Flip (algebraic geometry)

In mathematics, specifically in algebraic geometry, a flip is a certain kind of codimension-2 surgery operation arising naturally in the attempt to construct a minimal model of an algebraic variety.

The minimal model program can be summarised very briefly as follows: given a variety $X$, we construct a sequence of contractions $X\; =\; X\_1\; ightarrow\; X\_2\; ightarrow\; cdots\; ightarrow\; X\_n$, each of which contracts some curves on which the canonical divisor $K\_\{X\_i\}$ is negative. Eventually, $K\_\{X\_n\}$ 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 $X\_i$ may become 'too singular', in the sense that the canonical divisor $K\_\{X\_i\}$ is no longer Cartier, so the intersection number $K\_\{X\_i\}\; cdot\; C$ with a curve $C$ is not even defined. The (conjectural) solution to this problem is the "flip". Given a problematic $X\_i$ as above, the flip of $X\_i$ is a birational map (in fact an isomorphism in codimension 1) $f:\; X\_i\; ightarrow\; X\_i^+$ to a variety whose singularities are 'better' than those of $X\_i$. So we can put $X\_\{i+1\}\; =\; X\_i^+$, and continue the process.

The question of existence of flips (for varieties whose singularities are not too severe) appears to have been settled by the results of Birkar-Cascini-Hacon-McKernan. 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.

References

* Birkar, C., Cascini, P., Hacon, C., McKernan, J., 'Existence of minimal models for varieties of log general type'.
* Kollar, J., 'Flips, flops, minimal models, etc.', Surv. In Diff. Geom. 1 (1991), 113-199.
* Kollár, J. and Mori, S., "Birational Geometry of Algebraic Varieties", Cambridge University Press, 1998. ISBN 0-521-63277-3
* cite journal
last = Corti
first = Alessio
title = What Is...a Flip?
journal = Notices of the American Mathematical Society
year = 2004
month = December
volume = 51
issue = 11
pages = pp.1350–1351
url = http://www.ams.org/notices/200411/what-is.pdf
format = PDF
accessdate = 2008-01-17