Birational maps of Severi-Brauer surfaces, with applications to Cremona groups of higher rank

Abstract

We prove that any group of cardinality at most the one of $\mathbb{C}$ is a quotient of any Cremona group of rank at least $4$. This provides a definitive answer to the question of what the quotients of Cremona groups can be. As a consequence, this gives a negative answer to the question of I. Dolgachev of whether Cremona groups of all ranks are generated by involutions. As another application, we show that higher Cremona groups do not enjoy some classical group-theoretic properties (namely, the Hopfian property) which are satisfied by Cremona groups of rank $2$. Finally, we discover that the $3$-torsion of the Cremona group of rank at least $4$ is not countable. To deduce these properties of higher Cremona groups, we first describe the group of birational transformations of a non-trivial Severi-Brauer surface $S$ over a perfect field, proving in particular that if $S$ contains a point of degree $6$, then its group of birational self-maps is not generated by elements of finite order as it admits a surjective group homomorphism to~$\mathbb{Z}$. We then use this result to study Mori fibre spaces over the field of complex numbers, for which the generic fibre is a non-trivial Severi-Brauer surface.