We first prove some basic properties of Okounkov bodies and give a characterization of Nakayama and positive volume subvarieties of a pseudoeffective divisor in terms of Okounkov bodies. Next, we show that each valuative and limiting Okounkov bodies of a pseudoeffective divisor which admits the birational good Zariski decomposition is a rational polytope with respect to some admissible flag. This is an extension of the result of Anderson-Küronya-Lozovanu about the rational polyhedrality of Okounkov bodies of big divisors with finitely generated section rings.
Bibliographical noteFunding Information:
Supported by National Research FoundationNRF-2016R1C1B2011446. IBS-R003-D1, Institute for Basic Science in Korea.
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- Asymptotic invariant
- Okounkov body
- Pseudoeffective divisor
- Zariski decomposition