Tori and surfaces violating a local-to-global principle for rationality
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 count...
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| Language: | English |
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Académie des sciences
2024-10-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.602/ |
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| author | Kunyavskiĭ, Boris |
| author_facet | Kunyavskiĭ, Boris |
| author_sort | Kunyavskiĭ, Boris |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-0d042d83f66645348d2f5f593c6874d1 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-0d042d83f66645348d2f5f593c6874d12025-02-07T11:22:49ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-10-01362G884184910.5802/crmath.60210.5802/crmath.602Tori and surfaces violating a local-to-global principle for rationalityKunyavskiĭ, Boris0Department of Mathematics, Bar-Ilan University, 5290002 Ramat Gan, IsraelWe 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.602/Algebraic torustoric varietyrational surfaceconic bundlerationalityBrauer group |
| spellingShingle | Kunyavskiĭ, Boris Tori and surfaces violating a local-to-global principle for rationality Comptes Rendus. Mathématique Algebraic torus toric variety rational surface conic bundle rationality Brauer group |
| title | Tori and surfaces violating a local-to-global principle for rationality |
| title_full | Tori and surfaces violating a local-to-global principle for rationality |
| title_fullStr | Tori and surfaces violating a local-to-global principle for rationality |
| title_full_unstemmed | Tori and surfaces violating a local-to-global principle for rationality |
| title_short | Tori and surfaces violating a local-to-global principle for rationality |
| title_sort | tori and surfaces violating a local to global principle for rationality |
| topic | Algebraic torus toric variety rational surface conic bundle rationality Brauer group |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.602/ |
| work_keys_str_mv | AT kunyavskiiboris toriandsurfacesviolatingalocaltoglobalprincipleforrationality |