Introduction

Summary

Prom the beginnings of algebraic geometry it has been understood that birationally equivalent varieties have many properties in common. Thus it is natural to attempt to find in each birational equivalence class a variety which is simplest in some sense, and then study these varieties in detail.

Each irreducible curve is birational to a unique smooth projective curve, thus the investigation of smooth projective curves is equivalent to the study of all curves up to birational equivalence.

For surfaces the situation is more complicated. Each irreducible surface is birational to infinitely many smooth projective surfaces. The theory of minimal models of surfaces, developed by the Italian algebraic geometers at the beginning of the twentieth century, aims to choose a unique smooth projective surface from each birational equivalence class. The recipe is quite simple. If a smooth projective surface contains a smooth rational curve with self-intersection –1, then it can be contracted to a point and we obtain another smooth projective surface. Repeating this procedure as many times as possible, we usually obtain a unique ‘minimal model’. In a few cases we obtain a model that is not unique, but these cases can be described very explicitly.

A search for a higher dimensional analogue of this method started quite late. One reason is that some examples indicated that a similar approach fails in higher dimensions.

The works of Reid and Mori in the early 1980s raised the possibility that a higher dimensional theory of minimal models may be possible if we allow not just smooth varieties but also varieties with certain mild singularities.