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

TL;DR

This paper applies the log minimal model program to the Kontsevich space of stable maps to P^3, providing modular interpretations of intermediate spaces and establishing that this space is a Mori dream space.

Contribution

It performs the log minimal model program for ,0(, P^3, 3), giving modular descriptions of all intermediate models and showing the space is a Mori dream space.

Findings

One component of the Hilbert scheme is the flip of the moduli space.

The space admits a modular interpretation at each step.

The moduli space is a Mori dream space.

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.