On the Boundedness of Globally $F$-split varieties

TL;DR

This paper explores the use of $F$-split and globally $F$-regular conditions to establish boundedness results in algebraic geometry over fields of positive characteristic, focusing on threefold Mori fibre spaces.

Contribution

It introduces new techniques using $F$-split conditions to achieve birational boundedness results, reducing the problem to prime Fano varieties.

Findings

Proves birational boundedness for a class of varieties in positive characteristic.

Develops detailed analysis of threefold Mori fibre spaces over positive dimensional bases.

Reduces complex boundedness problems to the study of prime Fano varieties.

Abstract

This paper proposes the use of $F$-split and globally $F$-regular conditions in the pursuit of BAB type results in positive characteristic. The main technical work comes in the form of a detailed study of threefold Mori fibre spaces over positive dimensional bases. As a consequence we prove the main theorem, which reduces birational boundedness for a large class of varieties to the study of prime Fano varieties.