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

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.