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

This work is inspired by conversations with Izzet Coskun and Joe Harris. We run 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 to all the intermediate spaces appearing in the process. In particular, we show that 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 an easy corollary we obtain that ${\bar{\mathcal{M}}}_{0,0}({\mathbb{P}}^3,3)$ is a Mori dream space.