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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-10-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.602/ |
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| Summary: | 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. |
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| ISSN: | 1778-3569 |