4 - Surface Singularities of the Minimal Model Program

Summary

In this chapter we discuss various classes of surface singularities that are important for the minimal model program.

The first section contains an essentially complete description of all log canonical surface pairs. These results provide important signposts toward the higher dimensional theory. Many of these results are used later.

The aim of section 2 is to classify Du Val singularities by equations. These are precisely the canonical surface singularities. From this point of view they were classified by [DV34]. Du Val singularities appear naturally in many different contexts, see [Dur79] for a survey.

Simultaneous resolution of Du Val singularities is considered in section 3. This result was established by [Bri66, Bri71]. Our presentation is modelled on [Tyu70].

The natural setting for sections 2-3 is local analytic geometry. In all cases it is possible to work with algebraic varieties, but only at the price of some rather artificial constructions. In applying these results we therefore have to rely on some basic comparison theorems between analytic and algebraic geometry.

Section 4 considers elliptic surface singularities. Their theory was developed by [Rei76, Lau77]. These results are crucial for the treatment of 3-dimensional canonical singularities given in section 5.3.

In the last section we construct miniversal deformation spaces for isolated hypersurface singularities.

While all these results are very useful, and for the moment indispensable, for the 3-dimensional minimal model program, they are a hindrance from the point of view of the general theory. Many of the methods reducing 3-dimensional questions to surface problems, and the 2-dimensional results used in the process, do not generalize to higher dimensions.