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Some properties of the group of birational maps generated by the automorphisms of \({\mathbb {P}}^n_{\mathbb {C}}\) and the standard involution

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Abstract

We give some properties of the subgroup \(G_n({\mathbb {C}})\) of the group of birational self-maps of \({\mathbb {P}}^n_{\mathbb {C}}\) generated by the standard involution and the group of automorphisms of \({\mathbb {P}}^n_{\mathbb {C}}\). We prove that there is no nontrivial finite-dimensional linear representation of \(G_n({\mathbb {C}})\). We also establish that \(G_n({\mathbb {C}})\) is perfect, and that \(G_n({\mathbb {C}})\) equipped with the Zariski topology is simple. Furthermore if \(\upvarphi \) is an automorphism of \({{\mathrm {Bir}}}({\mathbb {P}}^n_{\mathbb {C}})\), then up to birational conjugacy, and up to the action of a field automorphism \({\upvarphi }_{\vert G_n({\mathbb {C}})}\) is trivial.

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Acknowledgments

I would like to thank D. Cerveau for his helpful and continuous listening. Thanks to the referee that helps me to improve the exposition. Thanks to I. Dolgachev for pointing out me that Coble introduced the group \(G_n({\mathbb {C}})\) in [15], and to J. Blanc, J. Diller, F. Han, M. Jonsson, J.-L. Lin for their remarks and comments.

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  1. Institut de Mathématiques de Jussieu Paris Rive Gauche, UMR 7586, Université Paris Diderot, Bâtiment Sophie Germain, Case 7012, 75205, Paris Cedex 13, France

    Julie Déserti

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Correspondence to Julie Déserti.

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The author is supported by ANR Grant “BirPol” ANR-11-JS01-004-01.

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Déserti, J. Some properties of the group of birational maps generated by the automorphisms of \({\mathbb {P}}^n_{\mathbb {C}}\) and the standard involution. Math. Z. 281, 893–905 (2015). https://doi.org/10.1007/s00209-015-1512-x

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