Reduction maps and minimal model theory

TL;DR

This paper investigates the minimal model program using reduction maps, demonstrating that the existence of good minimal models can be detected via the trivial reduction map, linking conjectures to the existence of certain curves.

Contribution

It introduces a new approach to the minimal model program by connecting the existence of minimal models to properties of reduction maps and curves on the variety.

Findings

Existence of good minimal models is detectable via the trivial reduction map.

Main conjectures relate to the existence of specific curves on the variety.

Reduction maps provide a new perspective on the minimal model program.

Abstract

We use reduction maps to study the minimal model program. Our main result is that the existence of a good minimal model for a klt pair $(X,\Delta)$ can be detected on the base of the $(K_{X}+\Delta)$-trivial reduction map. Thus we show that the main conjectures of the minimal model program can be interpreted as a natural statement on the existence of curves on $X$.