Mathematische Annalen

, Volume 356, Issue 3, pp 1183–1202 | Cite as

Okounkov bodies and toric degenerations



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.



I am grateful to Sara Billey, José Luis González, Sándor Kovács, Rob Lazarsfeld, Ezra Miller, Mircea Mustaţă, and Rekha Thomas for useful comments and fruitful discussions, and especially to Megumi Harada and Kiumars Kaveh for several valuable suggestions. I also thank Bill Fulton and Ben Howard for helping me learn about toric degenerations.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.Institut de Mathématiques de JussieuParisFrance

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