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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.

Let Mg be the moduli space of smooth, genus g curves over an algebraically closed field K of zero characteristic. Denote by Mg(G) the subset of Mg of curves δ such that G (as a finite non-trivial group) is 2010 Mathematics Subject Classification. 14H37, 14H50, 14H45. ) >.(3) if σ has order d(d − 2) then δ is K-isomorphic to X d + Y d−1 Z + Y Z d−1 = 0 and for d = 4, 6 we haveis an irreducible locus with one element, andThe automorphism groups for d = 4, 6 are given explicitly in §3.3, Proposition 15.(4) if σ has order d 2 − 3d + 3 then δ is K-isomorphic to the Klein curve K d :) is irreducible with one element, andWe refer to Remark 18 of §3.4 for the classical case d = 4.Remark 2. The above situations do not fit with curves that have large automorphism group in the classical definition. For example, the curve δ : X d + Y d−1 Z + XZ d−1 = 0 is defined over Q, δ/Aut(δ) is a projective line and the morphism δ → δ/Aut(δ) is ramified at two points of ramification index (d − 1) 2 and at d − 1-points of

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