  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.
