Orbits of rational n-sets of projective spaces und

arXiv:math/0701836v1[math.CO]29Jan2007Orbitsofrationaln-setsofprojectivespacesundertheactionofthelineargroupRicardMart´ı,EnricNart∗,1UniversitatAut`onomadeBarcelona,DepartamentdeMatem`atiques08193Bellaterra,Barcelona,SpainAbstractForafixeddimensionNwecomputethegeneratingfunctionofthenumberstN(n)(respectivelytN(n))ofPGLN+1(k)-orbitsofrationaln-sets(respectivelyrationaln-multisets)oftheprojectivespacePNoverafinitefieldk=Fq.ForN=1,2theseresultsprovideconcreteformulasfortN(n)andtN(n)asapolynomialinqwithintegercoefficients.Keywords:finitefield,rationaln-set,projectivespace,genusthreecurveIntroductionTherearemanyexamplesofdeeppropertiesofgeometricobjectsrelyingoncombinatorialpropertiesofunorderedstructures.Inparticular,n-setsofpro-jectivespaceshavedeservedtheattentionofgeometerssincealongtime.TheworkofCobleatthebeginningofthelastcenturyisanoutstandingcontri-butiontothestudyofgeometricstructuresunderlyingn-setsofprojectivespaces.Arevisionofthisworkinmodernlanguagecanbefoundinthebook[3]ofDolgachevandOrtland.Ifweareinterestedinarithmeticpropertiesofthegeometricobjectsassociatedton-sets,weneedtoconsiderrationaln-sets;thatis,n-setsdefinedoverthegroundfiedkweareinterestedin.Letkbeafixedalgebraicclosureofthefieldk;ann-setS={P1,...,Pn}ofaprojectivespacePN(k)isk-rationalif∗CorrespondingauthorEmailaddress:[e-mail]:[phone](EnricNart).1SupportedbytheprojectMTM2006-11391fromtheSpanishMECPreprintsubmittedtoElsevier2February2008SisinvariantundertheactionoftheabsoluteGaloisgroup{σ(P1),...,σ(Pn)}={P1,...,Pn},∀σ∈Gal(k/k).Thus,SisthedisjointunionoforbitsofpointsofPN(k)undertheactionoftheGaloisgroup.WedenotebyPNn!(k):=PN(k)n!Gal(k/k),PNn!!(k):=PN(k)n!!Gal(k/k),therespectivesetsofk-rationaln-setsandn-multisetsofPN(k).Ifweapplytoarationaln-setak-automorphismofPNweobtainanequivalentn-set,inthesensethattheunderlyinggeometricobjectsofbothn-setswillhavethesamegeometricandarithmeticproperties.TheaimofthispaperisthecomputationofthenumberofPGLN+1(k)-orbitsofrationaln-setsandn-multisetsofprojectivespacesPNdefinedoverafinitefieldk.Thatis,wewanttofindclosedformulasforthenumbers:tN(n):=PGLN+1(k)\PNn!(k),tN(n):=PGLN+1(k)\PNn!!(k).Thereisanextensiveliteratureontheenumerationoforbitsofpointwiserationaln-sets;thatis,n-setsS={P1,...,Pn}suchthatallPiarek-rationalpoints(havehomogeneouscoordinatesink).Thisisduetothefactthattheseorbitsareincorrespondencewithisometryclassesoflinearcodes[5],[1,Sec.3.2],[6],[8],[9].However,toourknowledgetheenumerationofrationaln-setshasnotbeenconsideredsofar,withtheexceptionof[7],wherethenumberst1(n)werecomputed.Letusillustrateboththeroleofglobal(notpointwise)rationalityandtheactionofthelineargroupwithanexample.Itiswell-knownthatthehyper-ellipticcurvesoveranalgebraicallyclosedfield(ofzerooroddcharacteristic)areparameterizedbyn-setsofP1;ifS={P1,...,Pn}isan-setofP1andweattachtoeachPianaffinecoordinateai,wecanconsiderthehyperellipticcurvegivenbytheWeierstrassequationy2=(x−a1)(x−a2)···(x−an).Ifwewanttoclassifyhyperellipticcurvesdefinedoveranon-algebraicallyclosedfieldkweareledtoconsiderk-rationaln-setsofP1.IfSisapointwiserationaln-settheaboveconstructionprovidesacurvewithallWeierstrasspointsdefinedoverk.Thesecurvesareasmallpartofthefamilyofhyperel-lipticcurvesdefinedoverk,givenbyWeierstrassequationsy2=f(x),withf(x)anarbitraryseparablepolynomialwithcoefficientsink.Finally,itis2easytocheckthatrationaln-setsinthesameorbitbytheactionofPGL2(k)determinek-isomorphiccurves.Thus,thenumberst1(n)countessentiallyk-isomorphismclassesofhyperellipticcurvesdefinedoverk[7].Anotherinterestingexampleisgivenbythe7-setsoftheprojectiveplane.OveranalgebraicallyclosedfieldcertainPGL3(k)-orbitsof7-setsclassifynon-hyperellipticcurvesofgenusthreewithafixed2-levelstructure[3,Chap.IX].Thus,ourenumerationresultsmayprovideinformationonthenumberofk-rationalpointsofthemodulispaceofsuchobjects,andfromthesenumbersonecanderivefurthergeometricandarithmeticinformationonthisspace.OurmainresultisacomputationofthegeneratingfunctionofthetN(n),tN(n)forfixedN(Theorem4.4).Itiswell-knownthatthegeneratingfunctionofthenumberoforbitsofpointwiserationaln-setsofPN(k)canbeexpressedintermsofthecycleindexofP´olya[1,3.2.16].Thiscycleindexisapolynomialinseveralvariablesthatcarriesallinformationaboutthelengthsofthecy-clesofallelementsofPGLN+1(k)actingaspermutationsofPN(k).Thus,thisinstrumentisnotabletoprovideinformationonrationaln-sets.However,in[9]wehavefoundarefinementofthecycleindexthatleadstomoreeffectiveformulasforthegeneratingfunctionofthenumberoforbitsofpointwisera-tionaln-setsofPN(k).Insection4weextendtheseideastorationaln-sets.WeintroducecertainequivalencerelationinthesetCofconjugacyclassesofΓ:=PGLN+1(k);ifSisthequotientset,foreachclassα∈S(wecallαasubtype)weconstructaposetL(α)ofcertainγ-invariantlinearsubvarietiesofPN(k)classifiedundertheactionoftheGaloisgroup.ThenodesVofthisposetL(α)carryaweight(dimV,expV,degV)ofthreenumericalvaluesdi-mension,exponentanddegree.Wecounttheelementsinthesamesubtypeαintheweightedsum:Mα:=Xγ∈α|Γγ|−1,whereΓγisthecentralizerofγinΓ.Then,we