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On smooth 4-fold flops. (English) Zbl 0958.14034

A flop is one of the elementary birational transformations between higher dimensional complex varieties. It is a small alteration, affecting the variety only in codimension 2. Conjecturally a series of flops relates the non unique minimal models in a birational class of higher dimensional varieties. In dimension three, flops preserve the analytic structure of the variety, so that a flop of a smooth threefold is again smooth. This is no longer true for fourfolds. In the short note under review the author gives an explicit toric construction to show that every isolated, Gorenstein, terminal cyclic quotient singularity can be obtained as the result of a series of flops of a smooth fourfold.

MSC:

14J35 \(4\)-folds
14E30 Minimal model program (Mori theory, extremal rays)
14E05 Rational and birational maps
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14B05 Singularities in algebraic geometry
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