Higher dimensional foliated Mori theory

TL;DR

This paper advances higher dimensional foliated Mori theory by establishing foundational results and applying them to describe the cone of curves and progress toward a minimal model program for rank 2 foliations on threefolds.

Contribution

It introduces new results in higher dimensional foliated Mori theory and applies them to structure theorems and minimal model program progress for rank 2 foliations.

Findings

Structure theorem for the Kleiman-Mori cone of curves

Progress toward minimal model program for rank 2 foliations

Results relating $K_{\mathcal{F}}$ to the cone of curves

Abstract

We develop some basic results in a higher dimensional foliated Mori theory, and show how these results can be used to prove a structure theorem for the Kleiman-Mori cone of curves in terms of the numerical properties of $K_{\mathcal{F}}$ for rank 2 foliations on threefolds. We also make progress toward realizing a minimal model program for rank 2 foliations on threefolds.