Subcanonical curves: on the loci of odd Weierstrass semigroups and the dimension of the loci for certain families of semigroups

Abstract

A celebrated result of KontsevichZorich ensures that the moduli space (...) of pointed curves of genus g whose marked point is subcanonical has three irreducible components. In this work we present an explicit method to construct a compactification of the loci which corresponds to a general point of an irreducible component of (...), namely the loci of pointed curves whose symmetric Weierstrass semigroup is odd. The construction is an extension of Stoehrs techniques using equivariant deformation of monomial curves given by Pinkham by exploring syzygies. As an application we prove the rationality of the loci for genus six. Byfixing a family of semigroups of multiplicity 6, we also compute the dimension of the moduli space of pointed curve whose Weierstrass semigroup at the marked point belogns to the fixed family.