The minimal model program can be summarised very briefly as follows: given a variety, we construct a sequence of contractions, each of which contracts some curves on which the canonical divisor is negative. Eventually, should become nef, which is the desired result. The major technical problem is that, at some stage, the variety may become 'too singular', in the sense that the canonical divisor is no longer a Cartier divisor, so the intersection number with a curve is not even defined. The solution to this problem is the flip. Given a problematic as above, the flip of is a birational map to a variety whose singularities are 'better' than those of. So we can put, and continue the process. Two major problems concerning flips are to show that they exist and to show that one cannot have an infinite sequence of flips. If both of these problems can be solved, then the minimal model program can be carried out. The existence of flips for 3-folds was proved by. The existence of log flips, a more general kind of flip, in dimension three and four were proved by whose work was fundamental to the solution of the existence of log flips and other problems in higher dimension. The existence of log flips in higher dimensions has been settled by. On the other hand, the problem of termination—proving that there can be no infinite sequence of flips—is still open in dimensions greater than 3.
Definition
If is a morphism, and K is the canonical bundle of X, then the relative canonical ring of f is and is a sheaf of graded algebras over the sheaf of regular functions on Y. The blowup of Y along the relative canonical ring is a morphism to Y. If the relative canonical ring is finitely generated then the morphism is called the flip of if is relatively ample, and the flop of if K is relatively trivial. In applications, is often a small contraction of an extremal ray, which implies several extra properties:
The exceptional sets of both maps and have codimension at least 2,
and are birational morphisms onto Y, which is normal and projective.
All curves in the fibers of and are numerically proportional.
Examples
The first example of a flop, known as the Atiyah flop, was found in. Let Y be the zeros of in, and letV be the blowup of Y at the origin. The exceptional locus of this blowup is isomorphic to, and can be blown down to in two different ways, giving varieties and. The natural birational map from to is the Atiyah flop. introduced Reid's pagoda, a generalization of Atiyah's flop replacingY by the zeros of.
