, Volume 356, Issue 3, pp 1183-1202
Date: 30 Nov 2012

Okounkov bodies and toric degenerations

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Let \(\varDelta \) be the Okounkov body of a divisor \(D\) on a projective variety \(X\) . We describe a geometric criterion for \(\varDelta \) to be a lattice polytope, and show that in this situation \(X\) admits a flat degeneration to the corresponding toric variety. This degeneration is functorial in an appropriate sense.

The author was partially supported by NSF Grants DMS-0502170 and DMS-0902967, and also by the Clay Mathematics Institute as a Liftoff Fellow.