On twists of smooth plane curves

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.
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https://hal.archives-ouvertes.fr/hal-01618759
Contributor : Marie-Annick Guillemer <>
Submitted on : Wednesday, October 18, 2017 - 2:36:19 PM
Last modification on : Thursday, November 15, 2018 - 11:56:48 AM

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Eslam Badr, Francesc Bars Cortina, Elisa Lorenzo Garcia. On twists of smooth plane curves. Mathematics of Computation, American Mathematical Society, 2017, 88 (315), pp.421-438. ⟨10.1090/mcom/3317⟩. ⟨hal-01618759⟩

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