On homomorphisms between Cremona groups
Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 53-100.

We look at algebraic embeddings of the complex Cremona group in n variables Cr n to the group of birational transformations Bir(M) of an algebraic variety M. First we study geometrical properties of an example of an embedding of Cr 2 into Cr 5 that is due to Gizatullin. In a second part, we give a full classification of all algebraic embeddings of Cr 2 into Bir(M), where M is a variety of dimension 3 and generalize this result partially to algebraic embeddings of Cr n into Bir(M), where the dimension of M is n+1, for arbitrary n. In particular, this yields a classification of all algebraic PGL n+1 ()-actions on smooth projective varieties of dimension n+1 that can be extended to rational actions of Cr n .

On s’intéresse aux plongements algébriques du groupe de Cremona complexe à n variables Cr n dans des groupes de transformations birationnelles Bir(M) d’une varété algébrique M. D’abord on regarde un plongement de Cr 2 dans Cr 5 qui était découvert par Gizatullin. Puis on donne une classification de tous les plongements algébriques de Cr 2 dans Bir(M) pour des variétés M de dimension 3 et on généralise partiellement ce résultat aux plongements algébriques de Cr n dans Bir(M), où la dimension de M est n+1 (pour tout n). On obtient notamment une classification de toutes les action régulières de PGL n+1 () sur des variétés projectives lisses de dimension n+1 qui s’étendent vers des actions rationnelles de Cr n .

Received:
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Accepted:
Published online:
DOI: 10.5802/aif.3151
Classification: 14E07,  14L30,  32M05
Keywords: Cremona group, rational group actions, algebraic group actions
Urech, Christian 1

1 IRMAR Université de Rennes 1 35042 Rennes (France)
License: CC-BY-ND 4.0
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Urech, Christian. On homomorphisms between Cremona groups. Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 53-100. doi : 10.5802/aif.3151. https://aif.centre-mersenne.org/articles/10.5802/aif.3151/

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