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 can be summarised very briefly as follows: given a variety ${\displaystyle X}$, we construct a sequence of contractions ${\displaystyle X=X_{1}\rightarrow X_{2}\rightarrow \cdots \rightarrow X_{n}}$, each of which contracts some curves on which the canonical divisor ${\displaystyle K_{X_{i}}}$ is negative. Eventually, ${\displaystyle K_{X_{n}}}$ 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 ${\displaystyle X_{i}}$ may become 'too singular', in the sense that the canonical divisor ${\displaystyle K_{X_{i}}}$ is no longer Cartier, so the intersection number ${\displaystyle K_{X_{i}}\cdot C}$ with a curve ${\displaystyle C}$ is not even defined.

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