p elementary subgroups of the Cremona group

Arnaud Beauville

Description

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) .

• torus has


• preserves gp


• homographies z


• then bav


• let cp ?


• identify ?


• now let


• quaternion algebra over


• cremona group

Sujets

• Beauville
• Cremona group

Informations

• Publié par: pefav
• Nombre de lectures: 9
• Langue: English

Extrait

p-elementarysubgroupsoftheCremonagroup ArnaudB EAUVILLE Introduction Letkbeanalgebraicallyclosedfield.TheCremonagroupCr k isthegroup ofbirationaltransformationsof P k 2 ,orequivalentlythegroupofk-automorphisms ofthefieldk(x,y).Thereisanextensiveclassicalliteratureaboutthisgroup,in particularaboutitsfinitesubgroups–seetheintroductionof[dF]foralistof references. TheclassificationofconjugacyclassesofelementsofprimeorderpinCr k has beengivenamoderntreatmentin[B-B]forp=2andin[dF]forp ≥ 3(seealso [B-Bl]).Inthisnotewegoonestepfurtherandclassifyp-elementarysubgroups– thatis,subgroupsisomorphicto( Z /p) r forpprime.Wewillmostlydescribesucha subgroupasagroupGofautomorphismsofarationalsurfaceS:weidentifyGto asubgroupofCr k bychoosingabirationalmapϕ:S99K P 2 .Thentheconjugacy classofGinCr k dependsonlyonthedata(G,S). Theorem. − LetGbeasubgroupofCr k oftheform( Z /p) r withpprime 6 =char(k).Then: a)Assumep ≥ 5.Thenr ≤ 2,andifr=2Gisconjugatetothep-torsion subgroupofthediagonaltorus 1 ofPGL 3 (k)=Aut( P 2 ). b)Assumep=3.Thenr ≤ 3,andifr=3Gisconjugatetothe3-torsion subgroupofthediagonaltorusofPGL 4 (k),actingontheFermatcubicsurface X 0 3 +...+X 3 3 =0. c)Assumep=2.Thenr ≤ 4,andifequalityholdsGisconjugatetooneof thefollowingsubgroups: c 1 )the2-torsionsubgroupofthediagonaltorusofPGL 5 (k),actingonthe 44 PP quarticdelPezzosurfacein P 4 withequations i=0 X i 2 = i=0 λ i X i 2 =0forsome distinctelementsλ 0 ,...λ 4 ofk. c 2 )thesubgroupofCr k spannedbytheinvolutions: 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,λ 2 } ,u=x 2 +x − 2 .Alternatively,thissubgroup actsontherationalsurfacey 2 − α(x)z 2 − β(x)=0ink ∗ × k 2 ,withβ=α − λ 2 , bychangingthesignofx,y,zandchangingxinx − 1 . 1 BythediagonaltorusofPGL r (k)wemeanthesubgroupofprojectivetransformations (X 0 ,...,X r ) 7→ (t 0 X 0 ,...,t r X r )fort 0 ,...,t r ink ∗ . 1