Galois theory monodromy groups and flexes of plane cubic curves - PDF document

Contents1Abstract12GaloisandMonodromyGroups32.1GeneralFibresofaDegreedRationalMap...............32.2TheGaloisGroup.............................42.3TheMonodromyGroup..........................63FlexesofPlaneCurves93.1LinearSystemsofPlaneCurves.....................93.2TheClassicalPl¨uckerFormulas.....................123.3EllipticCurvesandComplexTori....................213.4TheMonodromyGroupoftheNineFlexes................303.5LocatingtheFlexes............................334References34 1AbstractIn1979,J.Harris'paperGaloisGroupsofEnumerativeProblemswaspublished,which”[...]isconcernedwiththesolvabilityofcertainenumerativeproblemsinalge-braicgeometry”.TheemphasisofmythesiswillbetherstproblemHarrisintroduces,whichconsistsoftwoquestions:GivenacomplexplanecurveCofdegreed,howmanyexesexistonthiscurve?Secondly,canwendthem?WewillatrstintroducetheGaloisandthemonodromygroupofadominantmor-phismbetweenvarieties,whichturnouttobeidentical.Therstgroupwillbedenedintermsoftheinducedextensionoffunctionelds-andinfactreectthesecondquestionabove-whereasthemonodromygroupallowsforaneffectivecomputationbyconsideringloopsinthebasespace(whichwillbetheparameterspaceofthecurvesweconsider)andtheircorrespondinglifttothetotalspace.WewillthenturnourattentiontotherstquestionandusetheclassicalPl¨uckerfor-mulastooutlinethefactthatasmoothplanecurveCofdegreedhasprecisely3d(d�2)exescountedwithmultiplicities.Therelevantspecialcasethatanyellipticcurvehasninedistinctexesisprovendirectly.Theseformulaswilltellusabitmoreaboutthegeometryofmildlysingularplanecurvesastheyrelatevariousnumericalinvariantstoeachother:Namely,thedegree,thenumberofexes,thenumberofbitangents,thenumberofcusps,thenumberofdoublepointsandthecorrespondingvaluesforthedualcurveofC.ThisisanalgebraiccurveinthedualprojectiveplanethatisdenedasthesetofalltangentlinestoC.Inthenextsection,wedescribeellipticcurvesascomplextori,whichwillbeneededwhenwecomputetheGaloisgroupofthenineexes.Hence,wewillintroduceel-lipticfunctionsandperiodlatticestoconstructagroupisomorphismfromasuitableone-dimensionalcomplextorustoanygivenellipticcurveinWeierstrassform.WewillobservethatthenineexesofageneralplanecubicChavethestructureofatwo-dimensionalvectorspaceoverF3,andthemonodromygroupintroducedintherstchapterturnsouttopreservemuchofthisstructure:Infact,itisthesolvablegroupofafne-lineartransformationswithdeterminantoneonF23.Finally,wewillseethattheexesofanellipticcurveinWeierstrassformmaybeexplicitlycalculated.GermanSummary.DieseBachelorarbeitbefasstsichmitdemerstenProblem,dasJ.Harris'inseinem1979ver¨offentlichtenWerkGaloisGroupsofEnumerativeProblemsvorstelltundwelchesauszweiFragenbesteht:WievieleWendepunkteexistierenaufeinerkomplexen,ebenenalgebraischenKurveCvomGraddundk¨onnenwirdiesenden?Hierzuf¨uhrtHarrisdieGalois-undMonodromiegruppeeinesdominantenMorphis-muszwischenkomplexenVariet¨atenein,welchesichalsidentischherausstellen.DieersteGruppewirdmithilfederinduziertenErweiterungderFunktionenk¨orperdeniertundreektiertdiezweiteobigeFrage,wobeidieMonodromiegruppeindersp¨aterenAnwendungf¨urdieBerechnungverwendetwird.DieersteFragewirdmithilfederPl¨uckerformelnbeantwortet,dieverschiedenenu-merischeInvariantenvonebenenKurvenmiteinanderinVerbindungsetzen.Derf¨urunsrelevanteSpezialfall,dasseineelliptischeKurvegenauneunverschiedeneWen-depunktebesitzt,wirdgesondertbewiesen.1 ZurBerechnungderMonodromiegruppederneunWendepunkteeinerelliptischenKurvestellenwirdiesealskomplexenTorusdar.HierzuwerdenwirelliptischeFunk-tionenundGittereinf¨uhren,umeinenGruppenisomorphismusvoneinemgeeigneteneindimensionalenkomplexenTorusaufeinegegebeneelliptischeKurveinWeierstrass-formzukonstruieren.IndemletztenTeilstellenwirfest,dassdieMonodromiegruppevielvonderUnter-gruppenstrukturderneunWendepunktebewahrtundwirberechnensiealsdieau¨osbareGruppeASL2(F23).ZuletztwerdenwirdieWendepunkteeinerelliptischenKurveinWeierstrassformexplizitanhandderKoefzientenihrerGleichungermitteln.Acknowledgements.Iwouldliketothankmyadvisor,Prof.Dr.DanielHuybrechts,forhisthoroughsupportaswellasforhishighlyinstructivelectureswhichmadethisthesispossible.IamalsothankfultoMichaelKemenyforreadilyansweringmyques-tionsandforgivinggreatgeneraladvice.Someremarksonnotation.ThegroundeldisalwaysassumedtobeCunlessstatedotherwise.Afnen-space(overthecomplexnumbers)isdenotedAn,andprojectiven-spacewillreferredtoasPn:=(Cn+1nf0g)=C.ThecoordinateringandthefunctioneldofavarietyXarecalledK[X]andK(X),respectively.2 Thus,themappinducesanisomorphismpa:Ka'�!Kbythecompositionf7!fp�1,wherewenotethattheinverseofpisonlylocallydened.Furthermore,wemayembedthefunctioneldofXintoKbyrestriction:f:K(X)�!K:Similarly,byembeddingrationalfunctionsonYintoKaandthenapplyingpa,weobtainaninjectionfa:K(Y)�!K:DeneKa=fa(K(Y))KandletLbethesubeldofKgeneratedbythesub-eldsKa.Ifwelet˜gi=f(gi)˜fa=fa(f)˜P(T)=Td+˜g1Td�1+:::+˜gdtheneachfunction˜fasatises˜P(˜fa)=˜fda+˜g1˜fd�1a+:::+˜gd=0:Simplynotethatfa(P(f))=0bydenitionofP,andso-expandingtheleft-handside-theclaimfollowssincefa(p(gi))=˜gi.ToseethatLisindeedthesplittingeldofP,itsufcestoshowthatallthe˜faaredistinctsincePhasdegreedandLisbydenitionthesmallesteldcontainingallthe˜fa.Butif˜fa1=˜fa2,theninparticularf(qa1)=˜fa1(p)=˜fa2(p)=f(qa2),whichisimpossiblebecauseotherwisealltherationalfunctionsinK(Y)wouldcoincideatqa1andqa2,whichisfalse(simplynotethatforanytwodistinctpointsonavariety,thereexistsapolynomialfunctionvanishingonpreciselyoneofthem.Furthermore,noneoftheqaiwillbepolesoffforageneralp).WemaythusdeneG=Gal(L=K(X)),andeveryautomorphisminGinducesapermutationofthedrootsf˜fagofP.Consequently,wehaveaninclusionG,!Sd.2.3.TheMonodromyGroupThemonodromygroupisconstructedmerelyintermsofacoveringmapbetweentopologicalspacesandanactionofthefundamentalgroupofthebasespaceonthebreofp.Thenextresultshowsthatwemayregardacertainrestrictionofpasad-sheetedcovering.Theorem2.5.ThereexistsasufcientlysmallZariskiopensubsetUXwithpreimageV=p�1(U)Ysothatp:V!Uisanunbranchedcoveringmapwithrespecttotheanalytictopology.Proof.UsingLemma2.1,wereadilyobtainanunramiedrestrictionofp.FurtherrestrictionofpviaLemma2.4yieldsalocalbiholomorphismp:V!U,i.e.foreachpointq2p�1(p),thereexistsaneighbourhoodVqwhichismappedbiholomorphicallyontosomeopenneighbourhoodUpofp.ItremainstoshowthatwemayshrinkUpsuchthatalldpreimagesVqaremappedtothesameopenneighbourhood,whichfollowsfromthefactthatpreimagesunderpofcompactsetsremaincompact.Thelatterstatementistruebypropernessofpandcanbefoundin[1,A.14.6]. 6 Fig.1-LiftingalooptothetotalspaceandthemonodromyactionWhileitisobviousthat˜ga(1)mustlieinG,thecrucialpointisthatitmaynotcoincidewithqa,soeachliftingdenesapermutationofthebre:fg:G�!Gqa7�!˜ga(1):LetIdenotetheunitinterval[0,1].Thecoveringhomotopytheorem(cf.[2,Theorem3.4.])statesthat,foranylocallyconnectedspaceWandhomotopyF:WI�!Xwithprescribedliftingf:Wf0g�!Y,thereexistsauniquehomotopyG:WI�!Ymakingthefollowingdiagramcommute:Wf0gVWIU.........................................................................................................................................F................................................................................................................................................................................................................................................................................................f......................................................................................................................................................p.........................................................................................................................G8 IfFisahomotopyrelativetoW0foranarbitraryW0W,thensoisG.Inparticular,lettingthemapFbeahomotopyoftwopathsg1andg2rel.¶I(i.e.W=I),anytwoliftings˜g1and˜g2with˜g1(0)=˜g2(0)willsatisfy˜g1(1)=˜g2(1).Therefore,fgonlydependsonthehomotopyclassofg,sowehaveagrouphomomorphismp1(U;p)�!Sd[g]7�!fgTheimageMofthismapiscalledthemonodromygroupofp.Toconcludethischapter,wehavethefollowingresult.Theorem2.7.ThemonodromyandGaloisgroupareidentical.Wereferto[8,p.689]fordetails.Theideaisthatanalyticcontinuationofagermh2KaalongaloopdenesanautomorphismofLleavingtheembeddedfunctioneldf(K(X))xed,andhenceanyloopinUwhichgivesrisetoanelementofthemon-odromygroupalsopermutestherootsof˜P.Conversely,onehastoshowthateveryautomorphisminGisobtainedbyanalyticcontinuation.HarrisdoesthisbyprovingthatthexedeldofMGisf(K(X)),fromwhichG=Malreadyfollowsbydeni-tionoftheGaloisgroup.3FlexesofPlaneCurvesThischapterisdedicatedtothemonodromygroupofthenineexesofaplanecubiccurve,aproblemthathasyettobemadeprecise.Intherstpart,theissueofdeningthevarietiesIdandWd-thelatteronebeingthespaceofplanecurvesofdegreed-aswellasthemorphismIdp�!Wdisaddressed.Therstobjectivewillbetocomputethedegreeofthismap,orequivalently,thecardinalityofageneralbre.Sincethebreofacurvewillbijectontothesetofitsinectionpoints,theproblemofdeterminingthedegreeisreducedtothecomputationofthenumberofexesonageneralplanecurve,whichwewilldealwithinSection3.2.BeforewenallycancomputethemonodromygroupofpasASL2(Z=3),wewilldescribeellipticcurvesascomplextoribymeansoftheWeierstrassÃ-function.3.1.LinearSystemsofPlaneCurvesBeforesettingupthemainproblem,werecallsomebasicresultsanddenitions.AplanecurveCoverthecomplexnumbersisthezerolocusofahomogeneouspolynomialF2C[X0;X1;X2]d,i.e.C=V(F)P2:Morepreciselythen,Ciscalledaplanecurveofdegreed.Foranirreduciblepoly-nomialG,thelociV(G)andV(Gn)coincideforapositivepowern,sowemayassumethatthepowersoftheirreduciblefactorsinthedecompositionofapolynomialFareone.WewillalwaysassumethisinSection3.2aboutthePl¨uckerformulas.Now,letCandC0betwoplanecurveswithnocommoncomponentdenedbyhomogeneous9 polynomialsFandG,respectively.WedenetheintersectionmultiplicityofCandC0atapointp2P2asIp(C;C0)=dimCOP2;p=(F;G)where,foranalgebraicvarietyXandp2X,OX;pisthestalkatpofthestructuresheafofX,i.e.thelocalringofregularfunctionsatpwithmaximalidealmp=ff2OX;p:f(p)=0g.NotethatIp(C;C0)1ifandonlyifp2C\C0bytheverydenition.Itisalsoworthwhiletonotethatintersectionmultiplicitiesmaybecomputedintermsofafnecoordinates:Ifp=[p0:p1:p2]andp06=0-i.e.pliesintheafneopensetf[x:y:z]:x6=0g-thendimCOP2;p=(F;G)=dimCOA2;(p1;p2)=(F(1;X1;X2);G(1;X1;X2)):Here,wemadeuseofthefactthatafneandprojectivefunctioneldsareisomorphic,cf.[9,p.75].ThefollowingclassicalstatementabouttheintersectiontheoryofplanealgebraiccurvesisindispensableforusandcanbefoundasTheorem4.8in[9].Theorem3.1.(B´ezout)LetCandC0betwoplanealgebraiccurvesofdegreedandd0respectively,whichhavenocommoncomponents.ThenCandC0intersectindd0pointscountedwithmultiplicities,i.e.åp2P2Ip(C;C0)=dd0:ThecurvesCandC0aresaidtointersecttransverselyatp2C\C0ifbothcurvesaresmoothatpandTpC\TpC0=fpg,whereTpCandTpC0denotetherespectivetangentspacesatp.ByNakayama'sLemma(cf.[9,p.119]),thisisthecaseifandonlyifIp(C;C0)=1.Finally,asmoothpointp2CiscalledaexorinectionpointofCifIp(C;TpC)3,andpisasimpleexifequalityholds.WedenotebyWdthesetofallprojectiveplanecurvesofdegreed,i.e.Wd=P(C[X0;X1;X2]d).Notethattwohomogeneouspolynomialswhichdifferbyanonzeroscalardenethesamecurvebythisdenition,butwe-contrarytothenextsection-stilldistinguishtwocurvesfromoneanotheriftheprimepowersoccurringinbothfactorizationsdonotcoincide.Lemma3.2.TheparameterspaceWdisthecomplex-projectivespaceofdimension�2+d2
�1.Proof.Foranyk-vectorspaceofdimensionn,wehavethatdimkSd(V)=n+d�1d;whereSd(V)denotesthed-thsymmetricalgebraofV.SinceC[X0;X1;X2]d=Sd(C3),wemayconcludethatdimCC[X0;X1;X2]d=�2+dd
bytheaboveequationandthereforeidentifyWdwithP(2+d2)�1=Pd(3+d) 2. LetN=�2+d2
.10 Corollary3.6.Ageneralplanecurveofdegreedisirreducible.Proof.Thisfollowsfromthefactthatanyreducibleplanecurveissingular,whichisobviousbyB´ezout'stheorem:IfC=V(F1)[V(F2)isadecompositionintoirreduciblecomponents,thenthereexistssomep2P2suchthatF1(p)=F2(p)=0.Now,¶(F1F2) ¶Xi=(¶F1)F2 ¶Xi+(¶F2)F1 ¶XiandsopisasingularpointofC. Wewillnowsetupthemorphismp.RecallthatthedualprojectivespacePnisthespaceoflinesinPn.Now,letI0P2P2bethesubsetI0=f(p;l):p2lg.Lemma3.7.ThesetI0isanirreduciblethreefold.Proof.Thisisanapplicationofamuchmoregeneraltheorem.WeletG(r+1;n+1)betheGrassmanianvariety,i.e.thesetoflinearr+1-dimensionalsubspacesoftheafnen+1-spaceconsideredasaprojectivevarietyviathePl¨uckerembedding.WethendeneI(r;d;n)=f(X;e)2P(n+dd)�1G(r+1;n+1):eXg;thatis,thesetoftuples(X;e)whereXisaprojectivehypersurfaceofdegreedinPncontainingthelinearsubspacee.NotethatG(1;n+1)isisomorphictoPn(cf.[4,p.107]),sowehaveI0=I(0;1;2).Lemma12.6in[4]saysthatI(r;d;n)isanirreduciblesubvarietyofP(n+dd)�1G(r+1;n+1)ofdimension(r+1)(n�r)+n+dd�r+dd�1;soweobtainthedesiredresult. DeneIdWdI0asId=f(C;p;l):Ip(C;l)3g,thatis,thesetoftriples(C;P;l)suchthatPisaexofCwithtangentlinel.Ifweleth:Id!I0betheprojectiononthesecondfactor,weseethatIdisirreduciblebyLemma12.7in[4]andLemma3.7.Letp:Id!Wdbetheprojectionontherstfactor.Thebreofacurveisinone-to-onecorrespondencewiththesetofitsinectionpoints,henceweseethatthecardinalityofthegeneralbre-andthus,thedegreeofp-isgivenasthenumberofexesonageneralplanecurveofdegreed.Bytheprevioustheorem,wemayassumesuchacurvetobesmooth,whichturnsouttobehelpfulforplanecubiccurves.Inthiscase,everysmoothcubiccurve-thatis,everyellipticcurve-haspreciselyninedistinctexes,sotheproblemofdeterminingthedegreeofpisreducedtoprovingthisstatement.3.2.TheClassicalPl¨uckerFormulasOnemayaskoneselfhowmanypointsexhibit-ingacertainpropertycanbefoundonaplanealgebraiccurve.Inourcase,wewouldliketocounttheexes,butonemaysimilarlyaskhowmanybitangentsorcertainsin-gularitiesagivenplanecurveChas.ForageneralcubiccurveC,theanswertothesequestionsisparticularlyeasy:SuchacurvewillbesmoothbytheprevioussectionandpossessnobitangentsbyB´ezout'stheorem.Furthermore,everyexmustobviouslybesimple-alsobyB´ezout-whichimpliesthatthereareexactlyninedistinctexesonCasweshallprovesoon.12 Lemma3.10.Letfandpbeasbefore.Theorderoffinpisboundedvia0ordp(C)deg(C);withordp(C)�0ifandonlyifp2C.Furthermore,smoothnessinpisequivalenttoordp(C)=1.Inotherwords,Cissingularinpifandonlyifordp(C)�1.Lemma3.11.LetC=V(f)A2beanalgebraiccurveandlalinethroughp2C.Thenordp(C)Ip(C;l),andthereareatmostordp(C)linesthroughpforwhichthisinequalityisstrict.Proof.Afteralineartransformation,wemayassumethatpistheorigin.Letf=nåk=rf(k)betheTaylorexpansionoffaroundpwithr=ordp(f)andd=deg(f),andletlbeparametrizedbyg(T)=(l1T;l2T).Considerg(T):=f(g(T))=dåk=rf(k)(l1;l2)Tk:ThusIp(C;l)=ordp(g)ordp(f).Notethatthisinequalityisstrictifandonlyiff(r)(l1;l2)=0,andf(r)hasatmostrdistinctzeroesinP1byLemma3.8. Wemaynowgeneralizeourdenitionoftangentlinesandobtainadescriptionofacertaintypeofsingularityintheprocess.Denition3.12.LetCandlbeasbefore,p2C\landr=ordp(C).ThelinelistangenttoCatpifrIp(C;l).Bytheaboveproposition,thereareatmostrsuchlines.Thepointpisanordinaryr-foldpointifthismaximumisattained,i.e.therearerdistincttangentlinesatp.Lemma3.13.LetC=V(f)A2beanalgebraiccurvewhichissmoothattheoriginwithtangentlinel=V(X2).Supposethats=IP(C;l)¥(i.e.lisnotcontainedinC).Thenf(X1;X2)=Xs1g(X1)+X2h(X1;X2)withg(0)6=0andh(0;0)6=0.Proof.Letr=ord(0;0)(f),d=deg(f)andconsidertheTaylorexpansionaroundtheoriginf(X1;X2)=åkrf(k)=åkråm+n=kamnXm1Xn2:Letlbeparametrizedbyg(X)=(X;0)anddeneg(X):=f(g(X))=åkrf(k)(1;0)Xk:14 Euler'sformulastatesthat,foranyhomogeneouspolynomialG2k[X0;X1;:::;Xn]d,wehavenåi=0¶G ¶Xi=d
G:ApplyingthistoeachFi,wemayreplacetherstrowofthelattermatrixandobtaind�1 X0det0@F0F1F2F01F11F21F02F12F221A=d�1 X20X1X2det0@X0F0X1F1X2F2X0F01X1F11X2F21X0F02X1F12X2F221A=d�1 X20det0@X0F0+X1F1+X2F2F1F2X0F01+X1F11+X2F21F11F21X0F02+X1F12+X2F22F12F221A:UsingEuler'sformulaagaintoreplacetherstcolumn,wegetthedesiredequality. Corollary3.17.TheHessianH(C)containsallthesingularpointsofC.Proof.Letp2Cbesingular,wherewemayassumethatp06=0sincetheHessianisindependentofthecoordinates.ThenthestatementisimmediatebyLemma3.16,sincetherstrow�dFF1F2
(p)vanishes. Theorem3.18.[5,Section4.5]LetC=V(F)P2beanirreduciblecurveofdegreed2.Thenasmoothpointp2Cisaexifandonlyifp2H(C).Inparticular,forasmoothcurveC,theintersectionC\H(C)consistsofallexesonC.Finally,CandH(C)intersecttransverselyineverysimpleex.Proof.Letp=[1:0:0]2Cbeasmoothpointwithtangentlinet=V(X2).Lemma3.13allowsustowriteF(1;X1;X2)asf(X1;X2)=Xk1g(X1)+X2h(X1;X2)withg(0)6=0,h(0;0)6=0andk=Ip(C;t)2.Hence,wewriteXk1g=a2X21+a3X31+:::andh=b+b1X1+b2X2+:::,b6=0,wherethedotsstandfortermsofhigherorder.ComputingderivativesandusingLemma3.16,weobtaindetHF(p)=(d�1)20@00b02a2b1bb12b21A=�2(d�1)2b2a2:Now,recallthatpisaexifandonlyifk3,whichisequivalenttoa2=0.Thisconcludestherstandsecondstatement.ItremainstoprovethatifIp(C;t)=3,thenIp(C;H(C))=1.Firstofall,weexpandthedeterminantofHF(1;X1;X2)andobtain(d�1)(ff11f22+f1f21(d�1)f2+f2(d�1)f1f12�(d�1)f2f11f2�(d�1)f21f22�ff12f21);wherethelowercasefdenotesthedehomogenizedpolynomial.Wedene˜f=f22f11+f21f22�2f12f2f116 andnotethatIp(C;H(C))=I0(V(f);V(˜f))(inthedenitionof˜f,weomitthosetermscontainingf).Ashortcalculationrevealsthatin˜f,themonomialX1hasthecoefcient6a3b,whichdoesnotvanishifpisasimpleex.Inparticular,V(˜f)issmoothatpandthetangentlinesofbothcurvesaredistinct,whichprovesthestatementbyRemark3.14:Simplyrecallthatsmoothnessatapointpisequivalenttohavingorderoneinp. Corollary3.19.AgeneralplanecubiccurveChasninedistinctexes.Proof.Considerthepolynomialdet(HF).SimilarlytoTheorem3.5,oneprovesthatdet(HF)isnotidenticallyzeroforanopenanddensesubsetoftheparameterspaceWd,orinotherwords,H(C)isacubiccurveforageneralC.ThecubicConlyhassimpleexesbyB´ezout,soCandH(C)intersecttransverselyineachpointC\H(C)byTheorem3.18.Hence,applyingB´ezoutagain,weseethatChasninedistinctexes. Onemayprovethepreviousresultforallsmoothcubiccurves,notjustageneralone.Toensurethatdet(HF)isnonvanishing,weneedtheexistenceofatleastonepointthatisnotaex,whilsttherestoftheproofremainsunchanged.Thisobservationimmediatelyyieldsaslightgeneralization.Remark3.20.AsmoothplanecurveCofdegreedsuchthateveryexissimplehasprecisely3d(d�2)distinctexes.WenowmakeseveralpreparationsfortheformulationandproofofthePl¨uckerrela-tions.Unlessstatedotherwise,thecurvesareallowedtobereducibleagain.Denition3.21.LetC=V(F)P2beanalgebraiccurvewithdeg(F)2.Forapointq=[q0:q1:q2]2P2,deneFq=q0¶F ¶X0+q1¶F ¶X1+q2¶F ¶X2andCq=V(Fq),therstpolarofCwithrespecttoq.Theorem3.22.LetCandqbeasbeforewiththeadditionalassumptionthatCcontainsnolinesthroughq.ThenCqisanalgebraiccurveofdegreedeg(C)�1thathasnocommoncomponentwithC.TheintersectionC\CqconsistsofallthesingularitiesofCaswellasthepointsonCwhosetangentspassthroughq.Proof.Letd=deg(C)andq=[1:0:0]afterasuitablecoordinatetransformation.Supposedeg(Fq)d�1,whichimmediatelyimpliesFq=0.Hence,F(X0;X1;X2)=F(1;X1;X2)2C[X1;X2]ishomogeneousofdegreed,soLemma3.8tellsusthatCcontainsalinethroughq(infact,Cdegeneratesintoaunionoflines).Now,supposeFandFqshareacommonprimefactorG.ThenF=GHandGH0=Fq=¶F ¶X0=H¶G ¶X0+G¶H ¶X0;henceGalsodividesH¶G ¶X0.If¶G ¶X0isnonzero,thenGdividesHbecauseitisprimebyassumptionand¶G ¶X0isofstrictlysmallerdegree.Thus,G2isadivisorofF,but17 thisisnotpossible:WemayalwayschooseFtobeminimal,inthesensethatitisnotdivisiblebythesquareofanyprime(cf.Section3.1).Ontheotherhand,if¶G ¶X0vanishes,thenG2C[X1;X2]isalinearfactor,whichimpliesthatCcontainsalinethroughq,contradictingourassumption.ThestatementabouttheintersectionC\Cqisobviousfromthedenitions. Theorem3.23.[5,Proposition4.3]Letq=2Candp2Cbeapointwithsimpletangentlinel(i.e.Ip(C;l)=2)passingthroughq.ThenthecurveCanditspolarCqintersecttransverselyinp.Denition3.24.WedenethedualcurveofCasC=fl2P2:listangenttoCg:WedenetheclassofCasthemaximalnumberoftangentstosmoothpointsofCthatpassthroughaxedpointinP2,anddenotethisvaluedifCisacurveofdegreed.Wewillneedthreeresultsonthedualcurve:Thatitisalgebraic,thatitisirreducibleifCitselfwasirreducibleandthatdualizingCyieldsC.Weexpecttheclassofacurvetoberelatedtothedualcurve,whichwillbeformalizedbytheclassformula.Firstofall,thealgebraicityofCreliesonthefollowinglemma,whichisproveninChapter8of[5].Lemma3.25.LetCP2beanalgebraiccurveandletpbesomepointcontainedinC.AlinelP2istangenttoCatpisandonlyifthereisasequenceofsmoothpointsfpngn2NCconvergingtopsuchthatl=limn!¥TpnC:Theorem3.26.(Duality)(1)Thedualcurveisalgebraic.(2)IfCisirreducible,thensoisC.Furthermore,C=C.Proof.(1)LetC=V(F),degF=d.WeconsidertheintersectionofCwithanarbitrarylinel=V(y0X0+y1X1+y2X2),wherewemayassumethaty26=0.Hence,wecansolvetheequationoflforX2andthusobtainapolynomialG(X0;X1):=yd2F(X0;X1;�1 y2(y0X0+y1X1))=b0Xd1+b1Xd�11X0+:::+bdXd0thezerolocusofwhichcoincideswithC\l.Notethateachbiishomogeneousofdegreediny0;y1andy2-whichwenowinterpretasvariables-andsotheresultantofg(X1):=G(1;X1)andg0isanelementofC[Y0;Y1;Y2]2d2�d.DeneC0:=V(b0Dg)=V((�1)d(d�1) 2res(g;g0))P2;whichisanalgebraiccurveofdegree2d2�d.Thisisnotyetthedualcurve,butcontainsit:Ifl2C,thatis,listangenttoCinsomepointp,thenIp(C;l)�1andthusGhasamultiplezero.Ifthispointis[0:1],thenG(0;1)=b0(y)=0,otherwiseghasamultiplezeroandsoitsdiscriminantDgvanishes.Inbothcases,l2C0follows.Aswewillnowsee,C0containstwosortsoflineswhichwemusteliminate.18 TheclassformulathenfollowsfromthefactthatCandIpCintersectwithmultiplici-ties2and3inasimpledoublepointorsimplecusp,respectively.Thiscalculationcanbefoundin[5,p.91].Fortheinectionpointformula,wenotethatConlyhassimpleexesbyassumptiononthedualcurveandconsiderC\H(C).Recallthatbothcurvesintersecttransverselyinasimpleex,henceweget3d(d�2)=åp2CnsIp(C;H(C))+åp2CsingIp(C;H(C))=s+åp2CsingIp(C;H(C)):Itthusremainstocomputetheintersectionnumbers,namelythatCandH(C)inter-sectwithmultiplicity6and8inasimpledoublepointorsimplecusp,respectively(cf.[5,p.92]). AgeneralizationofthoseformulastoencompassarbitrarysingularcurvescanbefoundasTheorem2in[3,Chapter9.1].Inanycase,theexesofhigherorderhavetobecountedwithmultiplicities:Thatis,asmoothpointpwithIp(C;TpC)�3countsasIp(C;TpC)�2exes.Hence,toprovethatall3d(d�2)exesofageneralplanecurveCaredistinct,onewouldnecessarilyhavetoshowthatCjusthassimpleexes.3.3.EllipticCurvesandComplexToriThissectionfollowschaptersII,IIIandVIof[11].Recallthatanellipticcurve-thatis,asmoothplanecubic-carriesthestructureofanabeliangroup(cf.[11,Paragraph2]).WedenealatticeLCtobetheZ-modulegeneratedbytwoR-linearlyindepen-dentcomplexnumbers.OuraiminthissectionwillbetoshowthatanellipticcurvecanbeexpressedasatorusC=Lforsomesuitablelattice,inthesensethatthereexistsanisomorphismbetweenbothgroups.Morepreciselythen,wefocusonthosetheoremsanddenitionsin[11]necessarytoprovethisassertion.WerstofallintroducetheWeierstassÃ-function,andthenwederivesomeofitselementaryproperties.Denition3.32.AnellipticfunctionrelativetothelatticeLisameromorphicfunctionf:C!C[f¥gsuchthatf(z+w)=f(z)forallz2Candw2L.TheeldofallellipticfunctionsrelativetoL,denotedC(L),maythereforebeseenasmeromorphicfunctionsonthetorusC=L.Denition3.33.AfundamentalparallelogramforLisasetD=fa+t1w1+t2w2:t1;t22[0;1)gwherea2Candfw1;w2gisaZ-basisofL.Elementarycomplexanalysisshowsthatforanyholomorphicfunctionf(z)onanopendomainandanyisolatedsingularityw2Cthatisnotessential,thereexistsasmallestnumberk2Zsuchthat(z�w)kf(z)hasaliftablesingularityinw.Wethendeneordw(f)=�kastheorderoffinw,whichiseasilyseentobeadiscretevalua-tion.Hence,polesoffarepreciselythosepointsinwhichfhasnegativeorder.21 Theorem3.37.TheWeierstrassÃ-functionisholomorphiconCnLwithdoublepolesineachlatticepoint.Itisanevenellipticfunctionofordertwo,anditsderivativeÃ0isanunevenellipticfunctionoforderthree.Proof.WeshowthattheseriesdeningtheÃ-functionconvergesabsolutelyanduni-formlyoneverycompactsubsetofCnL.Bytheinverseandtheusualtriangleinequality,wehavethat1 (z�w)2�1 w2=z(2w�z) w(z�w)2=jzjj2�z wj jwj3j1�z wj2jzj(2+jz wj) jwj3(1�jz wj)2:Sincetheunderlyingsetiscompact,wemayassumethatjzjrforsomepositiver.Furthermore,assumingthatjwj2r(whichimpliesjzj jwj1 2),wecanestimateallbutanitenumberofterms:jzj(2+jz wj) jwj3(1�jz wj)2r5 2 jwj31 410r jwj3:Asintheproofofthepreviouslemma,åw2Lw6=01 jwj3converges,whichestablishestheclaim.Thus,byatheoremofWeierstrass,ÃisaholomorphicfunctiononCnLandwemaydifferentiateterm-wisetoobtainÃ0:Ã0(z)=�2åw2L1 (z�w)3:ItsufcestoshowperiodicitywithrespecttoabasisofL,sowexabasisvectorwandconsiderÃ0(z+w)=�2åw2L1 (z+w�w)3=Ã0(z);whichprovesellipticityofÃ0.Consequently,g(z):=Ã(z+w)�Ã(z)isaconstantfunctionbecauseitsderivativevanishes.Observingthat1 2w=2LandthatÃisanevenfunction,wehaveg(�1 2w)=Ã(1 2w)�Ã(�1 2w)=0;sog(z)=0everywhere:ThisprovesthatÃisalsoelliptic.Theremainingstatementsareobvious. Remark3.38.TheeldofellipticfunctionsrelativetoalatticeLisgivenbyC(Ã;Ã0),i.e.everyellipticfunctionisexpressibleasarationalfunctionintheWeierstrassÃ-functionanditsderivative.Inthenextstep,theLaurentexpansionoftheÃ-functionaroundtheoriginwillbecomputedandexpressedintermsoftheEisensteinseries.ThismakesitpossibletodeduceadifferentialequationthatissatisedbyÃanditsderivative,whichiscloselyrelatedtoellipticcurvesintheirWeierstrassnormalform.23 Lemma3.39.TheLaurentseriesoftheWeierstrassÃ-functioninaneighbourhoodofzeroisgivenbyÃ(z;L)=1 z2+¥ån=1(2n+1)G2(n+1)(L)z2n:Proof.Thefunctiong(z):=Ã(z)�1 z2isevenandholomorphicaroundtheorigin.AllitsTaylorcoefcientswithanoddindexmustthereforevanish,soithasaseriesexpansionoftheformg(z)=¥ån=0a2nz2n;a2n=g(2n)(0) (2n)!:(2)Wehavea0=0sinceg(0)=0,andforn�0,weinductivelyobtaing(n)(z)=dn dznåw2Lw6=01 (z�w)2�1 w2=(�1)n(n+1)!åw2Lw6=01 (z�w)n+2:Inparticular,thecoefcientsaregivenasa2n=(�1)2n(2n+1)! (2n)!åw2Lw6=01 w2(n+1)=(2n+1)G2(n+1)(L);andso-substitutingthesetermsinequation(2)-theproofiscomplete. Lemma3.40.TheWeierstrassÃ-functionrelativetoLanditsderivativesatisfytherelationÃ0(z)2=Ã(z)3�g2Ã(z)�g3;(3)whereg2=60G4(L)andg3=140G6(L).Proof.ConsidertheLaurentexpansionÃ(z)=z�2+3G4z2+5G6z4+:::.OnecanderivethisexpressionandsquareittoobtaintheseriesforÃ02:Ã0(z)=�2z�3+6G4z+20G6z3+:::;Ã0(z)2=4z�6�24G4z�2�80G6+::::Similarly,wehavethatÃ(z)2=z�4+6G4+10G6z2+:::;Ã(z)3=z�6+9G4z�2+15G6+::::ItissufcienttoexpandÃ3andÃ02onlyuptothethirdnonzerotermsincethoseofhigherorderareholomorphicinzandwillbecomeirrelevantinthefollowingargument.Leta;b;c2CandconsiderÃ0(z)+aÃ(z)3+bÃ(z)+c=4+a z6+�24G4+b+9aG4 z2�80G6+15aG6+c+:::Forthisexpressiontovanish,weinparticularneeda=�4whichthenimpliesb=60G4andc=140G6.Thelefthandsideisnowaholomorphicellipticfunctionwhichiszeroforz=0,andthereforeitgloballyvanisheswhichprovestheassertion. 24 x=wandx=�w.Hence,weeitherhavezwmodL,inwhichcasethereisnothingtoshow,orz�wmodL.ThelattercaseimpliesÃ0(z)=Ã0(�w)=�Ã0(w)=�Ã0(z);soÃ0(z)=0.Now,letfw1;w2gbeabasisofLandw2fw1 2;w2 2;w1+w2 2g.Then2w2LandÃ0(w)=Ã0(w�2w)=Ã0(�w)=�Ã0(w);soÃ0vanishesatthosethreedistinctpoints.SinceÃ0isanellipticfunctionoforderthree,itcannothaveanyotherroot,so2zmustbeoneofthem.Inparticular,2ziscontainedinL,whichgiveszwmodLandhenceprovesinjectivityofF0.Finally,themapC0i�!C(p1;p2)7�![1:p1:p2]isontoexceptforthosepointsinthecodomainwithvanishingrstcoordinatep0.If[0:p1:p2]2C,thennecessarilyp1=0andp26=0bythehomogeneousequationofC,soonly[0:0:1]doesnotlieintheimageofiF0.Hence,Fisbijective. ToprovethatFisagrouphomomorphism,wewillintroduceasecondgroupstruc-turePic0(C),onethatwemaydeneforanysmoothcurve.Inourcase-whenCiselliptic-thecurveCendowedwiththegeometricgroupstructureandPic0(C)willbeisomorphic,whichisakeystepinprovingourmaintheorem.Denition3.45.LetCbeanalgebraiccurve.ThedivisorgroupofCisthefreeabeliangroupgeneratedbythepointsofC,thatisDiv(C)=Mp2CZ:AnelementofDiv(C)willbedenotedbyD=åp2Cnp(p);whereallbutnitelymanycoefcientsvanish.ThedegreeofadivisorDisthesumdegD=åp2Cnp.Thedivisorsofdegree0formasubgroupofDiv(C)whichwecallDiv0(C).WenowassumeCtobesmoothandconsiderthelocalringOC;pwithmaximalidealmP=ff2OC;p:f(p)=0g.Recallthatthereisanaturalisomorphismofvectorspaces(cf.[9,Theorem3.14])TpC=(mp=m2p);soinparticular,dimCmp=m2p=1forasmoothpointp.ByNakayamaslemma,mpisaprincipalidealgeneratedbysomet2mp.Weemploytworesultsfromcommutativealgebra:26 Remark3.46.(1)ForanynoetherianintegraldomainAandanynon-unitt2A,wehavethat¥Tk=1(tk)=(0).(2)Let(A;m)bealocalintegraldomainwithquotienteldK=Quot(A)andm=(t)6=0.If¥Tk=1(tk)=(0),thenthefollowingtwostatementshold:1.Foreachnon-zeroa2A,thereexistsauniquek�0andaunitasuchthata=atk(tiscalledauniformizingparameter).2.Themapv:K!Zwhichsendseachatok�0(asin1.)andeacha=a0tok�k02ZisadiscretevaluationwithvaluationringA.Wemaythusconcludethat,foreachnon-singularpointp2C,themapordp:K(C)�!Zg=h7�!ordp(g)�ordp(h);whereordp(f)=maxfk:f2mkpgforf2OC;p,isadiscretevaluationonthefunctioneldK(C)=Quot(OC;p)withvaluationringOC;p.Denition3.47.Foranonzerorationalfunctionf2K(C),wedenethedivisorasso-ciatedtofbydiv(f)=åp2Cordp(f)(p)2Div(C):WecalladivisorDprincipalifD=div(f)forsomef2K(C).Notethatsinceordpisadiscretevaluation,themapdiv:K(C)�!Div(C)f7�!div(f)isagrouphomomorphism.Inparticular,thesetofprincipaldivisorsformasubgroupofDiv(C).Denition3.48.TwodivisorsDandD0arecalledlinearlyequivalentifD�D0=div(f)forsomef2K(C),i.e.iftheirdifferenceisprincipal.ThePicardgroupofC,calledPic(C),isthequotientofDiv(C)bythesubgroupofprincipaldivisors,i.e.Pic(C)=Div(C)=div(K(C)):TheprincipaldivisorsformasubgroupofDiv0(C),sowemaydenePic0(C)=Div0=div(K(C));thedegreezeropartofthePicardgroupofC.27 forsomerationalfunctionF2C(X;Y).WemayinterpretFasanelementofK(C)andthusobtaindiv(F)=(F(z1+z2))+(F(z1))+(F(z2))+(F(0)):Hence,F(z1+z2)=F(z1)+F(z2)followsfromCorollary3.54. OurlastconcerninthissectionistheexistenceofalatticewhoseEisensteinseries60G4and140G6haveprescribedvalues.Thisisdoneviathemodularfunctionjandwillonlybecitedhereforbrevity.Theorem3.57.Letg2andg3becomplexnumberssuchthatD=4g32�27g236=0.ThenthereexistsalatticeLCsuchthatg2=60G4(L)andg3=140G6(L).ThisresultcanbefoundasTheorem5.1in[11].RecallthatD6=0isalwaysfullledbyanellipticcurveCg2;g3,sotheproofofTheorem3.43iscomplete.3.4.TheMonodromyGroupoftheNineFlexesBySection3.2,wehavethatthemonodromygroupofthemapI3p!W3isasubgroupofS9.WeinvestigatethesubsetofexesofanellipticcurveC,whichweshalldenotebyG,morecloselybeforeweexplicitlycomputethemonodromygroup.LetC=V(F=�X0X22+4X31�g2X20X1�g3X30)P2beanellipticcurveinWeier-strassform.Thecompositionlaw(cf.[11])endowsCwiththestructureofanabeliangroup,wherewemayletO=[0:0:1]-thepointatinnity-actastheneutralelement.ThisturnsGintoasubgroupofC,asthefollowingthreeresultsshow.Lemma3.58.ThepointOisaexofC.Moreprecisely,IO(C;TOC)=3.Proof.Considerthepartialderivatives¶f ¶X0=�2g2X0X1�3g3X20�X22;¶f ¶X1=12X21�g2X20;¶f ¶X2=�2X0X2:ThetangentlineofCatapointPisgivenbyTPC=V¶f ¶X0(P)X0+¶f ¶X1(P)X1+¶f ¶X2(P)X2;soinparticular,TOCisthezerolocusof�X0.Now,TOC\C=f[0:p1:p2]g\C=f[0:0:1]g;henceOisaexofCbyB´ezout'stheorem. Lemma3.59.ApointP2Cisaexifandonlyif3P=O.Inotherwords,theinectionpointsofanellipticcurvearepreciselyits3-torsionpoints.Proof.LetP2C.ThenthethirdpointofintersectionofTPCandCisgivenas�2P,hence3P=OisequivalenttoC\TPC=fPg.ViaB´ezout,thelatterconditiontranslatestoIP(C;TPC)=3,whichconcludesthestatement. 30 ofgtoI3with˜gi(1)=(C;Qi;ti).Now,ifweletP(t)bethethirdpointofintersectionoftheline P1(t)P2(t)withCandl(t)=TP(t)C(t),wemayexplicitlydenealiftingviaa(t)=(C(t);P(t);l(t)):Clearly,a(0)=(C;P3;l3),sobyuniquenessoftheliftedpath(cf.Lemma2.6),a=˜g3andsoQ3isthethirdpointofintersectionof Q1Q2withC. Corollary3.63.ThemonodromygroupisasubgroupofASL2(Z=3).Proof.WehaveMAGL2(F3)bythepreviouslemma,soitremainstoassertthatnomatricesofdeterminanttwoarecontainedinM(cf.[8,p.693]). Asfurtherpreparationforthemaintheorem,wehavethefollowingelementaryresults.Lemma3.64.LetXbeaZariski-opensubsetofPnorAn.ThenXispathconnectedwithrespecttotheanalytictopology.Proof.Theprojectivecasefollowsfromtheafnecaseviathestandardopencovering.Thus,supposethatxandyareelementsofXAn.Letlbetheuniquelinethroughbothpointsandnotethatitissufcienttondapathwithinl\X,whichisanopensubsetofl.Hence,thecomplementln(l\X)isclosedinlandthusnite.AnylineinafnespaceishomeomorphictoR2,soweconcludethatthereexistsanitesetofpointsSsuchthatl\X=R2nS.Thelattersetisobviouslypathconnected,sotheproofiscomplete. Lemma3.65.LetYbeapathconnectedspaceandletp:Y!Xbeacoveringmap.Thenthemonodromyofpactstransitivelyonthebreofanypointx2X.Proof.Letgbeapathbetweentwopointsy1;y22p�1(x).Then[pg]2p1(X;x)isaloopwhichtautologicallyliftstog,andsotheinducedpermutationofthebresendsy1toy2. Asanimmediateconclusion,wehave:Corollary3.66.Themonodromyofthenineexesactstransitively. Theorem3.67.M=ASL2(Z=3).Proof.LetC2Ubeasmoothplanecubic.Afteralineartransformation,wemayassumethattheneutralelementisO=[0:0:1]withtangentlineV(X0).TheprecedingsectionguaranteestheexistenceofalatticeL=w1;w2&#x]TJ/;བ ;.9;‘ ;&#xTf 1;�.90;
0 ;&#xTd [;ZsuchthatF:C=L�!C:Moreover,thesubgroupofexesisV:=1 3L L=F�1(G).WerstofallclaimthatSL2(Z=3)iscontainedinMandthatthisispreciselythestabilizerofOinM,i.e.Stab(O):=fg2M:g:O=Og=SL2(Z=3):32 K(W3).Now,therootsofthisminimalpolynomialcorrespondtothepointsinageneralbrep�1(C),whichbijectontotheexesofC.Hence,weseethateachexmaybeexpressedintermsofthefunctioneldK(W3).WenowcalculatethecoordinatesofanellipticcurveinWeierstrassformintermsofitscoefcientsbyelementarymeans.LetF(X0;X1;X2)=�X0X22+4X31�g2X20X1�g3X30andconsiderthepartialderivativesF00=�2g2X1�6g3X0,F11=24X1,F22=�2X0,F10=�2g2X0,F12=0andF20=�2X2.Hence,thedeterminantoftheHessianhastheformdet(H(F))=F00F11F22�F02F11F20�F01F10F22=96g2X0X21+288g3X20X1�96X1X22+8g22X30:Wehavealreadyshownthatthepointatinnity[0:0:1]isaex,soitsufcestoconsidertheafneplaneX0=1.Hence,weseekthecommonzeroesofthepolynomialsf(X1;X2):=F(1;X1;X2);h(X1;X2):=1 8H(F)(1;X1;X2)=12g2X21+36g3X1�12X1X22g22:WesubstituteX22=4X31�g2X1�g3inhandobtain0=12g2X21+36g3X1�12X1(4X31�g2X1�g3)+g22,0=X41�g2 2X21�g3X1�g22 48;whichisanequationofdegreefourandhencesolvable.Itfollowsthatalleightremain-ingexeslieonfourverticallinesintheafneplanef[1:X1:X2]g.4References[1]E.Bombieri,W.Gubler:HeightsinDiophantineGeometry,CambridgeUniversityPress,2006.[2]G.E.Bredon:GeometryandTopology,Springer,1993.[3]E.Brieskorn,H.Kn¨orrer:PlaneAlgebraicCurves,Birkh¨auser,1986.[4]I.Dolgachev:IntroductiontoAlgebraicGeometry,http://www.math.lsa.umich.edu/idolga/lecturenotes.html,2010.[5]G.Fischer:PlaneAlgebraicCurves,AMSStudentMathematicalLibrary,2001.[6]W.Fulton:AlgebraicCurves-AnIntroductiontoAlgebraicGeometry,AddisonWesleyLongman,1969.[7]J.Harris:AlgebraicGeometry-AFirstCourse,Springer,2010.[8]J.Harris:GaloisGroupsofEnumerativeProblems,DukeMathematicalJournalVol.46No.4p.685-724,1979.34 [9]K.Hulek:ElementaryAlgebraicGeometry,AMSStudentMathematicalLibrary,2003.[10]D.Huybrechts:ComplexGeometry,Springer,2004.[11]J.H.Silverman:TheArithmeticofEllipticCurves,Springer,2ndEdition.35