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Mathematics of Computation

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On twists of smooth plane curves


Authors: Eslam Badr, Francesc Bars and Elisa Lorenzo García
Journal: Math. Comp. 88 (2019), 421-438
MSC (2010): Primary 11G30, 11D41, 14H37, 14H50, 14H45, 12F12
DOI: https://doi.org/10.1090/mcom/3317
Published electronically: March 15, 2018
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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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Additional Information

Eslam Badr
Affiliation: Department of Mathematics, Faculty of Science, Cairo University, 12613 Giza, Egypt
Email: eslam@sci.cu.edu.eg

Francesc Bars
Affiliation: Departament Matemàtiques, Edif. C, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia, Spain
Email: francesc@mat.uab.cat

Elisa Lorenzo García
Affiliation: Laboratoire IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes, Cedex, France
Email: elisa.lorenzogarcia@univ-rennes1.fr

DOI: https://doi.org/10.1090/mcom/3317
Keywords: Non-singular plane curves, automorphism groups, twist
Received by editor(s): February 22, 2017
Received by editor(s) in revised form: July 2, 2017, and July 28, 2017
Published electronically: March 15, 2018
Additional Notes: The first author (during his PhD study at UAB) and second author were supported by MTM2016-75980-P
Article copyright: © Copyright 2018 American Mathematical Society

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