the geometry of the compactification of the hurwitz scheme

《the geometry of the compactification of the hurwitz scheme》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、GEOMETRYOFTHEHURWITZSCHEMETheGeometryoftheCompactificationoftheHurwitzSchemebyShinichiMOCHIZUKI*TableofContentsTableofContentsIntroduction§1.DifferentTypesofHurwitzSchemes§2.Irreducibility§3.LogAdmissibleCoverings§3A.BasicDefinitions§3B.FirstProperties§3C.GlobalModuli§3D.Admis

2、sibleHurwitzCoverings§4.ConstructionoftheBoundaryComponentsAppendixto§4PictorialAppendix§5.CohomologyCalculations§6.TheMainFibration§6A.TheExcessDivisorsintheMainFibration§6B.IntersectionTheoryCalculations§6C.RamificationIndices§7.TheCoefficientMatrix§8.ArithmeticApplicationsB

3、ibliographyIntroductionThepurposeofthispaperistostudythegeometryoftheHarris-Mumfordcompactifi-cationoftheHurwitzscheme.TheHurwitzschemeparametrizescertainramifiedcoveringsReceivedJune11,1993.1991MathematicsSubjectClassification:14H10*ResearchInstituteforMathematicalSciences,Ky

4、otoUniversity,Kyoto606,Japan1f:C→P1oftheprojectivelinebysmoothcurves.Thus,fromtheveryoutset,oneseesthatthereareessentiallytwowaystoapproachtheHurwitzscheme:(1)WestartwithP1andregardtheobjectsofinterestascoveringsofP1;(2)WestartwithCandregardtheobjectsofinterestasmorphismsfr

5、omCtoP1.OnefindsthatonecanobtainthemostinformationabouttheHurwitzschemeanditscompactificationbyexploitinginterchangeablythesetwopointsofview.OurfirstmainresultisthefollowingTheorem:Letb,d,andgbeintegerssuchthatb=2d+2g−2,g≥5andd>2g+4.LetHbetheHurwitzschemeoverZ[1]parametrizingc

6、overingsoftheprojectivelineofb!degreedwithbpointsoframification.ThenPic(H)isfinite.Remark:ThenumberginthestatementoftheTheoremisthegenusofthe“curveCupstairs”ofthecoveringsinquestion.Note,however,thattheHurwitzschemeH,andhencealsothegenusg,arecompletelydeterminedbybandd.ThisTh

7、eoremisstatedin§6.7,ofthetext.NotethatalthoughinthestatementoftheTheoremhereintheintroduction,wespokeof“the”Hurwitz“scheme,”thereareinfactseveraldifferentHurwitzschemesusedintheliterature,someofwhichare,infact,notschemes,butstacks.FordetailsabouttheparticulartypeofHurwitzsch

8、emeforwhichthemaintheoremisproved,wereferthereadertotheexactstatementin§6.7,aswell