Counterexamples to the MMP for 1-foliations in positive characteristic

TL;DR

This paper demonstrates that key components of the Minimal Model Program do not hold for 1-foliated surface pairs with canonical singularities in positive characteristic, providing counterexamples to established conjectures.

Contribution

It provides explicit counterexamples showing the failure of the MMP for 1-foliations in positive characteristic, challenging existing assumptions in algebraic geometry.

Findings

Failure of the cone theorem in positive characteristic

Failure of the base point free theorem in positive characteristic

Non-existence of Mori fibre spaces for certain foliated pairs

Abstract

We show that many statements of the Minimal Model Program, including the cone theorem, the base point free theorem and the existence of Mori fibre spaces, fail for 1-foliated surface pairs $(X,\mathcal{F})$ with canonical singularities in characteristic $p>0$.