Surfaces of general type with $p_g=q=1, K^2=8$ and

arXiv:math/0311508v3[math.AG]14Mar2005SURFACESOFGENERALTYPEWITHpg=q=1,K2=8ANDBICANONICALMAPOFDEGREE2FRANCESCOPOLIZZIAbstract.Weclassifytheminimalalgebraicsurfacesofgeneraltypewithpg=q=1,K2=8andbicanonicalmapofdegree2.Itwillturnoutthattheyareisogenoustoaproductofcurves,i.e.ifSissuchasurface,thenthereexisttwosmoothcurvesC,FandafinitegroupGactingfreelyonC×FsuchthatS=(C×F)/G.WedescribetheC,FandGthatoccur.InparticularthecurveCisahyperelliptic-biellipticcurveofgenus3,andthebicanonicalmapφofSiscomposedwiththeinvolutionσinducedonSbyτ×id:C×F−→C×F,whereτisthehyperellipticinvolutionofC.Inthiswayweobtainthreefamiliesofsurfaceswithpg=q=1,K2=8whichyieldthefirst-knownexamplesofsurfaceswiththeseinvariants.Wecomputetheirdimensionandweshowthattheyarethreegenericallysmooth,irreduciblecomponentsofthemodulispaceMofsurfaceswithpg=q=1,K2=8.Moreover,wegiveanalternativedescriptionofthesesurfacesasdoublecoversoftheplane,recoveringaconstructionproposedbyDuVal.0.IntroductionIn[Par03]R.PardiniclassifiedtheminimalsurfacesSofgeneraltypewithpg=q=0,K2S=8andarationalinvolution,i.e.aninvolutionσ:S−→SsuchthatthequotientT:=S/σisarationalsurface.AlltheexamplesconstructedbyPardiniareisogenoustoaproduct,i.e.thereexisttwosmoothcurvesC,FandafinitegroupGactingfaithfullyonC,FandwhosediagonalactionisfreeontheproductC×F,insuchawaythatS=(C×F)/G.Pardini’sclassificationcontainsfivefamiliesofsuchsurfaces;inparticular,fourofthemareirreduciblecomponentsofthemodulispaceofsurfaceswithpg=q=0,K2S=8,andrepresentthesurfaceswiththeaboveinvariantsandnon-birationalbicanonicalmap.Inthispaperwedealwiththeirregularcase,infactwestudythecasepg=q=1,K2S=8.Surfaceswithpg=q=1aretheminimalirregularsurfacesofgeneraltypewiththelowestgeometricgenus,thereforeitwouldbeveryinterestingtoobtaintheircompleteclassifi-cation;forsuchareason,theyarecurrentlyanactivetopicofresearch.However,suchsurfacesarestillquitemysterious,andonlyafewfamilieshavebeenhithertodiscovered.IfSisasurfacewithpg=q=1,then2≤K2S≤9;thecaseK2S=2isstudiedin[Ca81],whereas[CaCi91]and[CaCi93]dealwiththecaseK2S=3.ForhighervaluesofK2Sonlysomesporadicexamplesweresofarknown;see[Ca99],whereasurfacewithK2S=4andonewithK2S=5areconstructed.Whenpg=q=1,therearetwobasictoolsthatonecanuseinordertostudythegeometryofS:theAlbanesefibrationandtheparacanonicalsystem.Firstofall,q=1impliesthattheAlbanesevarietyofSisanellipticcurveE,hencetheAlbanesemapα:S−→Eisaconnectedfibration;wewilldenotebyFthegeneralfibreofαandbyDate:February1,2008.1991MathematicsSubjectClassification.14J29,14J10,14H37.Keywordsandphrases.Surfacesofgeneraltype,bicanonicalmap,isotrivialfibrations,Galoiscoverings.1g=g(F)itsgenus.Letusfixazeropoint0∈E,andforanyt∈EletuswriteKS+tforthelinebundleKS+Ft−F0.ByRiemann-Rochandsemicontinuitytheoremwehaveh0(S,KS+t)=1forgeneralt∈E,hencedenotingbyCttheonlyelementinthecom-pletelinearsystem|KS+t|weobtaina1−dimensionalalgebraicfamily{K}={Ct}t∈EparametrizedbytheellipticcurveE.WewillcallittheparacanonicalsystemofS;accordingto[Be88],itistheirreduciblecomponentoftheHilbertschemeofcurvesonSalgebraicallyequivalenttoKSwhichdominatesE.Theindexι=ι(K)ofthepara-canonicalsystem{K}isthenumberofdistinctcurvesof{K}throughageneralpointofS.Theparacanonicalmapω:S−→E(ι),whereE(ι):=SymιE,isdefinedinthefollowingway:ifx∈Sisageneralpoint,thenω(x)=t1+···+tι,whereCt1,...,Ctιaretheparacanonicalcurvescontainingx.Thebestresultthatonemightobtainwouldbetoclassifythetriples(K2,g,ι)suchthatthereexistsaminimalsurfaceofgeneraltypeSwithpg=q=1andtheseinvariants.SincebytheresultsofGiesekerthemodulispaceMχ,K2ofsurfacesofgeneraltypewithfixedχ(OS),K2Sisaquasiprojectivevariety,itturnsoutthatthereexistonlyfinitelymanysuchtriples,butacompleteclassificationisstillmissing.Bytheresultsof[Re88],[Fr91]and[CaCi91]itfollowsthatthebicanonicalsystem|2KS|ofaminimalsurfaceofgeneraltypewithpg=q=1isbase-pointfree,whencethebi-canonicalmapφ:=φ|2K|:S−→PK2SofSisamorphism.Moreoversuchamorphismisgenericallyfiniteby[Xi85a],soφ(S)isasurfaceΣ.WewillsaythatasurfaceScontainsagenus2pencilifthereisamorphismf:S−→B,whereBisasmoothcurveandthegeneralfibreΦoffisasmoothcurveofgenus2.NoticethatinthiscasethebicanonicalmapφofSisnotbirational,since|2KS|cutsoutonΦasubseriesofthebicanonicalseriesofΦwhichiscomposedwiththehyperellipticinvolution.InthiscasewesaythatSpresentsthestandardcaseforthenon-birationalityofthebicanonicalmap;otherwise,namelyifφisnotbirationalbutSdoesnotcontainanygenus2pencils,wesaythatSpresentsthenon-standardcase.BytheresultsofBombieri(laterimprovedbyReider,see[Bo73]and[Re88])itfollowsthat,ifK2S≥10andthebicanonicalmapisnotbira-tional,thenScontainsagenus2pencil.Hencethereexistonlyfinitelymanyfamiliesofsurfacesofgeneraltypepresentingthenon-standardcase,andonewouldliketoclassifyallofthem;however,thisproblemisstillopen,althoughmanyexamplesareknown.Inthepaper[Xi90]G.Xiaogavetwolistsofpossibilitiesforthebicanonicalimageofsuchasurface;lateronseveralauthorsinvestigatedtheirrealoccurrence.Formoredetailsaboutthisargument,wereferthereadertothepaper[Ci97].Noexamplesofsurfaceswithpg=q=1andpresentingthenon-standardcasewerehithertokno