Maximally irregularly fibred surfaces of general t

arXiv:math/0209120v3[math.AG]21Oct2004MaximallyirregularlyfibredsurfacesofgeneraltypeMartinM¨ollerAbstract.WegeneraliseamethodofXiaoGangtoconstruct’prototypes’offibredsurfaceswithmaximalirregularitywithoutbeingaproduct.Thisenablesus,inthecaseoffibregenusg=3todescribethepossiblesingularfibresandtocalculatetheinvariantsofthesesurfaces.Wealsoprovestructuretheoremsonthemodulispaceforfibredsurfaceswithfibregenusg=2andg=3.MathematicsSubjectClassification2000.14J10,14J29,14D06Keywords.fibredsurface,modulispaceofsurfacesofgeneraltype,highirregularity,fixedpart,degeneratefibresIntroductionComplexfibredsurfacesf:X→Bofsmallfibregenusmaybestudiedbydifferenttechniquesaccordingtotheirirregularityq(X)=H1(X,OX).IfbdenotesthegenusofthebaseBandgthegenusofafibre,theirregularityofafibredsurfaceissubjecttob≤q≤g+b.Incaseq=g+bthefibrationistrivial,i.e.Xisbirationaltoaproduct.Xiao([Xi85])andSeiler([Sei95])examinedsurfacesXwithg=2andirregularityq(X)=busingthefactthatthesesurfacesaredoublecoveringsofruledsurfaces.Xiaoalsostudies([Xi85])surfaceswithg=2andq=b+1usingthefixedpartoftheJacobianfibration.Weextendthistechniquetosurfaceswithq=b+g−1,whichwecallmaxi-mallyirregularlyfibred.Maximalirregularityimpliesg≤7by([Xi87a])andmaximallyirregularfibrationswithg≤4areknowntoexist([Pi89]).Weconstructa’prototype’formaximallyirregularlyfibredsurfaceswithg=3,i.e.afibredsurfacesuchthatanyothersurfacewiththesameinvariantsarisesviapullbackbycoveringofthebasecurves(seeDef.1.3fortheprecisedefinition).ComparedtoXiao’scaseadditionaldifficultiesariseatthehyperellipticlocusduetothefailureofinfinitesimalTorelli.Theprototypeenablesustodeterminethedegeneratefibresandinvariantsofmaximallyirregularlyfibredsurfaceswithg=3.Thetechniquesapplyinprinciplealsoforg=4(and,ifsurfacesexist,alsoforg≥5),butthesecasesadditionallyneedananswertoaSchottkytypeproblem,asexplainedattheendofthepaper.Finallyweshowthattheseconstructionsgluetogetherinfamilies.Wethusobtainstruc-tureresultsforcomponentsoftheGiesekermodulispaceofsurfacesofgeneraltype.Thesurfacesadmittingamaximallyirregularfibrationformconnectedcomponentsofthemodulispace.Thesecomponentsfibreoveramodulispaceofabelianvarieties.Thefibresaremodulispacesofstablemappings.1Thepaperisorganisedasfollows:In§1werecallsomefactsonthefixedpartoftheJacobianfibration.Wearethenabletogivetheprecisedefinitionofprototype.OmittingsometechnicalconditionsontheabelianvarietyAoneofthemaintheoremscanthenberoughlystatedasfollows:Theorem1.4’Supposed≥3andletAbea(1,d)-polarisedabelianvarietyofdimension2.ThenthereisafibredsurfaceS(A,d)→B(A,d),suchthatanymaximallyirregularfibrationX→BwithfixedpartAandfibregenus3isobtainedviabasechangeB→B(A,d)fromS(A,d).ThebasecurveB(A,d)isadoublecoveringofthemodularcurveX(d).ItsproofreliesonaparametrisationoftheJacobiansofcurvesintowhichtheabelianvarietyAinjects.ThisresultisstatedasTheorem1.6anditsproofispresentedin§2.§3containstheproofofTheorem1.4togetherwiththecomputationoftheinvariantsofmaximallyirregularfibrations.Finallyin§4weusetheseresultstoderivesomestructureresultsforthecorrespondingcomponentsofthemodulispace.Mostoftheresultscanbefoundintheauthor’sthesis([Mo02]).AcknowledgementsTheauthorthankshisthesisadvisorF.Herrlichformanystimulatingdiscussionsandalotofpatience.HealsothanksE.ViehwegforworthfulremarksconcerningTorelli’stheorem.SomeresultsinthesamedirectionwereobtainedindependentlybyJ.-X.Cai.Theauthorthankshimandtherefereeforhissuggestions.NotationWeworkoverthecomplexnumbersthroughout.Forafibredsurfacef:X→Bwedenoteb=g(B)thebasegenusandg=g(F)thefibregenus.Theirregularityq(X)=H1(X,OX)issubjecttob≤q≤b+g.Wesaythefibrationfisoftype(g,b).LetCg(·)(resp.Ag(·))denotethemodulifunctorforsmoothcurvesofgenusg(resp.abelianvarietiesofdimensiong)andletMg(resp.Ag))denotethecorrespondingcoarsemodulispaces.Whendiscussingsurfacesofgeneraltype,wedenotebyXitscanonicalmodel,i.e.anormalsurfacewithKXampleandatmostrationaldoublepoints.Ifnecessary,Sdenotesthecorrespondingsmoothminimalmodel.ArelativecanonicalmodelofaflatfamilyofsurfacesofgeneraltypeoversomebaseThasacanonicalmodelinthefibreovereachcomplexpointofT.Eachflatfamilyofsurfacesofgeneraltypeisbirationaltoarelativecanonicalmodel(see[Tv72]).WedenotebyS(·)themodulifunctorwhichassociateswithaschemeTthesetofflatfamiliesofrelativecanonicalmodelsofsurfacesofgeneraltypeoverT.21PrototypesforfibredsurfacesAfibrationofasurfacef:X→BisasurjectionontoasmoothcurveBwithconnectedfibres.Thefibrationiscalledregularifq(X)=bandirregularotherwise.Ifq(X)=b+g,thesurfaceXisbirationaltoaproductofthebasecurveandsmoothfibre.IfthemodulimapB→Mghasimagereducedtoapointthefibrationissaidtobeisotrivialorofconstantmoduli.Inthesequelweareinterestedinthecaseofnon-trivialfibrationswithmaximalirre-gularity,i.e.withq(X)=g+b−1.Wecallthemmaximallyirregularlyfibred.TheJacobiansofthefibresofsuchasurfacehavealargeabelianvarietyincommon,thefixedpart(’partiefixe’or’L/K-trace’in[La59]).WerecallsomefactsaboutthefixedpartofafibredfamilyX→B→Tofsurfaces.ThereaderlessinterestedinstatementsonthemodulispacemaythinkofT=SpecCinsections1−3.LetA=Pic0X/T/JacB/T.OverthelocusB′whereX→Bissmooth(wh