p elementary subgroups of the Cremona group

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p-elementary subgroups of the Cremona group Arnaud BEAUVILLE Introduction Let k be an algebraically closed field. The Cremona group Crk is the group of birational transformations of P2k , or equivalently the group of k-automorphisms of the field k(x, y) . There is an extensive classical literature about this group, in particular about its finite subgroups – see the introduction of [dF] for a list of references. The classification of conjugacy classes of elements of prime order p in Crk has been given a modern treatment in [B-B] for p = 2 and in [dF] for p ≥ 3 (see also [B-Bl]). In this note we go one step further and classify p-elementary subgroups – that is, subgroups isomorphic to (Z/p)r for p prime. We will mostly describe such a subgroup as a group G of automorphisms of a rational surface S : we identify G to a subgroup of Crk by choosing a birational map ? : S 99K P 2 . Then the conjugacy class of G in Crk depends only on the data (G,S) . Theorem .? Let G be a subgroup of Crk of the form (Z/p) r with p prime 6= char(k) .

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quaternion algebra over

cremona group

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Beauville Cremona group

p-elementarysubgroupsoftheCremonagroupArnaudBEAUVILLEIntroductionLetkbeanalgebraicallyclosedﬁeld.TheCremonagroupCrkisthegroupofbirationaltransformationsofPk2,orequivalentlythegroupofk-automorphismsoftheﬁeldk(x,y).Thereisanextensiveclassicalliteratureaboutthisgroup,inparticularaboutitsﬁnitesubgroups–seetheintroductionof[dF]foralistofreferences.TheclassiﬁcationofconjugacyclassesofelementsofprimeorderpinCrkhasbeengivenamoderntreatmentin[B-B]forp=2andin[dF]forp≥3(seealso[B-Bl]).Inthisnotewegoonestepfurtherandclassifyp-elementarysubgroups–thatis,subgroupsisomorphicto(Z/p)rforpprime.WewillmostlydescribesuchasubgroupasagroupGofautomorphismsofarationalsurfaceS:weidentifyGtoasubgroupofCrkbychoosingabirationalmapϕ:S99KP2.ThentheconjugacyclassofGinCrkdependsonlyonthedata(G,S).Theorem.−LetGbeasubgroupofCrkoftheform(Z/p)rwithpprime6=char(k).Then:a)Assumep≥5.Thenr≤2,andifr=2Gisconjugatetothep-torsionsubgroupofthediagonaltorus1ofPGL3(k)=Aut(P2).b)Assumep=3.Thenr≤3,andifr=3Gisconjugatetothe3-torsionsubgroupofthediagonaltorusofPGL4(k),actingontheFermatcubicsurfaceX03+...+X33=0.c)Assumep=2.Thenr≤4,andifequalityholdsGisconjugatetooneofthefollowingsubgroups:c1)the2-torsionsubgroupofthediagonaltorusofPGL5(k),actingonthe44PPquarticdelPezzosurfaceinP4withequationsi=0Xi2=i=0λiXi2=0forsomedistinctelementsλ0,...λ4ofk.c2)thesubgroupofCrkspannedbytheinvolutions:1α(u)λ(u)y−α(u)(x,y)7→(−x,y),(x,y)7→(,y),(x,y)7→(x,),(x,y)7→(x,)xyy−λ(u)forsomeα,λink(u)withα∈/{0,λ2},u=x2+x−2.Alternatively,thissubgroupactsontherationalsurfacey2−α(x)z2−β(x)=0ink∗×k2,withβ=α−λ2,bychangingthesignofx,y,zandchangingxinx−1.1BythediagonaltorusofPGLr(k)wemeanthesubgroupofprojectivetransformations(X0,...,Xr)7→(t0X0,...,trXr)fort0,...,trink∗.1

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