On Cremona transformations of prime order

Arnaud Beauville

Description

On Cremona transformations of prime order Sur les transformations de Cremona d'ordre premier Arnaud Beauville Laboratoire J.-A. Dieudonne, UMR 6621 du CNRS, Universite de Nice, Parc Valrose, F-06108 Nice cedex 2 Jeremy Blanc Section de Mathematiques, Universite de Geneve, 2-4 rue du Lievre, Case postale 240, CH-1211 Geneve 24 Abstract We prove that an automorphism of order 5 of the Del Pezzo surface S5 of degree 5 is conjugate through a birational map S 99K P2 to a linear automorphism of P2. This completes the classification of conjugacy classes of elements of prime order in the Cremona group. Resume Nous demontrons qu'un automorphisme d'ordre 5 de la surface de Del Pezzo S de degre 5 est conjugue via une application birationnelle S 99K P2 a un automorphisme lineaire de P2. Cette observation complete la classification des classes de conjugaison d'elements d'ordre premier dans le groupe de Cremona. 1. The result The Cremona group Bir(P2) is the group of birational transformations of P2C, or equivalently the group of C-automorphisms of the field C(x, y). There is an extensive classical literature about this group, in particular about its finite subgroups; we refer to the introduction of [dF] for a list of references. The classification of conjugacy classes of elements of prime order p in Bir(P2) has been given a modern treatment in [B-B] for p = 2 and in [dF] for p ≥ 3.

• transformation ?


• rational surface


• surface s5 ?


• automorphisme d'ordre


• unique conjugacy


• order ≥


• prime order


• automorphisme lineaire de p2

Sujets

• Cremona
• Beauville
• Rational surface

Informations

• Publié par: pefav
• Nombre de lectures: 8
• Langue: English

Extrait

On Cremona transformations of prime order

Sur les transformations de Cremona d’ordre premier Arnaud Beauville LaboratoireJ.-A.Dieudonne´,UMR6621duCNRS,Universit´edeNice,ParcValrose,F-06108Nicecedex2

Je´r´emyBlanc SectiondeMath´ematiques,Universite´deGene`ve,2-4rueduLi`evre,Casepostale240,CH-1211Gen`eve24

Abstract We prove that an automorphism of order 5 of the Del Pezzo surfaceS 5 of degree 5 is conjugate through a 2 2 birational mapS 99KP to a linear automorphism of P . This completes the classification of conjugacy classes of elements of prime order in the Cremona group. R´esum´e Nousd´emontronsqu’unautomorphismed’ordre5delasurfacedeDelPezzoSnededegr´e5sectnoujuge´ivua 2 2 application birationnelleS 99KP ihmsompruaota`nurede´eaielin P cnoitavresboettesiascllate`eplomcfitaoin.C desclassesdeconjugaisond’e´lementsd’ordrepremierdanslegroupedeCremona.

1. Theresult

2 2 TheCremona groupBir(P) is the group of birational transformations ofP, or equivalently the group C ofC-automorphisms of the fieldC(x, y). There is an extensive classical literature about this group, in particular about its finite subgroups; we refer to the introduction of [dF] for a list of references. 2 The classification of conjugacy classes of elements of prime orderpin Bir(P) has been given a modern treatment in [B-B] forp= 2 and in [dF] forp≥3. Let us recall the main results. The linear transformations of given order are contained in a unique conjugacy class (see the Proposition below). Apart from these therearethreefamiliesofconjugacyclassesofinvolutions(thefamousdeJonquie`res,BertiniandGeiser involutions), then two families of conjugacy classes of order 3, given by automorphisms of special Del

Email addresses:[e-mail](Arnaud Beauville),[e-mail]´´e(J.)

Preprint submitted to Elsevier Science

23rd June 2004