The Dimension of the Moduli Space of Pointed Algebraic Curves of Low Genus

TL;DR

This paper explicitly computes the dimensions of moduli spaces of pointed algebraic curves with specified Weierstrass semigroups for low genus cases, providing detailed results up to genus seven.

Contribution

It offers explicit calculations of moduli space dimensions for pointed algebraic curves with given semigroups, especially for genus up to seven, filling gaps in existing classifications.

Findings

Computed dimensions for many genus ≤7 cases

Determined dimensions for all semigroups of genus seven

Provided explicit descriptions of moduli spaces for low genus curves

Abstract

We explicitly compute the moduli space pointed algebraic curves with a given numerical semigroup as Weierstrass semigroup for many cases of genus at most seven and determine the dimension for all semigroups of genus seven.