Finite generation of adjoint rings after Lazic: an introduction

  • Alessio Corti

    Imperial College London, UK
Finite generation of adjoint rings after Lazic: an introduction cover
Download Chapter PDF

A subscription is required to access this book chapter.

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.