On homomorphisms between Cremona groups

We look at algebraic embeddings of the Cremona group in n variables $Cr_n(\mathbb{C})$ 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 $Cr_2(\mathbb{C})$ into $Cr_5(\mathbb{C})$ that is due to Gizatullin. In a second part, we give a full classification of all algebraic embeddings of $Cr_2(\mathbb{C})$ into Bir(M), where dim(M) = 3 and generalize this result partially to algebraic embeddings of $Cr_n(\mathbb{C})$ into Bir(M), where dim(M) = n + 1, for arbitrary n. In particular, this yields a classification of all algebraic $PGL_{n+1}(\mathbb{C})$-actions on smooth projective varieties of dimension n + 1 that can be extended to rational actions of $Cr_n(\mathbb{C})$