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arXiv · 1901.04145

Quotients of higher dimensional Cremona groups

Jérémy Blanc·Stéphane Lamy·Susanna Zimmermann
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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.

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Jérémy Blanc, Stéphane Lamy, Susanna Zimmermann. 2019-01-14. Quotients of higher dimensional Cremona groups. https://doi.org/10.4310/acta.2021.v226.n2.a1

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