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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 modulispace $\overline{M}_{0,n+3}$ of $(n+3)$-pointed rational curves is. We alsoshow that $\overline{M}_{0,n}(\mathbb{P}^1,1)$ is a log Fano variety for $n \le5$.

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