  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 $n \le
5$.
