7 - Semi-stable Minimal Models

Summary

This chapter is devoted to a special case of the MMP, called the semi-stable MMP. Instead of dealing with a threefold in itself, we view it as a family of surfaces over a curve. Semi-stability is a somewhat technical assumption requiring that the surfaces be not too complicated. Under this assumption we prove that 3–dimensional flips exist and so the corresponding MMP works.

The original proof of the existence of 3–dimensional flips [Mor88] and the more general approach of [Sho92] are both long and involved. While semi-stable flips are rather special, their study shows many of the interesting features of the general case. Moreover, the semi-stable MMP has some very interesting applications. As a consequence of the classification of 3-dimensional flips [KM92] we know that almost all flips are semi-stable, but this may be very hard to prove directly.

Section 1 establishes the general setting of the semi-stable MMP.

Section 2 contains a proof of the semi-stable reduction theorem of [KKMSD73] in dimension 3.

Sections 3 and 4 are devoted to semi-stable flips. First we consider the so-called special semi-stable flips. Then we show that the general case can be reduced to this one. Starting with any semi-stable flipping contraction, an auxiliary construction leads to another semi-stable MMP which involves only special semi-stable flips. This method was first used by [Sho92] in a somewhat different setting. Our approach is based on some ideas of Corti.

Three applications of the semi-stable MMP are considered in section 5. In all three cases the semi-stable MMP provides the solution to a crucial step of the problem. These points are explained in detail.