On surfaces with $p_g=q=2$ and non-birational bica

arXiv:math/0012211v2[math.AG]5Dec2001Onsurfaceswithpg=q=2andnon-birationalbicanonicalmap∗CiroCiliberto,MargaridaMendesLopesAbstractThepresentpaperisdevotedtotheclassificationofirregularsur-facesofgeneraltypewithpg=q=2andnonbirationalbicanonicalmap.Themainresultisthat,ifSissuchasurfaceandifSismini-malwithnopencilofcurvesofgenus2,thenSisadoublecoverofaprincipallypolarizedabeliansurface(A,Θ),withΘirreducible.ThedoublecoverS→AisbranchedalongadivisorB∈|2Θ|,havingatmostdoublepointsandsoK2S=4.1IntroductionIfasmoothsurfaceSofgeneraltypehasapencilofcurvesofgenus2,i.e.ithasamorphismtoacurvewhosegeneralfibreFisasmoothirreduciblecurveofgenus2,thenthelinebundleOS(KS)⊗OFisthecanonicalbundleonF,andthereforethebicanonicalmapφofScannotbebirational.Sincethispropertyis,ofcourse,ofabirationalnature,thesameremarkappliesifShasarationalmaptoacurvewhosegeneralfibreisanirreduciblecurvewithgeometricgenus2.Wecallthisexceptiontothebirationalityofthebicanonicalmapφthestandardcase.Anon-standardcasewillbetheoneofasurfaceofgeneraltypeSforwhichφisnotbirational,butthereisnopencilofcurvesofgenus2.Theclassificationofthenon-standardcaseshasalonghistoryandwerefertotheexpositorypaper[Ci]forinformationonthisproblem.Wewilljustmentionherethefactthatthenon-standardcaseswithpg≥4areallregular.∗2000MathematicsSubjectClassification:14J291Theclassificationofnon-standardirregularsurfaceshasbeenconsideredbyXiaoGangin[X1]andbyF.Cataneseandtheauthorsofthepresentpaperin[CCM].XiaoGangstudiedthegeneralproblemofclassifyingthenon-standardcasesbytakingthepointofviewoftheprojectivestudyoftheimageofthebicanonicalmap.Theoutcomeofhisanalysisisalistofnumericalpossibilitiesfortheinvariantsofthecaseswhichmightoccur.Morepreciseresultshavebeenobtainedin[CCM],wherethefirstsignificantcasepg=3hasbeenconsidered.Indeedin[CCM]itisshown,amongotherthings,thataminimalirregularsurfaceSwithpg=3presentsthenon-standardcaseifandonlyifSisisomorphictothesymmetricproductofasmoothirreduciblecurveofgenus3,thuspg=q=3andK2=6.Inthepresentpaperwestudythisproblemforsurfaceswithpg=q=2andweprovethefollowingresult,whichrulesoutasubstantialnumberofpossibilitiespresentedin[X1]:Theorem1.1LetSbeaminimalsurfaceofgeneraltypewithpg=q=2.ThenSpresentsthenon-standardcaseifandonlyifSisadoublecoverofaprincipallypolarizedabeliansurface(A,Θ),withΘirreducible.ThedoublecoverS→AisbranchedalongasymmetricdivisorB∈|2Θ|,havingatmostdoublepoints.OnehasK2S=4.Surfaceswithpg=q=2arestillfarfrombeingunderstood.Thelistofknownexamplesofsurfacesofgeneraltypewithpg=q=2isrelativelysmall(see[Z1],[Z2])andthereareseveralconstraintsfortheirexistenceknown.Hereweonlymentionthattherearevariousrestrictionsfortheexistenceofagenus2fibration(see[X2])andalsothatM.Manetti,workingontheSevericonjecture,showedinparticularthatifpg=q=2,KSisampleandK2S=4thenSisadoublecoverofitsAlbaneseimage(see[Ma]).Toproveourclassificationtheorem1.1wefirstshowthatthedegreeofthebicanonicalmapis2forsurfacespresentingthenon-standardcase,thenwestudythepossibilitiesforthequotientsurfacebytheinvolutioninducedbythebicanonicalmap,andfinallyweshowthattheuniquecasewhichreallyoccursistheonedescribedabove.Weuseadiversityoftechniques,whichmaybeusefulinothercontexts.Thepaperisorganizedasfollows.Insection2welistthepropertiesofsurfacesSwithpg=q=2thatweneed.Insection3wecharacterize,byasmalladaptationofaproofin[CCM],thesurfacesSpresentingthenon-standardcasewithK2S=9,andinparticularweverifythatthereis2nosuchsurfacewithpg=q=2.Insection4weestablishsomepropertiesoftheparacanonicalsystemandthenweusetheseresultsinsection5toconcludethatforthenon-standardcasesSwithpg=q=2thedegreeofthebicanonicalmapis2.ThusthereisaninvolutioniinducedbythebicanonicalmaponS.Weconsiderthequotientsurface˜Σ:=S/iandtheprojectionmapp:S→˜Σ.Insection6wediscussthevariouspossibilitiesfor˜Σ,showingthattheonlyonewhichcanreallyoccuristhat˜Σisaminimalsurfaceofgeneraltypewithpg(˜Σ)=2,q(˜Σ)=0,K2˜Σ=2andwith20nodes.Moreoverweshowthatthedoublecoverpramifiesexactlyoverthe20nodes.Finallyinsection7,usingthisdescription,andsomeresultsonPrymvarietiescontainedin[CPT],wefinallyprovetheorem1.1.AcknowledgementsThepresentcollaborationtakesplaceintheframe-workoftheeuropeancontractEAGER,no.HPRN-CT-2000-00099.Thesec-ondauthorisamemberofCMAFandoftheDepartamentodeMatem´aticadaFaculdadedeCiˆenciasdaUniversidadedeLisboa.WeareindebtedtoRitaPardiniforinterestingdiscussionsonthesub-jectofthispaperandinparticularforhavingpointedouttheuseoftheKawamata–Viehwegvanishingtheoreminsection6.Wethankprof.FabrizioCataneseforhavingcommunicatedtoussomeofhisideasonthissubject.WededicatethispapertothememoryofPaoloFrancia,withwhomwefirststartedonthissubject.NotationsandconventionsWeworkoverthecomplexnumbers.Allvarietiesareassumedtobecompactandalgebraic.Wedonotdistinguishbetweenlinebundlesanddivisorsonasmoothvariety,usingtheadditiveandthemultiplicativenotationinterchangeably.Linearequivalenceisdenotedby≡andnumericalequivalenceby∼.Aitnodeonasurfaceisanordinarydoublepoint(i.easingularityoftypeA1).TheexceptionaldivisorofaminimaldesingularizationofanodeisarationalirreduciblecurveA