Quotients of higher dimensional Cremona groups

TL;DR

This paper investigates the structure of large birational transformation groups of higher-dimensional varieties, revealing their non-simplicity and the existence of many homomorphisms to Z/2, especially for Cremona groups of rank at least 3.

Contribution

It constructs infinitely many homomorphisms from Bir(X) to Z/2 for certain varieties, showing these groups are not perfect or simple, and challenges previous assumptions about their generation.

Findings

Bir(X) admits infinitely many homomorphisms to Z/2.

Bir(X) is not perfect and not simple.

Cremona groups of rank n ≥ 3 are not generated solely by linear and Jonquières elements.

Abstract

We study large groups of birational transformations Bir(X), where X is a variety of dimension at least 3, defined over C or a subfield of C. Two prominent cases are when X is the projective space, in which case Bir(X) is the Cremona group of rank n, or when X 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 Bir(X) to Z/2, showing in particular that the group Bir(X) is not perfect and thus not simple. As a consequence we also obtain that the Cremona group of rank n at least 3 is not generated by linear and Jonqui\`eres elements.