Minimal Models of Canonical 3-Folds

Abstract

This paper introduces a temporary definition of minimal models of 3-folds (0.7), and studies these under extra hypotheses. The main result is Theorem (0.6), in which I characterise the singularities which necessarily appear on a minimal model, and prove the existence of a minimal model $S$ of a 3-fold of f.g. general type, by blowing up the canonical model $X$ studied in [C3-f], imitating closely the minimal resolution of Du Val surface singularities.

Apart from techniques familiar from [C3-f] (computations of the valuations of differentials; cyclic covers; crepant blow-ups of index 1 points which are not cDV), the main new element (Theorem (2.6)) is a method of blowing up the 1-dimensional singular locus, based on the Brieskorn–Tyurina result on the existence of simultaneous resolutions of a family of Du Val surface singularities, together with the elementary transformations in $(-2)$-curves of Burns and Rapoport. Part II is devoted to an exposition of these elementary transformations; much of this is folklore material, but it seems worthwhile to give a detailed account of what seems to be a key phenomenon of higher-dimensional birational geometry.

The canonical and terminal singularities introduced in [C3-f] and here have strong inductive properties, and there is some reason for believing that terminal singularities will provide the natural category for an inductive extension of Mori’s results: elementary contractions (when these exist) specified by extremal faces of the $K<0$ part of the Mori cone are always discrepant. I have included in §4 conjectures as to what the theory of minimal models and classification of 3-folds will look like in 3 or 4 years’ time, and a section of conjectures in §8 attempting to pin down the non-uniqueness of minimal models in the non-ruled case.