On the Moduli Space of Pointed Algebraic Curves of Low Genus II ---Rationality---

Abstract

We show that the moduli space $\mathcal{M}_{g,1}^N$ of pointed algebraic curves of genus $g$ with a given numerical semigroup $N$ is an irreducible rational variety if $N$ is generated by less than five elements for low genus ($ g \leq 6$) except one case. As a corollary to this result, we get a computational proof of the rationality of the moduli space $\mathcal{M}_{g,1}$ of pointed algebraic curves of genus $g$ for $1 \leq g \leq 3$. If $g \leq 5$, we also have that $\mathcal{M}_{g,1}^N$ is an irreducible rational variety for any semigroup $N$ except two cases. It is known that such a moduli space $\mathcal{M}_{g,1}^N$ is non-empty for $g \leq 7$.