Weierstrass semigroups on Castelnuovo curves

TL;DR

This paper introduces Castelnuovo semigroups, studies the associated loci of marked curves with these semigroups, and analyzes their irreducibility and dimension, revealing complex geometric structures.

Contribution

It defines Castelnuovo semigroups and characterizes the geometric properties of the loci of curves with these semigroups in moduli space.

Findings

Number of irreducible components determined

Dimensions of loci computed

Examples of reducible and non-equidimensional loci provided

Abstract

We define a class of numerical semigroups S, which we call Castelnuovo semigroups, and study the subvariety $M^S_{g,1}$ of $M_{g,1}$ consisting of marked smooth curves with Weierstrass semigroup S. We determine the number of irreducible components of these loci and determine their dimensions. Curves with these Weierstrass semigroups are always Castelnuovo curves, which provides the basic tool for our argument. This analysis provides examples of numerical semigroups for which $M^S_{g,1}$ is reducible and non-equidimensional.