Acta Mathematica

Volume 226 (2021)

Number 2

Quotients of higher-dimensional Cremona groups

Pages: 211 – 318



Jérémy Blanc (Universität Basel, Switzerland)

Stéphane Lamy (Université de Toulouse, France)

Susanna Zimmermann (Université d’Angers, France)


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.


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

2010 Mathematics Subject Classification

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

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