# On twists of smooth plane curves

Eslam Badr, Francesc Bars, Elisa Lorenzo García

Research output: Contribution to journalArticleResearchpeer-review

6 Citations (Scopus)

## Abstract

© 2018 American Mathematical Society. Given a smooth curve defined over a field k that admits a nonsingular plane model over 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: a curve defined over k admitting plane models but none defined over k. 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 we also show an example of a twist not admitting any non-singular plane model over k. As a consequence, we get explicit equations for a non-trivial Brauer-Severi surface. Finally, we obtain a theoretical result to describe all the twists of smooth plane curves with cyclic automorphism group having a model defined over k whose automorphism group is generated by a diagonal matrix.
Original language English 421-438 Mathematics of Computation 88 315 https://doi.org/10.1090/mcom/3317 Published - 1 Jan 2018

## Keywords

• Automorphism groups
• Non-singular plane curves
• Twist