the geometry of the compactification of the hurwitz scheme

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

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

3、raphyIntroductionThepurposeofthispaperistostudythegeometryoftheHarris-Mumfordcompactifi-cationoftheHurwitzscheme.TheHurwitzschemeparametrizescertainramifiedcoveringsReceivedJune11,1993.1991MathematicsSubjectClassification:14H10*ResearchInstituteforMathematicalSciences,KyotoUniver

4、sity,Kyoto606,Japan1f:C→P1oftheprojectivelinebysmoothcurves.Thus,fromtheveryoutset,oneseesthatthereareessentiallytwowaystoapproachtheHurwitzscheme:(1)WestartwithP1andregardtheobjectsofinterestascoveringsofP1;(2)WestartwithCandregardtheobjectsofinterestasmorphismsfromCtoP1.Onefi

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

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

7、7,ofthetext.NotethatalthoughinthestatementoftheTheoremhereintheintroduction,wespokeof“the”Hurwitz“scheme,”thereareinfactseveraldifferentHurwitzschemesusedintheliterature,someofwhichare,infact,notschemes,butstacks.FordetailsabouttheparticulartypeofHurwitzschemeforwhichthemainthe

8、oremisproved,wereferthereadertotheexactstatementin§6.7,aswell