Hypersurface model-fields of definition for smooth hypersurfaces and their twists

TL;DR

This paper investigates the existence of non-singular hypersurface models over a base field for smooth projective varieties and their twists, providing criteria and examples of when such models do or do not exist over the field.

Contribution

It introduces a criterion for the existence of hypersurface models over the base field for twists and presents examples illustrating the phenomenon where models do not exist over the field.

Findings

Examples of varieties with models over algebraic closure but not over the base field

A criterion to determine when a twist admits a hypersurface model over the base field

A theoretical description of twists of hypersurfaces with cyclic automorphism groups

Abstract

Given a smooth projective variety of dimension $n-1\geq 1$ defined over a perfect field $k$ that admits a non-singular hypersurface modelin $\mathbb{P}^n_{\overline{k}}$ over $\overline{k}$, 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 $n\geq 2$, 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.