Sobre l'estratificació de corbes planes llises per grup d'automorfismes

• Autores: Eslam Farag
• Directores de la Tesis: Francesc Bars Cortina (dir. tes.)
• Lectura: En la Universitat Autònoma de Barcelona ( España ) en 2017
• Idioma: español
• Tribunal Calificador de la Tesis: Joan Carles Lario Loyo (presid.) , Francesc Xavier Xarles Ribas (secret.) , Josep Maria Miret Biosca (voc.)
• Texto completo no disponible (Saber más ...)
• Resumen

  • Smooth projective curves over a field k with non-trivial automorphism group are always of deep interest in the literature. Following the philosophy of Diophantine equations theory, the simplest case is to consider smooth plane curves over k of geometric genus g≥3. In the thesis, we study the stratification of smooth plane curves by their automorphism groups and we deal with both algebraic and arithmetic geometry aspects.

    We first give a general study of the stratum, consisting of k ̅-isomorphism classes of smooth plane curves of fix genus g with a fixed automorphism subgroup G, where k ̅ is a fixed algebraic closure of k. In particular, a detailed study of the structure of their automorphism group and the associated defining equations is investigated.

    Second, let C be a smooth projective curve defined over k, which is also plane viewed as a smooth curve over k ̅. We aim to study fields of definition for non-singular plane models of C and also of its twists over k by considering the embedding into P_k ̅^2 instead of the one given by the canonical model into P_k ̅^(g-1). More concretely, we ask whether C is a smooth plane curve over k or not; and if the answer is yes, is every twist of C over k also a smooth plane curve over k?. For both questions the answer is no in general, it is not. We obtain results for the curves for which the above questions always have an affirmative answer, and we show different examples concerning the negative general answer. Interestingly, in the way to get these examples, we need to handle with non-trivial Brauer-Severi surfaces, and we are able to compute explicit equations of a non-trivial one. As far as we know, this is the first time that such equations are exhibited.

    Third, we obtain a so-called representative classification for the strata by automorphism group of smooth k ̅-plane curves of genus g=6, where k is perfect of characteristic p=0 or p>13. Interestingly, in the way to get these families, we find two remarkable phenomena that did not appear before. One is the existence of a non 0-dimensional final stratum of plane curves. At a first sight it may sound odd, but we will see that this is a normal situation for higher degrees and we will give an explanation for it. We explicitly describe representative families for all strata, except for the stratum with cyclic automorphism group of order 5(fortunately, we are still able to prove the existence of such family by applying a version of Lüroth’s theorem in dimension 2). Here we find the second difference with the lower genus cases where the known techniques do not fully work.

    Finally, let k be a perfect field of characteristic different from 2, and C be a smooth plane curve over k ̅ whose automorphism group of C is PGL_3 (k ̅)-conjugate to a diagonal group. It is known from the work of B. Huggins in her PhD thesis (2010, Berkeley) that the field of moduli for C, relative to the Galois extension k ̅/k does not need to be a field of definition. Motivated by these results, we wonder about characterizations of such curves not definable over their field of moduli. We distinguish between the two cases depending on whether the number of points of the projective plane P_k ̅^2 fixed by the automorphism group is finite or infinite. Our results can be usede as a constructive source of so many examples of smooth plane curves with cyclic automorphism where the field of moduli is not a field of definition.