Quotients of higher-dimensional Cremona groups

Jérémy Blanc Universität Basel, Switzerland Stéphane Lamy Université de Toulouse, France Susanna Zimmermann Université d’Angers, France

Group Theory and Lie Theory Algebraic Geometry mathscidoc:2203.17001

Acta Mathematica, 226, (2), 211-318, 2021.7
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 P^n, in which case Bir(X) is the Cremona group of rank n, or when X⊂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 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⩾3 is not generated by linear and Jonquières elements.
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  title={Quotients of higher-dimensional Cremona groups},
  author={Jérémy Blanc, Stéphane Lamy, and Susanna Zimmermann},
  booktitle={Acta Mathematica},
Jérémy Blanc, Stéphane Lamy, and Susanna Zimmermann. Quotients of higher-dimensional Cremona groups. 2021. Vol. 226. In Acta Mathematica. pp.211-318. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310111046284012922.
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