On homomorphisms between Cremona groups

Abstract

We look at algebraic embeddings of the Cremona group in $n$ variables $C{r}_n(C)$ 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 $C{r}_2(C)$ into $C{r}_5(C)$ that is due to Gizatullin. In a second part, we give a full classification of all algebraic embeddings of $C{r}_2(C)$ into $Bir(M)$, where $dim(M)=3$, and generalize this result partially to algebraic embeddings of $C{r}_n(C)$ into $Bir(M)$, where $dim(M)=n+1$, for arbitrary $n\ge 2$. In particular, this yields a classification of all algebraic $PG{L}_{n+1}(C)$-actions on smooth projective varieties of dimension $n+1$ that can be extended to rational actions of $C{r}_n(C)$.

arXiv comment33 pages