Quelques propriétés des transformations birationnelles du plan projectif complexe, une histoire pour S.
Séminaire de théorie spectrale et géométrie, Tome 27 (2008-2009), pp. 45-100.

On présente certaines (malheureusement pas toutes) propriétés connues du groupe de Cremona en faisant, lorsque c’est possible, un parallèle avec le groupe des automorphismes polynomiaux de 2 . Les propriétés abordées seront essentiellement de nature algébrique : théorème de génération, sous-groupes finis, sous-groupes de type fini, description du groupe d’automorphismes du groupe de Cremona,... mais aussi de nature dynamique : classification des transformations birationnelles, centralisateur, dynamique d’un sous-groupe de Heisenberg... On évoque un peu les aspects concernant l’étude dynamique des itérés d’une transformation birationnelle ainsi que les problèmes de construction d’automorphismes de type entropique sur les surfaces rationnelles.

We present some (unfortunately not all) known properties of the Cremona group; when it’s possible we mentioned links with the most known group of polynomial automorphisms of the affine plane. The mentioned properties are essentially algebraic properties: generators, relations, finite subgroups, subgroups of finite type, automorphisms of the Cremona group, Tits alternative... but also dynamical properties: classification of birational maps, centralizer, dynamic of an Heisenberg subgroup... We deal with a little the dynamical study of the iterates of a birational map and also with the construction of automorphisms with positive entropy.

DOI : https://doi.org/10.5802/tsg.270
Classification : 14E07,  14E05
Mots clés : Transformations birationnelles, groupe de Cremona
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Déserti, Julie. Quelques propriétés des transformations birationnelles du plan projectif complexe, une histoire pour S.. Séminaire de théorie spectrale et géométrie, Tome 27 (2008-2009), pp. 45-100. doi : 10.5802/tsg.270. http://archive.numdam.org/articles/10.5802/tsg.270/

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