On twists of smooth plane curves

TL;DR

This paper investigates when twists of smooth plane curves admit non-singular plane models over a field $k$, providing conditions, examples, and algorithms to compute such twists, with applications to Brauer-Severi surfaces.

Contribution

It characterizes twists of smooth plane curves that admit plane models over $k$, improves algorithms for computing twists, and provides explicit examples and theoretical results.

Findings

Identifies conditions for twists to have plane models over $k$

Provides an algorithm to compute twists of non-hyperelliptic curves

Constructs explicit equations for a non-trivial Brauer-Severi surface

Abstract

Given a smooth curve defined over a field $k$ that admits a non-singular plane model over $\overline{k}$, a fixed separable closure of $k$, it does not necessarily have a non-singular plane model defined over the field $k$. We determine under which conditions this happens and we show an example of such phenomenon. Now, even assuming that such a smooth plane model exists, we wonder about the existence of non-singular plane models over $k$ for its twists. We characterize twists possessing such models and use such characterization to improve, for the particular case of smooth plane curves, the algorithm to compute twists of non-hyperelliptic curves wrote recently down by the third author. We also show an example of a twist not admitting such non-singular plane model. As a consequence, we get explicit equations for a non-trivial Brauer-Severi surface. Finally, we obtain a theoretical result to compute all the twists of smooth plane curves with cyclic automorphism group having a $k$-model whose automorphism group is generated by a diagonal matrix. Some examples are also provided.