Finite generation of adjoint rings after Lazic: an introduction

Abstract

This note is an introduction to all the key ideas of Lazic's recent proof of the theorem on the finite generation of adjoint rings [Laz09]. (The theorem was first proved in [BCHM09].) I try to convince you that, despite technical issues that are not yet adequately optimised, nor perhaps fully understood, Lazic's argument is a self-contained and transparent induction on dimension based on lifting lemmas and relying on none of the detailed general results of Mori theory. On the other hand, it is shown in [CL10] that all the fundamental theorems of Mori theory follow easily from the finite generation statement discussed here: together, these results give a new and more efficient organisation of higher dimensional algebraic geometry. The approach presented here is ultimately inspired by a close reading of the work of Shokurov [Sho03], I mean specifically his proof of the existence of 3-fold flips. Siu was the first to believe in the possibility of a direct proof of finite generation, and believing that something is possible is, of course, a big part of making it happen. All mathematical detail is taken from [Laz09]; my contribution is merely exegetic. I begin with a few key definitions leading to the statement of the main result.