When is $\overline{M}_{0,n}(\mathbb{P}^1,1)$ a Mori dream space?

TL;DR

This paper investigates the conditions under which the moduli space of n-pointed stable maps to the projective line is a Mori dream space, establishing a connection with the moduli space of rational curves and exploring its Fano properties.

Contribution

It proves that ar;M_{0,n}(\u00b7P^1,1) is a Mori dream space when ar;M_{0,n+3} is, and shows it is a log Fano variety for n  5, linking these properties.

Findings

ar;M_{0,n}(\u00b7P^1,1) is a Mori dream space if ar;M_{0,n+3} is.

ar;M_{0,n}(\u00b7P^1,1) is a log Fano variety for n  5.

The paper establishes a criterion connecting the Mori dream space property with the moduli space of rational curves.

Abstract

We prove that the moduli space of $n$-pointed stable maps $\overline{M}_{0,n}(\mathbb{P}^1,1)$ is a Mori dream space whenever the moduli space $\overline{M}_{0,n+3}$ of $(n+3)$-pointed rational curves is. We also show that $\overline{M}_{0,n}(\mathbb{P}^1,1)$ is a log Fano variety for $n \le 5$.