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This work is inspired by conversations with Izzet Coskun and Joe Harris. Werun the log minimal model program for the Kontsevich space of stable maps$\bar{\mathcal M}_{0,0}(\mathbb P^{3}, 3)$ and give modular interpretations toall the intermediate spaces appearing in the process. In particular, we showthat one component of the Hilbert scheme $\mathcal H_{3,0,3}$ is the flip of$\bar{\mathcal M}_{0,0}(\mathbb P^{3}, 3)$ over the Chow variety. Finally as aneasy corollary we obtain that $\bar{\mathcal M}_{0,0}(\mathbb P^{3}, 3)$ is aMori dream space.

Log Minimal Model Program for the Kontsevich Space of Stable Maps $\bar{\mathcal M}_{0,0}(\mathbb P^{3}, 3)$