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Pré-Publication, Document De Travail Année : 2016

On homomorphisms between Cremona groups

Résumé

We look at algebraic embeddings of the Cremona group in $n$ variables $Cr_n(C)$ to the groups of birational transformations $Bir(M)$ of an algebraic variety $M$. First we study geometrical properties of an example of an embedding of $Cr_2(C)$ into $Cr_5(C)$ that is due to Gizatullin. In a second part, we give a full classification of all algebraic embeddings of $Cr_2(C)$ into $Bir(M)$, where $dim(M)=3$ and generalize this result partially to algebraic embeddings of $Cr_n(C)$ into $Bir(M)$, where $dim(M)=n+1$, 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 $Cr_n(C)$.
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Dates et versions

hal-01286455 , version 1 (10-03-2016)

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  • HAL Id : hal-01286455 , version 1

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Christian Urech. On homomorphisms between Cremona groups. 2016. ⟨hal-01286455⟩
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