094v1[math.AG]11May2001Linkageonsingularrationalnormalsurfacesand3-foldswithapplicationtotheclassificationofcurvesofmaximalgenus.RitaFerraroFebruary8,2008DipartimentodiMatematicaUniversit`adiRomaTreLargoSanLeonardoMurialdo,1-00146Roma1IntroductionInalgebraicgeometryandcommutativealgebrathenotionoflinkagebyacom-pleteintersection,whichwewillherecallclassicallinkage,hasbeenforalongtimeaninterestingandactivetopic.Inthisnoteweprovideageneralizationofclassicallinkageinadifferentcontext.Namelywewilllookatresidualsintheschemetheoreticintersectionofarationalnormalsurfaceor3-foldwithtwohypersurfacesofdegreeaandb(ac.i.oftype(a,b)onthescroll,seeDef.2.14).Whenthescrollissingularac.i.oftype(a,b)onitmaynotbeGorenstein,i.e.itsdualizingsheafmaynotbeinvertible.Ifthisisthecase,classicallinkage,evenifsuitablygeneralized,doesnotapply.Themainpurposeofthisarticleistoestablishaframeworkallowingonetofindrelationsbetweenthedimensionofsomeimportantcohomologygroupsattachedtothetwolinkedschemes.InthelastpartofthepaperweshowhowtoapplytheseresultsandtechniquestotheclassificationofcurvesCinPnofdegreedandmaximalgenusG(d,n,s)amongthosenotcontainedinsurfacesofdegreelessthanacertainfixedones.Thiswastheoriginalmotivationofthiswork.Acompleteclassificationtheoremhasbeengivenforn=3byL.GrusonandC.Peskinein[GP],forn=4byL.ChiantiniandC.Cilibertoin[CC]andforn=5bytheauthorin[F3].Forn=3andn=4therespectiveclassificationTheoremshavebeenprovenwithtechniquesofclassicallinkagebutforn≥5thisisnolongerpossible.Forn≥5ands≥2n−1theclassificationprocedureconsistsintheprecisedescriptionofthelinkedcurvetoCbyacertainc.i.onarationalnormal3-foldX.InExample5.7wedescribethislinkedcurveintheeasiestcase,i.e.whenitisaplanecurve.InExample5.9weconstructexamplesofsmoothcurvesofmaximalgenusG(d,n,s)foreverydandsintherangeofExample5.7.1Turningtoadetailedpresentationoftheresults,letW⊂Pn−1bearationalnormalsurfaceandletX⊂Pnbearationalnormal3-foldinPn;throughoutthearticleWwillbeoftenageneralhyperplanesectionofX.LetZ1andZ2(resp.Y1andY2)bethetwolinkedschemesbyac.i.oftype(a,b)onW(resp.X).WebeginwiththecasewhenthescrollW(resp.X)issmooth.Inthiscaseonecanuseastraightforwardgeneralizationofclassicallinkage(inparticularofProp.2.5of[PS]).NamelytheconstructionthroughthemappingconeofalocallyfreeresolutionofOZ2(resp.OY2)inMod(W)(resp.Mod(X))fromalocallyfreeresolutionofOZ1(resp.OY1),allowsustofindthefollowingresults(Prop.3.1andProp.3.9):h0(IZ2|W⊗OW((i+2)H+KW)=h1(IZ1|W⊗OW((a+b−2−i)H))forimin{a,b},andh1(IY2|X⊗OX((i+3)H+KX))=h1(IY1|X⊗OX((a+b−3−i)H))foreveryi.HereHisahyperplanesectiondivisor(inbothWandX)andKW∼−2H+(n−4)R(resp.KX∼−3H+(n−4)R)isthecanonicaldivisorinW(resp.inX),whereRisadivisorintherulingofW(resp.X).Thefirstresultallowsustocomputeh0(IZ2|W⊗OW((i+2)+KW))forlowvaluesofiintermsoftheHilbertfunctionhZ1(a+b−2−i)ofthe0-dimensionalschemeZ1.IfY1isaritmeticallyCohenMacaulay,thesecondresultimpliesthath1(IY2|X⊗OX((i+3)H+KX))=0foreveryi,andthereforetherestrictionmapH0(IY2|X⊗OX(iH+(n−4)R))→H0(IZ2|W⊗OW(iH+(n−4)R))issurjectiveforeveryi;thatmeanswecanliftcurvesonWlinearlyequivalenttoiH+(n−4)RthroughageneralhyperplanesectionZ2ofY2tosurfacesonXlinearlyequivalenttoiH+(n−4)RpassingthroughY2.Thisconstructionisproblematicifthescrollissingular;howeveranaiveapproachyieldssomeinteresting,evenifweaker,results.Namely,inSection4wewillproveforY1arithmeticallyCohen-Macaulayandforimin{a,b}thefollowinginequalities:Theorem1.1(seeTheorems4.10and4.20)h0(IZ2|W⊗OW(iH+(n−4)R))≥h1(IZ1|W⊗OW((a+b−2−i)H))h0(IY2|X⊗OX(iH+(n−4)R))≥h1(IZ1|W⊗OW((a+b−2−i)H)).InthiscaseOW(iH+(n−4)R)(resp.OX(iH+(n−4)R))isthedivisorialsheafassociatedtoaWeildivisor∼iH+(n−4)RonW(resp.X).ThismeansthattheHilbertfunctionhZ1(a+b−2−i)givesusalowerboundforboththedimensionsofthevectorspacesofcurvesonWpassingthroughZ2andsurfacesonXpassingthroughY2linearlyequivalenttoiH+(n−4)R,forlowvaluesofi.Aswewillsee,thisissufficientformanyapplications.ThesametechniqueusedtoprovetheaboveresultsallowsustoprovealsoaformulawhichrelatesthearithmeticgeneraofthecurvesY1andY2,linkedbyacompleteintersectionYoftype(a,b)onthescrollX,inthecasethatthevertexofXisapoint:2Theorem1.2(seeTheorem4.30)pa(Y2)=pa(Y1)−pa(Y)+(a+b−3)·deg(Y2)+(n−4)·deg(R∩Y2)+1.(Thesameformulaiseasilyprovedwithclassicallinkagetechniquesinthesmoothcase,seeProp.3.11).MuchoftheselinkagetechniquesandtheclassificationforcurvesofmaximalgenusG(d,n,s)incasen=5appearedaspartofmydoctoraldissertation[F1].TheauthorthanksheradvisorCiroCiliberto.ThepaperhasbeenwrittenwhiletheauthorwassupportedbyaINDAMscholarship.2PreliminariesInthissection,wecollectthedefinitionsandnotationtobeusedinthispaper,andstatesomeofthebasicresultsoflinkagetheory.WeintroducethedefinitionsofgeometricandalgebraiclinkagebyaprojectiveschemeY,withoutsupposingYtobeacompleteintersection.Mostofthismaterialiswellknown;sometimes,however,duetolackofareference,wewillindicateaproof.Asourprimarytoolisthetheoryoflocallyfreeresolutionofsheaves,wewillincludeashortdiscussionofthis,indicatingthemainresultswewilluse.Moreoverwewillbrieflyintroducerationalnormalscrolls,inparticularwha