On twists of smooth plane curves

Badr, Eslam; Bars, Francesc; Lorenzo García, Elisa

doi:10.1090/mcom/3317

On twists of smooth plane curves
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Math. Comp. 88 (2019), 421-438 Request permission

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: 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.

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