Effective cones of quotients of moduli spaces of s

arXiv:math/0311358v2[math.AG]24Jan2006EFFECTIVECONESOFQUOTIENTSOFMODULISPACESOFSTABLEn-POINTEDCURVESOFGENUSZERO.WILLIAMF.RULLAAbstract.LetXn:=M0,n,themodulispaceofn-pointedstablegenuszerocurves,andletXn,mbethequotientofXnbytheactionofSn−monthelastn−mmarkedpoints.TheconesofeffectivedivisorsNE1(Xn,m),m=0,1,2,arecalculated.Usingthis,upperboundsfortheconesMov(Xn,m)generatedbydivisorswithmovinglinearsystemsarecalculated,m=0,1,alongwiththeinducedboundsontheconesofampledivisorsofMgandMg,1.Asanapplication,theconeNE1(M2,1)isanalyzedindetail.1.IntroductionInstudyinganobjectXinthecategoryofprojectivevarieties,itisusefultoknowtowhichotherobjectsXadmitssurjectivemorphisms,ormoregenerally,dominantrationalmaps.Thus,ofinterestarethesetsoflinebundleswhichhavebasepointfree,ormoregenerally,non-empty,linearseries.TheclosuresNef1(X)andNE1(X)ofthecorrespondingconesintherealvectorspaceN1(X)cansome-timesbedeterminedwithnothingmorethanbasicintersectiontheory.TheintermediateconeNef1(X)⊆Mov(X)⊆NE1(X),generatedbyalllinearseries|M|whichhavebaselocibl|M|ofcodimension≥2,isofgreaterinterestthanNE1(X)instudyingrationalmapsϕ|D|:X99KZ,sinceitisnaturaltoextendamapϕ|D|overaCartierdivisorE∈bl|D|byconsidering|M:=D−E|instead.Mov(X)similarlyhasanupperboundNem(X)whichispotentiallycalculableviabasicintersectiontheory.Thepurposeofthispaperistointroducethe“nem”coneandsomeofitsproperties,andcalculateitforcertainfinitequotientsofthemodulispacesM0,nofstablen-pointedcurvesofgenuszero.UntilrecentlyitwasconjecturedthattheconesNE1(M0,n)weregeneratedbytheirreduciblecomponentsofthelociofsingularcurvesΔ⊆M0,n.Thisisnowknowntobefalseforn≥6([HT02]or[Ver02]),buttheanalogousstatementistrueforthequotientsXn,0:=M0,n/Snbythenaturalactionofthesymmetricgroup.Anaturalquestion(S.Keel)is:forwhichmdoestheanalogousstatementholdforthequotientsXn,m:=M0,n/Sn−m,whereSn−mactsonthelastn−mmarkedDate:November19,2003.2000MathematicsSubjectClassification.Primary:14E05,14H10;Secondary:14E30.Keywordsandphrases.Modulispace,rationalcurve,birationalgeometry,classificationofmorphisms/rationalmaps.ThispaperisaproductofaVIGREseminaronM0,nconductedbyV.AlexeevattheUni-versityofGeorgia,Athens,duringtheSpringof2002.ThankstoS.Keelforposingthequestionmotivatingthepaper,andtohim,R.Varley,andE.Izadiforhelpandadvice.Thanksalsototherefereeformanyvaluablecomments.PORTA[POR]wasusedincalculatingseveralexamples.Xfigwasusedforthefigures.12W.F.RULLApoints?Thispaperprovidestheanswer:form≤2orn≤5(Propositions4.2,5.3,and7.2).Alsoincludedisasystematictreatmentandextensionofsomeoftheadhoccalculationsof[Rul01],relatingmodulispacesMg,ktovariousXn,m.Itistheauthor’sintenttousetheseresultsforongoingworkonM3andotherspaces.Thepaperisstructuredasfollows:§2establishesnotation,definitions,andbasicpropertiesofNem(X).In§3basesfortherealvectorspacesN1(Xn,m)(0≤m≤3)arecalculated.§4pullsbackthecounterexampleof[HT02]and[Ver02]inX6,3toallXn,m,n≥6andm≥3.§§5-8consistofintersection-theoreticalcalculationsofeffectiveandnemconesandexamples.In§9thecalculationsofNem(X2g+2,0)andNem(X2g+3,1)areusedtoimposeboundsontheconesofampledivisorsofMgandMg,1,respectively,viasurjectionsofthesespacesontothecorrespondinglociofhyperellipticcurves.Asanapplication,in§10theconeofeffectivedivisorsNE1(M2,1)ofthemodulispaceM2,1ofstablegenus2curveswithmarkedpointisstudiedindetail.2.Notation,definitions,andfunctorialpropertiesNotation.LetXbeaprojectivevariety,overa(nalgebraicallyclosed)fieldk.Thecharacteristicc=charkwillbearbitrary,exceptfortherestrictionsc6=2in§§9and10,andc=0fortheapplicationsofMoritheoryin§10.(1)Thedescription”Q-factorial”willimplynormality.(2)Thesymbol“≡”willbeusedtodenotenumericalequivalence(ofCartierdivisorsorone-cycles).(3)N1(X)isthegroupofone-cyclesmodulonumericalequivalence,withrealcoefficients[Kol96,p.122].(4)N1(X)isthegroupofCartierdivisorsmodulonumericalequivalence,withrealcoefficients[Kol96,p.123].(5)TheconeofeffectivedivisorsNE1(X)istheclosureoftheconeinN1(X)generatedby(classesof)linebundleshavingnon-zeroglobalsections.(6)AQ-CartierdivisorM∈NE1(X)willbecalled“moving”ifthereisapositiveintegernsuchthat(nMisCartierand)thebaselocusofthelinearseries|nM|isofcodimension≥2.TheconeofmovingdivisorsMov(X)istheclosureofthesubconeofNE1(X)generatedbymovingdivisors.(7)TheconeofnefdivisorsNef1(X)istheclosureofthesubconeofNE1(X)generatedbylinebundleshavingbasepointfreelinearsystems.FollowingMilesReid,wethinkoftheword“nef”asanacronymfor“numericallyeventuallyfree.”(8)TheconeofnefcurvesNef1(X)istheconeinN1(X)dualtoNE1(X).FollowingMilesReid,Definition.LetXbeaprojectivevariety.AQ-CartierdivisorM∈NE1(X)willbecalled“nem”(for“numericallyeventuallymoving”)ifforeveryprimedivisorD⊆X,M|D∈NE1(D).TheconeofnemdivisorsNem(X)istheclosureoftheconeinNE1(X)generatedbynemQ-Cartierdivisors.EFFECTIVECONESOFMODULISPACES3Remarks.(1)Clearlythenemconeisanupperboundonthemovingcone:Mov(X)⊆Nem(X).(2)Forsurfaces,thenef,moving,andnemconesarethesame.(3)IfD≡PαiDiwhereαi0andtheDiareprimedivisors,thenD∈Nem(X)iffD|Di∈NE1(Di)foreachi.(4)IfM∈Nem(X)isaprimedivisorandC∈Nef1(M),thenC∈Nef1(X).Weprovesomefunct