Quotients of higher-dimensional Cremona groups

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.

Keywords

Cremona groups, normal subgroups, conic bundles, Sarkisov links, BAB conjecture

2010 Mathematics Subject Classification

14E05, 14E07, 14E30, 14J45, 20F05, 20L05

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The first author acknowledges support by the Swiss National Science Foundation Grant “Birational transformations of threefolds” 200020_178807. The second author was partially supported by the UMICRM 3457 of the CNRS in Montréal, and by the Labex CIMI. The third author was supported by Projet PEPS 2018 ”JC/JC” and is supported by the ANR Project FIBALGA ANR-18-CE40-0003-01.

Received 11 February 2019

Received revised 4 December 2019

Accepted 4 March 2020

Published 2 July 2021