[Un déterminant sur applications birationnelles de surfaces de Severi-Brauer]
We define a determinant on the group of automorphisms of non-trivial Severi–Brauer surfaces over a perfect field. Using the generators and relations, we extend this determinant to birational maps between Severi–Brauer surfaces. By means of this determinant and a group homomorphism found in [BSY24], we can determine the abelianization of the group of birational transformations of a non-trivial Severi–Brauer surface. This is the first example of an abelianization of the group of birational transformations of a geometrically rational surface where the automorphisms are non-trivial. Using the abelianization, we find maximal subgroups of the group of birational transformations of a non-trivial Severi–Brauer surface over a perfect field.
Nous définissons un déterminant sur le groupe des automorphismes des surfaces de Severi-Brauer non triviales sur un corps parfait. À l’aide des générateurs et des relations, nous étendons ce déterminant aux applications birationnelles entre surfaces de Severi-Brauer. En utilisant ce déterminant et un homomorphisme de groupes trouvé dans [BSY24], nous pouvons déterminer l’abélianisé du groupe des transformations birationnelles d’une surface de Severi-Brauer non triviale. Il s’agit du premier exemple d’abélianisé d’un groupe de transformations birationnelles d’une surface géométriquement rationnelle dont les automorphismes sont non triviaux. Au moyen de cette abélianisé, nous trouvons des sous-groupes maximaux du groupe des transformations birationnelles d’une surface de Severi-Brauer non triviale sur un corps parfait.
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Keywords: Severi–Brauer surfaces, Galois cohomology, Sarkisov program, abelianization
Mots-clés : Surfaces Severi-Brauer, cohomologie galoisienne, programme de Sarkisov, abélianisation
Elias Kurz 1
CC-BY 4.0
Elias Kurz. A determinant on birational maps of Severi–Brauer surfaces. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 1275-1322. doi: 10.5802/jep.347
@article{JEP_2026__13__1275_0,
author = {Elias Kurz},
title = {A determinant on birational maps of {Severi{\textendash}Brauer} surfaces},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1275--1322},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.347},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.347/}
}
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