On homomorphisms between Cremona groups
Abstract
We look at algebraic embeddings of the Cremona group in $n$ variables $Cr_n(C)$ 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 $Cr_2(C)$ 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 $Cr_n(C)$.
 Publication:

arXiv eprints
 Pub Date:
 March 2016
 DOI:
 10.48550/arXiv.1603.03294
 arXiv:
 arXiv:1603.03294
 Bibcode:
 2016arXiv160303294U
 Keywords:

 Mathematics  Algebraic Geometry;
 14E07 (Primary);
 14L30;
 32M05 (Secondary)
 EPrint:
 33 pages