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1 Université de Bâle 2 IRMAR - Institut de Recherche Mathématique de Rennes

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

Keywords : Cremona group + birational geometry + group theory + algebraic group actions

Author: Christian Urech -

Source: https://hal.archives-ouvertes.fr/


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