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

Abstract

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.