Finite generation and geography of models

TL;DR

This paper develops a unified framework for the Minimal Model Program that encompasses classical cases and Mori Dream Spaces, clarifying the role of finite generation in running the MMP.

Contribution

It generalizes the MMP framework to a broader setting and identifies conditions beyond finite generation necessary for the MMP to succeed.

Findings

Finite generation of divisorial rings is necessary but not sufficient for the MMP.

The paper provides criteria to determine when the MMP can be performed.

A unified approach to classical MMP and Mori Dream Spaces is established.

Abstract

There are two main examples where a version of the Minimal Model Program can, at least conjecturally, be performed successfully: the first is the classical MMP associated to the canonical divisor, and the other is Mori Dream Spaces. In this paper we formulate a framework which generalises both of these examples. Starting from divisorial rings which are finitely generated, we determine precisely when we can run the MMP, and we show why finite generation alone is not sufficient to make the MMP work.