On the log minimal model program for $3$-folds over imperfect fields of characteristic $p>5$

TL;DR

This paper extends key results of the log minimal model program to three-dimensional varieties over imperfect fields of characteristic greater than five, with implications for algebraic geometry and tight closure theory.

Contribution

It establishes the existence of flips, cone theorem, contraction theorem, and log minimal models for 3-folds over imperfect fields of characteristic p>5, broadening the scope of the LMMP.

Findings

Proves existence of flips for 3-folds over imperfect fields

Establishes the cone and contraction theorems in this setting

Applications to tight closure in dimension 4

Abstract

We prove that many of the results of the LMMP hold for $3$-folds over fields of characteristic $p>5$ which are not necessarily perfect. In particular, the existence of flips, the cone theorem, the contraction theorem for birational extremal rays, and the existence of log minimal models. As well as pertaining to the geometry of fibrations of relative dimension $3$ over algebraically closed fields, they have applications to tight closure in dimension $4$.