Counterexamples to the MMP for 1-foliations in positive characteristic

TL;DR

The work investigates extending the Minimal Model Program to 1-foliations in characteristic $p>0$ and demonstrates that key results such as the cone theorem, base point free theorem, and the existence of Mori fibre spaces can fail. It develops a concrete construction via $p$-cyclic coverings $X=Y[\sqrt[p]{s}]$ yielding canonical rank-1 foliations with $-K_{\\mathcal{F}}$ ample, and analyzes the resulting Mori cones and birational behavior to produce counterexamples. The findings show that the MMP for foliations in positive characteristic exhibits severe pathologies, including nonpolyhedral Mori cones and nef divisors that are not semi-ample, while also connecting these phenomena to the Jacobson correspondence and inseparable morphisms. These results highlight the limits of MMP-type techniques in positive characteristic and point to special foliations, such as infinitesimal groupoid foliations, where the cone theorem may still hold, indicating a nuanced landscape for foliation MMP in low dimensions.

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 $p>0$.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1
• Definition 1
• Definition 2
• Remark 2
• Theorem 2.1: Jacobson's correspondence
• Proposition 1: PW22*Proposition 2.10
• Definition 3