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Maximal subgroups in the Cremona group

Part of: Linear algebraic groups and related topics Birational geometry Algebraic groups

Published online by Cambridge University Press:  03 November 2025

Andrea Fanelli ,
Enrica Floris  and
Susanna Zimmermann Open the ORCID record for Susanna Zimmermann [Opens in a new window]
Andrea Fanelli
Affiliation:
Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400 Talence, France andrea.fanelli@math.u-bordeaux.fr
Enrica Floris
Affiliation:
Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR 7348 du CNRS, Batiment H3 - Site du Futuroscope, 11 boulevard Marie et Pierre CURIE, TSA 61125, 86073 Poitiers Cedex 9, France, Institut Universitaire de France enrica.floris@univ-poitiers.fr
Susanna Zimmermann
Affiliation:
Mathematisches Institut, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland susanna.zimmermann@unibas.ch
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Abstract

We show that for any $n\geq5$ there exist connected algebraic subgroups in the Cremona group $\text{Bir}(\mathbb P^n)$ that are not contained in any maximal connected algebraic subgroup. Our approach exploits the existence of stably rational, non-rational threefolds.

Keywords

Cremona groupsalgebraic groupsstable rationality

MSC classification

Primary: 14E07: Birational automorphisms, Cremona group and generalizations
Secondary: 14L30: Group actions on varieties or schemes (quotients) 20G99: None of the above, but in this section

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 9 , September 2025 , pp. 2442 - 2460
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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