Hypersurface model-fields of definition for smooth hypersurfaces and their twists
Badr, Eslam (Cairo University. Department of Mathematics (Egypt))
Bars Cortina, Francesc (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Fecha:	2020
Resumen:	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.
Ayudas:	Ministerio de Economía y Competitividad MTM2016-75980-P
Derechos:	Tots els drets reservats.
Lengua:	Anglès
Documento:	Article ; recerca ; Versió publicada
Materia:	Fields of definition ; Hypersurface models ; Twists ; Automorphism groups
Publicado en:	Acta Arithmetica, Vol. 194 (March 2020) , p. 267-280, ISSN 1730-6264

DOI: 10.4064/aa180524-31-7

15 p, 454.5 KB

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