The p-rank stratification of Artin-Schreier

curves

Thep-rankstrati cationofArtin-Schreier

curves

RachelPriesandHuiJuneZhu

Abstract

WestudyamodulispaceASgforArtin-Schreiercurvesofgenusgoveranalgebraicallyclosed eldkofcharacteristicp.Westudythestrati cationofASgbyp-rankintostrataASg.sofArtin-Schreiercurvesofgenusgwithp-rankexactlys.WeenumeratetheirreduciblecomponentsofASg,sand ndtheirdimensions.Asanapplication,whenp=2,weprovethateveryirreduciblecomponentofthemodulispaceofhyperelliptick-curveswithgenusgand2-rankshasdimensiong 1+s.Wealsodetermineallpairs(p,g)forwhichASgisirreducible.

keywords:Artin-Schreier,hyperelliptic,curve,moduli,p-rank.

subjclass[2000]11G15,14H40,14K15

1Introduction

Letkbeanalgebraicallyclosed eldofcharacteristicp>0.AnArtin-Schreierk-curveisasmoothprojectiveconnectedk-curveYwhichisa(Z/p)-coveroftheprojectiveline.TheRiemann-HurwitzformulaimpliesthatthegenusgofYisoftheformg=d(p 1)/2forsomeintegerd≥0.Thep-rankofYistheintegerssuchthatthecardinalityofJac(Y)[p](k)isps.Itiswellknownthat0≤s≤g.BytheDeuring-Shafarevichformula,s=r(p 1)forsomeintegerr≥0.Inthispaper,westudyamodulispaceASgforArtin-Schreierk-curvesofgenusg.Westudyitsstrati cationbyp-rankintostrataASg,swhosepointscor-respondtoArtin-Schreiercurvesofgenusgwithp-rankexactlys.Throughout,weassumeg=d(p 1)/2ands=r(p 1)forsomeintegersd≥1andr≥0

curves

sincetheproblemistrivialotherwise.Wedenoteby · and · the oorandceilingofarealnumber,respectively.Weprove:

Theorem1.1.Letg=d(p 1)/2withd≥1ands=r(p 1)withr≥0.

1.ThesetofirreduciblecomponentsofASg,sisinbijectionwiththesetofpartitions{e1,...er+1}ofd+2intor+1positiveintegerssuchthateachej≡1modp.

2.TheirreduciblecomponentofASg,sforthepartition{e1,...er+1}hasdi-mension

d 1 ∑ (ej 1)/p .

j=1r+1

Theproofusesideasfrom[2,Section5.1],[7],and[12].AsanapplicationofTheorem1.1,wedetermineallcaseswhenASgisirreducible,usingthefactthateveryirreduciblecomponentofASghasdimensiond 1,[9,Cor.3.16].Corollary1.2.ThemodulispaceASgisirreducibleinexactlythefollowingcases:(i)p=2;or(ii)g=0org=(p 1)/2;or(iii)p=3andg=2,3,5.Whenp=2,themodulispaceASgisthesameasHg,themodulispaceofhyperelliptick-curvesofgenusg.By[8,Thm.4.1],Hgisirreducibleofdimension2g 1whenp=2.LetHg,s Hgparametrizehyperelliptick-curvesofgenusgwith2-ranks.Theorem1.1yieldsthefollowingdescriptionofHg,s.Thisalsogeneralizestheresultdim(Hg,0)=g 1whenp=2from[13,Prop.4.1].Corollary1.3.Letp=2andg≥1.TheirreduciblecomponentsofHg,sareinbijectionwithpartitionsofg+1intos+1positiveintegers.Everycomponenthasdimensiong 1+s.

Whenp=2,wealsogiveacompletecombinatorialdescriptionofhowtheirreduciblecomponentsofHg,s ttogetherinHg,Corollary4.6.

ThegeometryofASgismorecomplicatedwhenp≥3.Forexample,Theorem

1.1showsthat,for xedgands,theirreduciblecomponentsofASg,scanhavedifferentdimensionsandthusASg,sisnotpureingeneralwhenp≥3.

Hereissomemotivationforthispaper,whichalsogivesanotherillustrationhowthegeometryofASgismorecomplicatedwhenp≥3.Recallthatthemod-ulispaceAgofprincipallypolarizedabelianvarietiesoverkofdimensiongcanbestrati edbyp-rank.LetVg,s Agdenotethestratumofabelianvarietieswith

2

curves

p-ranks.By[11,1.6],everycomponentofVg,shascodimensiong sinAg.SupposeMisasubspaceofMg,themodulispaceofk-curvesofgenusg.OnecanaskwhethertheimageT(M)ofMundertheTorellimorphismisingeneralpositionrelativetothep-rankstrati cation.Anecessaryconditionforanaf r-mativeansweristhatcodim(T(M)∩Vg,s,T(M))=g s.Thishasbeenveri edwhenM=Mgin[5,Thm.2.3]andwhenM=Hgforp≥3in[6,Thm.1].Corol-lary1.3showsthatthisnecessaryconditionissatis edforM=Hgwhenp=2.Corollary3.11showsthatitisnotsatis edforM=ASgwhenp≥3.

Hereisanoutlineofthepaper.InSection2,wedescribethep-ranksofArtin-Schreiercurvesandtherelationshipbetweenirreduciblecomponentsandparti-tions.Section3containstheproofofthemainresult.One ndsTheorem1.1inSection3.4,Corollary1.2inSection3.5,andCorollary1.3inSection3.6.InSection4,weconsiderhowthecomponentsofASg,s ttogetherinsideASg.OnecanaskwhetherASg,s (p 1)isintheclosureofASg,sinASg.Wegiveanaf rmativeanswerinsomecases(inparticular,wheneverp=2)andanegativeanswerinothers.Thisinvolvesdeformationsofwildlyrami edcoverswithnon-constantbranchlocus.Weconcludewithsomeopenquestions.2PartitionsandArtin-Schreiercurves

2.1Partitions

Fixaprimep>0andanintegerd≥1,withdevenifp=2.Let dbethesetofpartitionsofd+2intopositiveintegerse1,e2,...witheachej≡1modp.Let d,rbethesubsetof dconsistingofpartitionsoflengthr+1.If E∈ d,letr:=r( E)betheintegersothat E∈ d,r.Write E={e1,...,er+1}withe1≤···≤er+1.Thereisanaturalpartialordering<on dsothat E< E′if E′isare nementof E,inotherwords,iftheentriesof E′canbepidedintodisjointsubsetswhosesumsareinbijectionwiththeentriesof http://doc.guandang.netingthispartialordering,onecanconstructadirectedgraphGd.Theverticesofthegraphcorrespondtotheparti-tions Ein d.Thereisanedgefrom Eto E′ifandonlyif E< E′,and E= E′, ′′<E ′forandthereisnopartitionlyingstrictlyinbetweenthem(i.e.,if E<E ′′in dthenE ′′= ′′=E ′).someEEorE ′inthedirectedgraphGdcanbeoftwotypes.The rsttypeAnedge E<E

hasr( E)=r( E′) 1.Inthiscase,oneentryeof Esplitsintotwoentriese1ande2of E′sothate=e1+e2andnoneofthethreeiscongruentto1modulop.Onecansummarizethisbywriting{e}→{e1,e2}.Thesecondtypehasr( E)=r( E′) 2.

3

curves

Inthiscase,oneentryeof Esplitsintothreeentriese1,e2,e3of E′sothate=e1+e2+e3andeachej≡(p+1)/2modp.Itfollowsthatnoneofthefouriscongruentto1modulop.Onecansummarizethisbywriting{e}→{e1,e2,e3}.Example2.1.Letp=3andd=10.HereisthegraphG10for 10.{12}

{6,6{3,9}

{2,2,8}

{2,2,2,6}{2,5,5}{2,2,3,5}

{2,2,2,3,3}

{2,2,2,2,2,2}

Weskiptheproofsofsomeofthefollowingstraightforwardresults.Lemma

2.2isusedin[1],whileLemmas2.3and2.4areusedinSection3.5.

Lemma2.2.Theset d,0isnonemptyifandonlyifp (d+1).Ifp (d+1 …… 此处隐藏：23102字，全部文档内容请下载后查看。喜欢就下载吧 ……