Existence of minimal models for threefold generalized pairs in positive characteristic

TL;DR

This paper proves the existence of minimal models and the termination of flips for three-dimensional pseudo-effective generalized pairs over algebraically closed fields of characteristic greater than 5, using ACC results.

Contribution

It establishes the existence of minimal models and termination of flips for threefold generalized pairs in positive characteristic, providing a new proof avoiding non-vanishing theorems.

Findings

Existence of minimal models for threefold generalized pairs in characteristic p>5.

Termination of flips for pseudo-effective threefold generalized pairs.

Proof relies on ACC for lc thresholds in low dimensions.

Abstract

Let $\mathbb{K}$ be an algebraically closed field of characteristic $p>5$. We show the existence of minimal models for pseudo-effective NQC lc generalized pairs in dimension three over $\mathbb{K}$. As a consequence, we prove the termination of flips for pseudo-effective threefold NQC lc generalized pairs over $\mathbb{K}$. This provides a new proof on the termination of flips for pseudo-effective pairs over $\mathbb{K}$ without using the non-vanishing theorems. A key ingredient of our proof is the ACC for lc thresholds in dimension $\leq 3$ and the global ACC in dimension $\leq 2$ for generalized pairs over $\mathbb{K}$.