On homomorphisms between Cremona groups

Urech, Christian

Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 53-100.

On s’intéresse aux plongements algébriques du groupe de Cremona complexe à $n$ variables ${Cr}_n$ dans des groupes de transformations birationnelles $Bir\left(M\right)$ 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 $Bir\left(M\right)$ pour des variétés $M$ de dimension $3$ et on généralise partiellement ce résultat aux plongements algébriques de ${Cr}_n$ dans $Bir\left(M\right)$, où la dimension de $M$ est $n+1$ (pour tout $n$). On obtient notamment une classification de toutes les action régulières de ${PGL}_{n+1}\left(ℂ\right)$ sur des variétés projectives lisses de dimension $n+1$ qui s’étendent vers des actions rationnelles de ${Cr}_n$.

We look at algebraic embeddings of the complex Cremona group in $n$ variables ${Cr}_n$ to the group of birational transformations $Bir\left(M\right)$ 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 $Bir\left(M\right)$, where $M$ is a variety of dimension 3 and generalize this result partially to algebraic embeddings of ${Cr}_n$ into $Bir\left(M\right)$, where the dimension of $M$ is $n+1$, for arbitrary $n$. In particular, this yields a classification of all algebraic ${PGL}_{n+1}\left(ℂ\right)$-actions on smooth projective varieties of dimension $n+1$ that can be extended to rational actions of ${Cr}_n$.

Classification : 14E07, 14L30, 32M05
Keywords: Cremona group, rational group actions, algebraic group actions
Mot clés : Groupe de Cremona, actions rationnelles des groupes

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