On twists of smooth plane curves

Abstract

Abstract. 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: 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.

The Picard group of Brauer-Severi varieties

“…k is equal to Υ. In [1] we constructed twists of smooth plane curves over k not having smooth plane model over k. These twists happened to be contained in Brauer-Severi surfaces as in Theorem 3.1. [1]] Given a smooth plane curve over k: C/k ⊆ P 2 with degree d ≥ 4, there exists a natural map Σ :…”

“…is the set of twists of C admitting a smooth plane model over k, where [P 2 k ] is the trivial class associated to the trivial Brauer-Severi surface of the projective plane over k. [1]). We can reinterpret the map Σ in Theorem 3.2 as the map that sends a twist C ′ to the Brauer-Severi variety B in Theorem 3.1.…”

“…The main idea is looking for smooth plane curves C of degree a multiple of 3, otherwise all their twists are smooth plane curves over k, see Theorem 2.6 in [1], and having an automorphism of the form [aZ : X : Y ] to define the twist C ′ given by the cocycle that sends a generator σ of the degree 3 cyclic extension L/k defining B to the automorphism [aZ : X : Y ].…”

“…Its group the Picard Pic(B) it is known to be isomorphic to Z. As far as we know, the first explicit equations defining a non-trivial Brauer-Severi surface in the literature are in [1]. After this, an algorithm to compute these equations for any Brauer-Severi variety is given in [7].…”

Hypersurface model-fields of definition for smooth hypersurfaces and their twists

“…Acknowledgments. The authors would like to express their gratitude to David Kohel who suggested to us the generalization of the work in [1] for hypersurfaces instead of plane curves, also to Joaquim Roé and Xavier Xarles for their useful comments and suggestions.…”