## Tokyo Journal of Mathematics

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

Tetsuo NAKANO

#### 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 $g \leq 7$.

#### Article information

Source
Tokyo J. Math., Volume 31, Number 1 (2008), 147-160.

Dates
First available in Project Euclid: 27 August 2008

https://projecteuclid.org/euclid.tjm/1219844828

Digital Object Identifier
doi:10.3836/tjm/1219844828

Mathematical Reviews number (MathSciNet)
MR2426799

Zentralblatt MATH identifier
1145.14025

#### Citation

NAKANO, Tetsuo. On the Moduli Space of Pointed Algebraic Curves of Low Genus II ---Rationality---. Tokyo J. Math. 31 (2008), no. 1, 147--160. doi:10.3836/tjm/1219844828. https://projecteuclid.org/euclid.tjm/1219844828

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