On the Moduli Space of Pointed Algebraic Curves of Low Genus ---A Computational Approach---

Abstract

We compute explicitly the moduli space $\mathcal{M}_{g,1}^N$ of pointed algebraic curves of genus $g$ with a given numerical semigroup $N$ when $g$ is small ($2 \leq g \leq 5$). It is known that such a moduli space $\mathcal{M}_{g,1}^N$ is non-empty for $g \leq 7$. The main results obtained in this note are the irreducibility and the determination of the dimension of $\mathcal{M}_{g,1}^N$ for $g \leq 5$ except a few cases. In particular, it turns out that many of these moduli spaces are unirational.