Tori and surfaces violating a local-to-global principle for rationality

Boris Kunyavskiĭ

doi:10.5802/crmath.602

Article de recherche - Théorie des nombres

Tori and surfaces violating a local-to-global principle for rationality
[Tores et surfaces violant le principe local-global pour la rationalité]

Boris Kunyavskiĭ ¹

¹ Department of Mathematics, Bar-Ilan University, 5290002 Ramat Gan, Israel

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 841-849.

Résumé
Abstract

Nous démontrons que même dans une classe des variétés où l’obstruction de Brauer–Manin est la seule obstruction à l’existence de points rationnels (le principe de Hasse) cette obstruction, même sous sa forme la plus forte invariante par rapport au changement de base, peut être insuffisant pour expliquer des contre-exemples au principe local-global pour la rationalité. Nous présentons des exemples de variétés toriques et de surfaces rationnelles sur un corps global arbitraire $k$ dont chacune est rationnelle partout localement mais n’est pas $k$ -rationnelle, en absence d’obstruction de Brauer à la rationalité.

We show that even within a class of varieties where the Brauer–Manin obstruction is the only obstruction to the local-to-global principle for the existence of rational points (Hasse principle), this obstruction, even in a stronger, base change invariant form, may be insufficient for explaining counter-examples to the local-to-global principle for rationality. We exhibit examples of toric varieties and rational surfaces over an arbitrary global field $k$ each of those, in the absence of the Brauer obstruction to rationality, is rational over all completions of $k$ but is not $k$ -rational.

Reçu le : 2023-05-09
Révisé le : 2023-12-16
Accepté le : 2023-12-22
Publié le : 2024-10-07

DOI : 10.5802/crmath.602

Classification : 14E08, 14G12, 14G25, 14J20, 14J26, 14M20, 11G35, 20G30
Keywords: Algebraic torus, toric variety, rational surface, conic bundle, rationality, Brauer group
Mot clés : Tore algébrique, variété torique, surface rationnelle, fibré en coniques, rationalité, groupe de Brauer

Affiliations des auteurs :

Boris Kunyavskiĭ ¹

¹ Department of Mathematics, Bar-Ilan University, 5290002 Ramat Gan, Israel

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Tori and surfaces violating a local-to-global principle for rationality},
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Boris Kunyavskiĭ. Tori and surfaces violating a local-to-global principle for rationality. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 841-849. doi : 10.5802/crmath.602. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.602/

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