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 each of those, in the absence of the Brauer obstruction to rationality, is rational over all completions of but is not -rational.
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 dont chacune est rationnelle partout localement mais n’est pas -rationnelle, en absence d’obstruction de Brauer à la rationalité.
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Keywords: Algebraic torus, toric variety, rational surface, conic bundle, rationality, Brauer group
Mots-clés : Tore algébrique, variété torique, surface rationnelle, fibré en coniques, rationalité, groupe de Brauer
Boris Kunyavskiĭ 1

@article{CRMATH_2024__362_G8_841_0, author = {Boris Kunyavski\u{i}}, title = {Tori and surfaces violating a local-to-global principle for rationality}, journal = {Comptes Rendus. Math\'ematique}, pages = {841--849}, publisher = {Acad\'emie des sciences, Paris}, volume = {362}, year = {2024}, doi = {10.5802/crmath.602}, language = {en}, }
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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