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Concept
Flip (mathematics)
Summary
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.
Minimal model program
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 (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 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 (conjectural) solution to this problem is the flip. Given a problematic as above, the flip of is a birational map (in fact an isomorphism in codimension 1) 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.
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.
Official source
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