Flip (mathematics)

In algebraic geometry, flips and flops are codimension-2 surgery operations arising in the minimal model program, given by blowing up along a relative canonical ring. In dimension 3 flips are used to construct minimal models, and any two birationally equivalent minimal models are connected by a sequence of flops. It is conjectured that the same is true in higher dimensions.
The minimal model program.
The minimal model program can be summarised very briefly as follows: given a variety formula_1, we construct a sequence of contractions formula_2, each of which contracts some curves on which the canonical divisor formula_3 is negative. Eventually, formula_4 should become nef (at least in the case of nonnegative Kodaira dimension), which is the desired result. The major technical problem is that, at some stage, the variety formula_5 may become 'too singular', in the sense that the canonical divisor formula_3 is no longer a Cartier divisor, so the intersection number formula_7 with a curve formula_8 is not even defined.
The (conjectural) solution to this problem is the "flip". Given a problematic formula_5 as above, the flip of formula_5 is a birational map (in fact an isomorphism in codimension 1) formula_11 to a variety whose singularities are 'better' than those of formula_5. So we can put formula_13, 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 formula_14 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 formula_16 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 (as an algebra over formula_16 ) then the morphism formula_19 is called the flip of formula_20 if formula_21 is relatively ample, and the flop of formula_20 if "K" is relatively trivial. (Sometimes the induced birational morphism from formula_1 to formula_24 is called a flip or flop.)
In applications, formula_20 is often a small contraction of an extremal ray, which implies several extra properties:
Examples.
The first example of a flop, known as the Atiyah flop, was found in .
Let "Y" be the zeros of formula_34 in formula_35, and let "V" be the blowup of "Y" at the origin. 
The exceptional locus of this blowup is isomorphic to formula_36, and can be blown down to formula_37 in two different ways, giving varieties formula_38 and formula_39. The natural birational map from formula_38 to formula_39 is the Atiyah flop.
 introduced Reid's pagoda, a generalization of Atiyah's flop replacing "Y" by the zeros of formula_42.