Hypersurface model-fields of definition for smooth hypersurfaces and their twists
Given a smooth projective variety of dimension n - 1 ≥ 1 defined over a perfect field k that admits a non-singular hypersurface model in Pnk- over k-, a fixed algebraic closure of k, it does not necessarily have a non-singular hypersurface model defined over the base field k. We first show an exampl...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2020 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:243408 |
| Acceso en línea: | https://ddd.uab.cat/record/243408 https://dx.doi.org/urn:doi:10.4064/aa180524-31-7 |
| Access Level: | acceso abierto |
| Palabra clave: | Fields of definition Hypersurface models Twists Automorphism groups |
| Sumario: | Given a smooth projective variety of dimension n - 1 ≥ 1 defined over a perfect field k that admits a non-singular hypersurface model in Pnk- over k-, a fixed algebraic closure of k, it does not necessarily have a non-singular hypersurface model defined over the base field k. We first show an example of such phenomenon: a variety defined over k admitting non-singular hypersurface models but none defined over k. We also determine under which conditions a non-singular hypersurface model over k may exist. Now, even assuming that such a smooth hypersurface model exists, we wonder about the existence of non-singular hypersurface models over k for its twists. We introduce a criterion to characterize twists possessing such models and we also show an example of a twist not admitting any non-singular hypersurface model over k, i.e. for any n ≥ 2, there is a smooth projective variety of dimension n - 1 over k which is a twist of a smooth hypersurface variety over k, but itself does not admit any non-singular hypersurface model over k. Finally, we obtain a theoretical result to describe all the twists of smooth hypersurfaces with cyclic automorphism group having a model defined over k whose automorphism group is generated by a diagonal matrix. The particular case n = 2 for smooth plane curves was studied by the authors jointly with E. Lorenzo García in [Math. Comp. 88 (2019)], and we deal here with the problem in higher dimensions. |
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