6 - Three-dimensional Flops

Summary

In this chapter we begin the detailed study of 3–dimensional birational geometry. Compared with algebraic surfaces, the main new feature of 3–fold geometry is the appearance of flips and flops. This chapter is devoted to the study of flops.

Section 1 is a general introduction to flips and flops. These results hold in all dimensions.

Flops of 3–dimensional varieties with terminal singularities are discussed in section 2. The existence of flops is proved using the classification of index 1 terminal 3–fold singularities. This is one of the main reasons why the existence of higher dimensional flops is unknown. We also consider sequences of flops. For the applications it is important to establish that our procedures to improve a variety do not result in an infinite sequence of flips or flops.

Section 3 is a continuation of our earlier studies of 3–fold canonical singularities. The main theorems (6.23) and (6.25) show how to simplify canonical singularities by explicit blow ups until we reach a variety with only ℚ-factorial terminal singularities. The proof uses the classification of index 1 canonical singularities established in section 5.3. The existence of these ℚ-factorial terminalizations follows from the 3–dimensional MMP, but we will use it to establish parts of the MMP.

In section 4 we use ℚ-factorial terminalizations to construct flops of varieties with canonical singularities. The method, called ‘crepant descent’, reduces the existence of canonical flops to terminal flops. This method works in many different situations and also in higher dimensions.

Flips and Flops

In this section we discuss general results concerning flips and flops. We mainly consider uniqueness and the connection of flips with the finite generation of certain algebras.