Morphisms between Cremona groups and characterization of rational varieties

Following the paper of Serge Cantat we will classify homomorphisms from $PGL_{r+1}(\mathbb C)$ to the group $Bir(M)$ of birational transformations of a complex projective variety $M$ of dimension $dim(M)\leq r$. As a byproduct, we show that (1) $Cr(m)$ is isomorphic, as an abstract group, to $Cr(n)$ if and only if $n = m$ and (2) $M$ is rational if and only if $PGL_{dim(M)+1}(\mathbb C)$ embeds as a subgroup of $Bir(M)$.