Irreducibility and stable rationality of the loci of curves of genus at most six with a marked Weierstrass point

Abstract

Given a numerical semigroup H ⊆ ( Z ≥ 0 , + ) H\subseteq (\mathbf {Z}_{\geq 0},+) , we consider the locus M g , 1 H \mathcal {M}_{g,1}^H of smooth curves of genus g g with a marked Weierstrass point of semigroup H H . We show that for all semigroups H H of genus g ≤ 6 g\leq 6 the locus M g , 1 H \mathcal {M}_{g,1}^H is irreducible and that for all but possibly two such semigroups it is stably rational.