3 - Examples

Summary

We studied log canonical surface singularities in Section 2.2 and gave examples of typical log terminal 3-fold singularities in Section 2.4. In Section 2.5 we proved that dlt singularities are rational in any dimension.

The aim of this chapter is to show, by many examples, that the above results are by and large optimal: log canonical singularities get much more complicated in dimension 3 and even terminal singularities are likely non-classifiable in dimension 4.

A first set of examples of the various higher dimensional singularities occurring in the minimal model program are given in Section 3.1. These are rather elementary, mostly cones, but they already illustrate how subtle log canonical pairs can be.

We consider in greater detail quotient singularities in Section 3.2. This is a quite classical topic but with many subtle aspects. These are some of the simplest log terminal singularities in any dimension but they are the most likely to come up in applications.

Section 3.3 gives a rather detailed classification of log canonical surface singularities. Strictly speaking, not all of Section 3.3 is needed for the general theory, but it is useful and instructive to have a thorough understanding of a class of concrete examples. Our treatment may be longer than usual, but it applies in positive and mixed characteristics as well.