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Okounkov bodies and toric degenerations

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Abstract

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.

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Acknowledgments

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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  1. Dave Anderson

    Present address: Institut de Mathématiques de Jussieu, 75013 , Paris, France

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  1. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Dave Anderson

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Correspondence to Dave Anderson.

Additional information

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

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Anderson, D. Okounkov bodies and toric degenerations. Math. Ann. 356, 1183–1202 (2013). https://doi.org/10.1007/s00208-012-0880-3

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  • DOI: https://doi.org/10.1007/s00208-012-0880-3

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