On homomorphisms between Cremona groups

TL;DR

This paper classifies algebraic embeddings of Cremona groups into birational transformation groups of algebraic varieties, revealing geometric properties and extending known classifications to higher dimensions.

Contribution

It provides a full classification of algebraic embeddings of $Cr_2(C)$ into $Bir(M)$ for 3-dimensional $M$, and extends results to higher dimensions, linking group actions to birational transformations.

Findings

Classified all algebraic embeddings of $Cr_2(C)$ into $Bir(M)$ for $ ext{dim}(M)=3$.

Extended classification results to embeddings of $Cr_n(C)$ into $Bir(M)$ for $ ext{dim}(M)=n+1$, $n extgreater{}2$.

Connected group actions of $PGL_{n+1}(C)$ to Cremona group embeddings.

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 $Cr_n(C)$.