Web of Science: 5 cites, Scopus: 6 cites, Google Scholar: cites
On twists of smooth plane curves
Badr, Eslam (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Bars Cortina, Francesc (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Lorenzo García, Elisa (Universiteit Leiden)

Data: 2019
Resum: Given a smooth curve defined over a field k that admits a non-singular 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. 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.
Ajuts: Ministerio de Economía y Competitividad MTM2013-40680-P
Drets: Tots els drets reservats.
Llengua: Anglès
Document: Article ; recerca ; Versió sotmesa a revisió
Matèria: Non-singular plane curves ; Automorphism groups ; Twist
Publicat a: Mathematics of Computation, Vol. 88, Issue 315 (March 2019) , p. 421-438, ISSN 0025-5718

DOI: 10.1090/mcom/3317


Preprint
19 p, 351.2 KB

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