Interseções completas e pontos de Weierstrass

Abstract

There are many advances related to the the rationality of the moduli space 𝒨H𝑔,1 parametrising pointed smooth projective curves of genus 𝑔 ≥ 0 and Weierstrass semigroup H at the marked point. The semigroups studied in the results have genus 𝑔 ≤ 6 or they are symmetric semigroups generated by at most four elements. This thesis is concerned to show two results about the geometry of 𝒨H 𝑔,1 that can be described as follows: given a numerical semigroup H of genus 𝑔 ≥ 1, if the monomial affine curve Spec k[H] is a complete intersection, then we can show that 𝒨H𝑔,1 admits a compactification that is isomorphic to the projetivization of the negatively graded part of the first cohomology moduli of k[H]. The complete intersection hypothesis can be interchanged by the hypothesis that the curve 𝑆𝑝𝑒𝑐 k[H] is a local complete intersection, but in this case we have to assume that 𝒨H𝑔,1 is non-empty. Under these new conditions we show the same conclusion as the first result. Many examples are made, including for families of semigroups. A classical result of realizable semi-groups is obtained independently through a simple application of the Jacobian criterion, and another result on the smoothing of monomial curves, without any obstructions, immediately declines from the second result described above.