Quotients of higher dimensional Cremona groups
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.
Cite
@article{arxiv.1901.04145,
title = {Quotients of higher dimensional Cremona groups},
author = {Jérémy Blanc and Stéphane Lamy and Susanna Zimmermann},
journal= {arXiv preprint arXiv:1901.04145},
year = {2021}
}
Comments
Item (RF4) in main definition 3.1 was modified: many thanks to Yang He for spotting the problem! Other minor changes. To appear in Acta Mathematica