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...

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Autores: Badr, Eslam|||0000-0002-3960-7243, Bars Cortina, Francesc|||0000-0003-4779-3995
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
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spelling Hypersurface model-fields of definition for smooth hypersurfaces and their twistsBadr, Eslam|||0000-0002-3960-7243Bars Cortina, Francesc|||0000-0003-4779-3995Fields of definitionHypersurface modelsTwistsAutomorphism groupsGiven 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. 22020-01-0120202020-01-01Articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/243408https://dx.doi.org/urn:doi:10.4064/aa180524-31-7reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengMinisterio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM2016-75980-Popen accesshttp://purl.org/coar/access_right/c_abf2Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.https://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:2434082026-06-06T12:50:31Z
dc.title.none.fl_str_mv Hypersurface model-fields of definition for smooth hypersurfaces and their twists
title Hypersurface model-fields of definition for smooth hypersurfaces and their twists
spellingShingle Hypersurface model-fields of definition for smooth hypersurfaces and their twists
Badr, Eslam|||0000-0002-3960-7243
Fields of definition
Hypersurface models
Twists
Automorphism groups
title_short Hypersurface model-fields of definition for smooth hypersurfaces and their twists
title_full Hypersurface model-fields of definition for smooth hypersurfaces and their twists
title_fullStr Hypersurface model-fields of definition for smooth hypersurfaces and their twists
title_full_unstemmed Hypersurface model-fields of definition for smooth hypersurfaces and their twists
title_sort Hypersurface model-fields of definition for smooth hypersurfaces and their twists
dc.creator.none.fl_str_mv Badr, Eslam|||0000-0002-3960-7243
Bars Cortina, Francesc|||0000-0003-4779-3995
author Badr, Eslam|||0000-0002-3960-7243
author_facet Badr, Eslam|||0000-0002-3960-7243
Bars Cortina, Francesc|||0000-0003-4779-3995
author_role author
author2 Bars Cortina, Francesc|||0000-0003-4779-3995
author2_role author
dc.subject.none.fl_str_mv Fields of definition
Hypersurface models
Twists
Automorphism groups
topic Fields of definition
Hypersurface models
Twists
Automorphism groups
description 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.
publishDate 2020
dc.date.none.fl_str_mv 2
2020-01-01
2020
2020-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/243408
https://dx.doi.org/urn:doi:10.4064/aa180524-31-7
url https://ddd.uab.cat/record/243408
https://dx.doi.org/urn:doi:10.4064/aa180524-31-7
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Ministerio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM2016-75980-P
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
reponame_str Dipòsit Digital de Documents de la UAB
collection Dipòsit Digital de Documents de la UAB
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repository.mail.fl_str_mv
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