Flips in low codimension : classification and quantitative theory
A flip is a birational map of 3-folds X- ---> X+ which is an isomorphism away from curves C- c X- and C+ c X+ and does not extend across these curves. Flips are the primary object of study of this thesis. I discuss their formal definition and history in Chapter 1. Flips are well known in toric geometry. In Chapter 2, I calculate how the numbers K3 and χ(nK) differ between X- and X+ for toric flips. These numbers are also related in a primary way by Riemann-Roch theorems but I keep that quiet until Chapter 5. In Chapter 3, I describe a technique, which I learned from Miles Reid, for constructing a flip as C* quotients of a local variety 0 E A, taken in different ways. The codimension of my title refers to the minimal embedding dimension of 0 E A. The case of codimension 0 turns out to be exactly the case of toric geometry as studied in Chapter 2. The main result of Chapter 3 classifies the cases when A c C5 is a singular hypersurface, that is, when A defines a flip in codimension 1. Chapters 4 and 5 concern themselves with computing new examples of flips in higher codimension and studying changes in general flips. I indicate one benefit of knowing how these changes work. The main results of Chapters 2 and 3 have been circulated informally as  and  respectively.