Flip (algebraic geometry)

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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\rightarrow X_2 \rightarrow \cdots \rightarrow 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 Xi 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 Xi as above, the flip of Xi is a birational map (in fact an isomorphism in codimension 1) f: X_i \rightarrow X_i^+ to a variety whose singularities are 'better' than those of Xi. 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.

[edit] 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
  • Corti, Alessio (December 2004). "What Is...a Flip?" (PDF). Notices of the American Mathematical Society 51 (11): pp.1350–1351.