Convex Bodies Associated to Linear Series

By R. Lazarsfeld, M. Mustaţă. 2008

544 citations recorded by the source corpus.

In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens the door to a number of extensions. The purpose of this paper is to initiate a systematic development of the theory, and to give a number of applications and examples.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ec35b701b22403b35c02b108270be4635dbf8e78

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In Geometry and Topology

Citations in the corpus

Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory (2009, methodology): “at beside following the ideas in [Lazarsfeld-Mustata08], we use results which apply to arbitrary semigroups of integral points. Another diﬀerence between the approach in the present paper and that of [Lazarsfeld-Mustata08] is that we use abstract valuations on algebras, as opposed to a valuation on ”
Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs (2015, background): “The following result is proved using Okounkov bodies [49, 55] (see also the first author’s appendix in [78]).”
Vanishing sequences and Okounkov bodies (2013, methodology): “When v is divisorial, we can even provide an explicit formula for the density of μL ,v in terms of restricted volumes [10], using the differentiability property of the volume function proved in [4,19].”
On K‐stability and the volume functions of ℚ‐Fano varieties (2015, background): “Let Δ(−KX) ⊂ R be the Okounkov body of −KX with respect to Z• in the sense of [34].”
Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs (2010, background): “First we recall the necessary facts on Okounkov bodies from [22].”
Crystal bases and Newton–Okounkov bodies (2011, result): “s of GL(n,C) ([G-C50]). On the other hand, following the pioneering works of A. Okounkov in [Ok96] and [Ok03], the author and A. G. Khovanskii ([K-Kh09], [K-Kh08]), as well as Lazarsfeld and Mustata ([L-M08]), developed a theory of convex bodies associated to linear series on algebraic varieties, an”
Filtrations and test-configurations (2011, background): “Remark 19 The special case of Theorem 16 where S contains an ample series, which is what is being used in the previous corollary, was first established in [18].”
K-STABILITY OF FANO MANIFOLDS WITH NOT SMALL ALPHA INVARIANTS (2016, unknown): “Consider the Okounkov body 1Z•(σ (−K X )− εF) ⊂ R>0 of σ (−K X )− εF with respect to Z• in the sense of [32].”