5 - Flips

Summary

One of the landmarks in birational geometry is the attainment of the existence of threefold flips by Mori. We elucidate his approach in detail in this chapter. Passing through the analytic category and the flop of the double cover, we reduce the existence to the general elephant conjecture on an irreducible extremal neighbourhood. The study of an extremal neighbourhood is performed with numerical invariants defined in terms of filtrations of the structure sheaf and the dualising sheaf. Locally at a point, the inverse image of the curve by the index-one cover turns out to be planar. We divide singular points into types according to this structure. Then we classify the set of singular points by deforming the neighbourhood. It is easy to prove that the general elephant is Du Val when it does not contain the exceptional curve. The hard case when it contains the curve requires a really delicate analysis of how the curve is embedded in the threefold. As discussed in the preceding chapter, an extremal neighbourhood is considered to be a one-parameter deformation of a principal prime divisor on it. We describe the associated surface morphism and build a threefold flip from it.