Projective 3-folds of general type with X=1

arXiv:math/0611898v2[math.AG]5Sep2007PROJECTIVE3-FOLDSOFGENERALTYPEWITHχ=1MENGCHENANDLEIZHUAbstract.Therearemanyexamplesof3-foldsofgeneraltypewithχ(O)=1foundbyFletcherandReidabouttwentyyearsago.FletcherhaseverprovedP12(X)≥1andP24(X)≥2forallminimal3-foldsXofgeneraltypewithχ(OX)=1.Inthispaper,weimproveonFletcher’smethod.Ourmainresultisthatϕmisbirationalontoitsimageforallm≥63.ToprovethiswewillshowPm≥1forallm≥14andP2l+18≥3foralll≥0.1.IntroductionToclassifyalgebraicvarietiesisoneofthemaingoalsofalgebraicgeometry.Inthispaperweareconcernedwiththeexplicitalgebraicgeometryofcomplexprojective3-foldsofgeneraltype.LetVbeasmoothprojective3-foldofgeneraltype.LetXbeaminimalmodelofV.Denotebyϕmthem-thpluricanonicalmap.Aclassicproblemistoseewhenϕmisbirationalontoitsimage.Re-centlyaremarkabletheorembyTsuji[26],Hacon-McKernan[13]andTakayama[24]saysthatthereisauniversalconstantr3suchthatϕmisbirationalforallm≥r3andforarbitrary3-foldsofgeneraltype.AverynewresultbyJ.A.Chenandthefirstauthorin[5]showsthatonemaytaker3=77.Therehavebeensomeconcreteknownboundsonr3already.Forexample,r3≤5(sharp)ifXisGorensteinbyJ.A.Chen,M.Chen,D.-Q.Zhang[4];r3≤8(sharp)ifeitherq(X)0byJ.A.Chen,C.D.Hacon[3]orpg(X)≥2byM.Chen[6];r3≤14(sharp)ifχ(OX)≤0byM.Chen,K.Zuo[9].Itisnaturaltostudya3-foldwithχ(O)≥1.Firstwetreatageneral3-foldandprovethefollowing:Theorem1.1.LetVbeanonsingularprojective3-foldofgeneraltype.AssumePm1(V)≥2andPm(V)≥1forallm≥m0≥2.Thenthepluricanonicalmapϕmisbirationalforallm≥max{m0+4m1+2,5m1+4}.Theorem1.1hasimprovedKoll´ar’sCorollary4.8in[17]andTheorem0.1of[8].ThefirstauthorwassupportedbytheProgramforNewCenturyExcellentTal-entsinUniversity(#NCET-05-0358)andtheNationalOutstandingYoungScientistFoundation(#10625103).ThesecondauthorwassupportedbyGraduateStudents’InnovationProjects(EYH5928004).12M.ChenandL.ZhuInthesecondpartweprovethefollowing:Theorem1.2.LetVbeanonsingularprojective3-foldofgeneraltypewithχ(OV)=1.Then(i)Pm(V):=h0(V,mKV)0forallm≥14;(ii)P18+2l(V)≥3forallintegerl≥0;(iii)ϕmisbirationalontoitsimageforallm≥63.Theorem1.2hasimprovedIano-Fletcher’sresultsin[11].Throughoutourpaperthesymbol≡standsforthenumericalequiv-alenceofdivisors,whereas∼denotesthelinearequivalenceand=QdenotestheQ-linearequivalence.2.PluricanonicalsystemsInthissectionwearegoingtotreatageneral3-foldofgeneraltype.Bythe3-dimensionalMMP(see[18,14,20]forinstance)wemaycon-sideraminimal3-foldXofgeneraltypewithQ-factorialterminalsingularities.2.1.Assumption.Assumethat,onasmoothmodelV0ofX,thereisaneffectivedivisorΓ≤m1KV0withnΓ:=h0(V0,OV0(Γ))≥2.NaturallyPm1≥2.Wewouldliketostudytherationalmapϕ|Γ|.AveryspecialsituationisΓ=m1KV0,meanwhileϕ|Γ|isnothingbutthem1-canonicalmap.2.2.Setup.FirstwefixaneffectiveWeildivisorKm1∼m1KX.Takesuccessiveblow-upsπ:X′→X(alongnonsingularcenters),whichexistsbyHironaka’sbigtheorem,suchthat:(i)X′issmooth;(ii)thereisabirationalmorphismπΓ:X′→V0;(iii)themovablepartMΓof|π∗Γ(Γ)|isbasepointfree;(iii)thesupportofπ∗(Km1)∪π∗Γ(Γ)isofsimplenormalcrossings.DenotebygthecompositionϕΓ◦πΓ.Sog:X′−→W′⊆PnΓ−1isamorphism.LetX′f−→Bs−→W′betheSteinfactorizationofg.Wehavethefollowingcommutativediagram:V0X′W′B-??@@@@@R------------fsπΓϕΓgDenotebyMkthemovablepartof|kKX′|foranypositiveintegerk0.Wemaywritem1KX′=Qπ∗(m1KX)+Eπ,m1=Mm1+Zm1,whereMm1isthemovablepartof|m1KX′|,Zm1thefixedpartandEπ,m1aneffectiveQ-divisorwhichisaQ-sumofdistinctexceptionalProjective3-foldsofgeneraltype3divisors.ByKX′−1m1Eπ,m1,wemeanπ∗(KX).So,wheneverwetaketheroundupofmπ∗(KX),wealwayshavepmπ∗(KX)q≤mKX′forallpositivenumbersm.SinceMΓ≤Mm1≤π∗(m1KX),wecanwriteπ∗(m1KX)=MΓ+E′ΓwhereE′ΓisaneffectiveQ-divisor.Setd:=dim(B).DenotebySagenericirreducibleelement(seeDefinition2.3)of|MΓ|.ThenSisasmoothprojectivesurfaceofgeneraltype.Whend=1,onecanwriteMΓ≡aΓSwhereaΓ≥nΓ−1.Definition2.3.Assumethatacompletelinearsystem|M′|ismovableonanarbitraryvarietyV.AgenericirreducibleelementS′of|M′|isdefinedtobeagenericirreduciblecomponentinageneralmemberof|M′|.ClearlyS′∼M′onlywhen|M′|isnotcomposedwithapencil.Beforeprovingthemainresultwebuildatechnical,butaquiteusefultheoremwhichisageneralizedformofTheorem2.6in[6].Theorem2.4.LetXbeaminimalprojective3-foldofgeneraltypewithQ-factorialterminalsingularities.Assumethat,onasmoothmodelV0ofX,thereisaneffectivedivisorΓ≤m1KV0withnΓ:=h0(V0,OV0(Γ))≥2.Keepthesamenotationasin2.2above.Onehasafibrationf:X′−→BinducedbyϕΓ.DenotebySagenericirre-ducibleelementof|MΓ|.Assumethat,onthesmoothsurfaceS,thereisamovablelinearsystem|G|(notnecessarilybasepointfree).LetCbeagenericirreducibleelementof|G|(soCcanbesingular).Setξ:=(π∗(KX)·C)X′∈Qandp:=(1ifdim(B)≥2aΓ(see2.2forthedefinition)otherwise.Thentheinequalitymξ≥2g(C)−2+α0(whereg(C)isthegeomet-ricgenusofC)holdsundertheassumptions(1)and(2)below.Fur-thermoreϕmofXisbirationalontoitsimageundertheassumptions(1),(2)′,(3)and(4)below.Assumptions,forapositiveintegerm:(1)Thereisarationalnumberβ0suchthatπ∗(KX)|S−βCisnumericallyequivalenttoaneffectiveQ-divisor;andsetα:=(m−1−m1p−1β)ξandα0:=pαq.(2)Theinequalityα1holds.(2)’Eitherα2orα1andCisnon-hyper-elliptic.(3)Thelinearsystem|mKX′|separatesdifferentgenerici