On surfaces of general type with p_g=q=1 isogenous

arXiv:math/0601063v3[math.AG]3Aug2008ONSURFACESOFGENERALTYPEWITHpg=q=1ISOGENOUSTOAPRODUCTOFCURVESFRANCESCOPOLIZZITothememoryofmycolleagueandfriendGiulioMinervini.Abstract.AsmoothalgebraicsurfaceSissaidtobeisogenoustoaproductofunmixedtypeifthereexisttwosmoothcurvesC,FandafinitegroupG,actingfaithfullyonbothCandFandfreelyontheirproduct,sothatS=(C×F)/G.Inthispaperweclassifythesurfacesofgeneraltypewithpg=q=1whichareisogenoustoanunmixedproduct,assumingthatthegroupGisabelian.Itturnsoutthattheybelongtofourfamilies,thatwecallsurfacesoftypeI,II,III,IV.ThemodulispacesMI,MII,MIVareirreducible,whereasMIIIisthedisjointunionoftwoirreduciblecomponents.InthelastsectionwestarttheanalysisofthecasewhereGisnotabelian,byconstructingseveralexamples.0.IntroductionTheproblemofclassificationofsurfacesofgeneraltypeisofexponentialcomputationalcomplexity,see[Ca92],[Ch96],[Man97];nevertheless,onecanhopetoclassifyatleastthosewithsmallnumericalinvariants.Itiswell-knownthatthefirstexampleofsurfaceofgeneraltypewithpg=q=0wasgivenbyGodeauxin[Go31];lateron,manyotherexampleswerediscovered.Ontheotherhand,anysurfaceSofgeneraltypeverifiesχ(OS)0,henceq(S)0impliespg(S)0.Itfollowsthatthesurfaceswithpg=q=1aretheirregularoneswiththelowestgeometricgenus,henceitwouldbeimportanttoachievetheircompleteclassification;sofar,thishasbeenobtainedonlyinthecasesK2S=2,3(see[Ca81],[CaCi91],[CaCi93],[Pol05],[CaPi05]).Asthetitlesuggests,thispaperconsiderssurfacesofgeneraltypewithpg=q=1whichareisogenoustoaproduct.ThismeansthatthereexisttwosmoothcurvesC,FandafinitegroupG,actingfreelyontheirproduct,sothatS=(C×F)/G.Wehavetwocases:themixedcase,wheretheactionofGexchangesthetwofactors(andthenCandFareisomorphic)andtheunmixedcase,whereGactsdiagonally.IntheunmixedcaseGactsseparatelyonCandF,andthetwoprojectionsπC:C×F−→C,πF:C×F−→Finducetwoisotrivialfibrationsα:S−→C/G,β:S−→F/G,whosesmoothfibresareisomorphictoFandC,respectively.IfSisisogenoustoaproduct,thereexistsauniquerealizationS=(C×F)/Gsuchthatthegenerag(C),g(F)areminimal([Ca00],Proposition3.13);wewillalwaysworkwithminimalrealizations.Surfacesofgeneraltypewithpg=q=0isogenoustoaproductappearin[Be96],[Par03]and[BaCa03];theircompleteclassificationhasbeenfinallyobtainedin[BaCaGr06].Someunmixedexampleswithpg=q=1havebeengivenin[Pol06];soitseemednaturaltoattackthefollowingMainProblem.Classifyallsurfacesofgeneraltypewithpg=q=1isogenoustoaproduct,anddescribethecorrespondingirreduciblecomponentsofthemodulispace.InthispaperwefullysolvetheMainProblemintheunmixedcaseassumingthatthegroupGisabelian.Ourresultsarethefollowing:TheoremA(seeTheorem4.1).IfthegroupGisabelian,thenthereexistexactlyfourfamiliesofsurfacesofgeneraltypewithpg=q=1isogenoustoanunmixedproduct.Ineverycaseg(F)=3,whereastheoccurrencesforg(C)andGareDate:August3,2008.1991MathematicsSubjectClassification.14J29(primary),14J10,20F65.Keywordsandphrases.Surfacesofgeneraltype,actionsoffinitegroupsoncurves.1I.g(C)=3,G=(Z2)2;II.g(C)=5,G=(Z2)3;III.g(C)=5,G=Z2×Z4;IV.g(C)=9,G=Z2×Z8.SurfacesoftypeIalreadyappearin[Pol06],whereasthoseoftypeII,III,IVprovidenewexamplesofminimalsurfacesofgeneraltypewithpg=q=1,K2=8.TheoremB(seeTheorem5.1).ThemodulispacesMI,MII,MIVareirreducibleofdimension5,4,2,respectively.ThemodulispaceMIIIisthedisjointunionoftwoirreduciblecomponentsM(1)III,M(2)III,bothofdimension3.ThecasewhereGisnotabelianismoredifficult,andacompleteclassificationisstilllacking(seeRemark7.4).However,wecanshedsomelightonthisproblem,byprovingTheoremC(seeTheorem7.1).LetS=(C×F)/Gbeasurfaceofgeneraltypewithpg=q=1,isogenoustoanunmixedproduct,andassumethatthegroupGisnotabelian.Thenthefollowingcasesoccur:G|G|g(C)g(F)S3634D4835D61273A41245S42494A560214TheexampleswithG=S3andD4alreadyappearin[Pol06],whereastheothersarenew.Itwouldbeinterestingtohaveadescriptionofthemodulispacesforthesenewexamples(seeRemark7.3).Whiledescribingtheorganizationofthepaperweshallnowexplainthestepsofourclassifica-tionprocedureinmoredetail.ThecrucialpointisthatintheunmixedcasethegeometryofthesurfaceS=(C×F)/GisencodedinthegeometryofthetwoG−coversh:C−→C/G,f:F−→F/G.Thisallowsusto”detopologize”theproblembytransformingitintoanequiv-alentproblemabouttheexistenceofapairofepimorphismsfromtwogroupsofFuchsiantypeintoG;thisisessentiallyanapplicationoftheRiemann’sexistencetheorem.Theseepimor-phismsmustsatisfysomeadditionalpropertiesinordertogetafreeactionofGonC×Fandaquotientsurfacewiththedesiredinvariants(Proposition3.1).Thegeometryofthemodulispacescanbealsorecoveredfromthesealgebraicdata(Propositions3.4and3.5).Inthenonabeliancasewefollowasimilarapproach(Proposition7.2).InSection1wefixthealgebraicsetup.ThereaderthatisonlyinterestedintheproofofTheoremsAandCmightskiptoSection2afterreadingSection1.1.Ontheotherhand,thecontentofSections1.2,1.3,1.4,1.5isessentialinordertounderstandtheproofofTheoremB.Theresultsin1.3arewell-known,whereasforthosein1.4and1.5wehavenotbeenabletofindanycompletereference;sowehadtocarryout”byhand”allthe(easy)computations.InSection2weestablishsomebasicresultsaboutsurfacesSofgeneraltypewithpg=q=1isogenoustoaproduct.Suchsurfacesarealwaysminim