2018/04/17 by Eslam Badr, Badr, Eslam, Francesc Bars +1
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1804.06118
openalex publication_date 2018/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a smooth projective variety of dimension n-1\≥ 1 defined over a\nperfect field k that admits a non-singular hypersurface modelin\n\ℙn_\k over \k, a fixed algebraic closure of\nk, it does not necessarily have a non-singular hypersurface model defined\nover the base field k. We first show an example of such phenomenon: a variety\ndefined over k admitting non-singular hypersurface models but none defined\nover k. We also determine under which conditions a non-singular hypersurface\nmodel over k may exist. Now, even assuming that such a smooth hypersurface\nmodel exists, we wonder about the existence of non-singular hypersurface models\nover k for its twists. We introduce a criterion to characterize twists\npossessing such models and we also show an example of a twist not admitting any\nnon-singular hypersurface model over k, i.e for any n\≥ 2, there is a\nsmooth projective variety of dimension n-1 over k which is a twist of a\nsmooth hypersurface variety over k, but itself does not admit any\nnon-singular hypersurface model over k. Finally, we obtain a theoretical\nresult to describe all the twists of smooth hypersurfaces with cyclic\nautomorphism group having a model defined over k whose automorphism group is\ngenerated by a diagonal matrix.\n