Quotients of higher-dimensional Cremona groups

Abstract

We study large groups of birational transformations $\operatorname{Bir}(X)$, where $X$ is a variety of dimension at least $3$, defined over $\mathbf{C}$ or a subfield of $\mathbf{C}$. Two prominent cases are when $X$ is the projective space $\mathbb{P}^n$, in which case $\operatorname{Bir}(X)$ is the Cremona group of rank $n$, or when $X \subset \mathbb{P}^{n+1}$ is a smooth cubic hypersurface. In both cases, and more generally when $X$ is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from $\operatorname{Bir}(X)$ to $\mathbf{Z}/2$, showing in particular that the group $\operatorname{Bir}(X)$ is not perfect, and thus not simple. As a consequence, we also obtain that the Cremona group of rank $n \geqslant 3$ is not generated by linear and Jonquières elements.