Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory

By Kiumars Kaveh, Askold Khovanskii. 2009

388 citations recorded by the source corpus.

Generalizing the notion of Newton polytope, we define the Newton-Okounkov body, respectively, for semigroups of integral points, graded algebras, and linear series on varieties. We prove that any semigroup in the lattice Z^n is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone. Applying this we obtain several results: we show that for a large class of graded algebras, the Hilbert functions have polynomial growth and their growth coefficients satisfy a Brunn-Minkowski type inequality. We prove analogues of Fujita approximation theorem for semigroups of integral points and graded algebras, which imply a generalization of this theorem for arbitrary linear series. Applications to intersection theory include a far-reaching generalization of the Kushnirenko theorem (from Newton polytope theory) and a new version of the Hodge inequality. We also give elementary proofs of the Alexandrov-Fenchel inequality in convex geometry and its analogue in algebraic geometry.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1916947beb7f95b774d5faf78f3c80a0995f6349

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Citations in the corpus

Crystal bases and Newton–Okounkov bodies (2011, background): “the algebra C[G/U] of a reductive group G, is a special case of the above. 1.2. Newton-Okounkov bodies. In this section we consider valuations on the rings of sections of line bundles. We recall from [KaKh12a] the construction of a Newton-Okounkov body and the related results on Hilbert functions. L”
Filtrations and test-configurations (2011, result): “In this generality, the following result is due to Kaveh and Khovanskii [16] (see also [3]).”
Essential bases and toric degenerations arising from birational sequences (2015, methodology): “nice properties of the multiplicative behavior of the dual canonical basis, there is the approach of Kaveh [K1] and Kiritchenko [Kir2], which has been inspired by the theory of Newton-Okounkov bodies [KK], and there is the approach in [FFL1], which has been inspired by a conjecture of Vinberg concer”
FAVOURABLE MODULES: FILTRATIONS, POLYTOPES, NEWTON–OKOUNKOV BODIES AND FLAT DEGENERATIONS (2013, background): “nt and maps U· [vM] isomorphically onto U·[vM⊙n] = U·[vM ⊗...⊗vM], which ﬁnishes the proof. 6. Newton-Okounkovbodiesand filtrations Our general reference for more details on Newton-Okounkov bodies is [KK]. Let Nbe a nilpotent Lie algebra and let M be a ﬁnite dimensional cyclic nilpotent module with ”
Toric geometry and the Semple–Nash modification (2009, neutral): “9 [20] there is δ0 ∈ ∩ int(σ̌ ) such that δ0+ σ̌ ∩M ⊂ .”
ON THE GEOMETRY OF SPHERICAL VARIETIES (2012, background): “singularities under deformations (see [53]). 4.3.3. Okounkov bodies and degree of line bundles. After Okounkov [115] and Lazarsfeld and Mustaţă [98], Kaveh and Khovanskii study, in a series of papers [76],[77], the relationshipbetween the geometryofvarieties(in particularthe growth of sections of li”
Convex bodies and multiplicities of ideals (2013, methodology): “y for mixed multiplicities of ideals using arguments similar to those of this paper. This proof then implies an Alexandrov-Fenchel inequality for covolumes of convex regions (in a similar way that in [Kaveh-Khovanskii09] and in [Khovanskii88] the authors obtain a proof of the usual Alexandrov-Fenche”
Integrable systems, toric degenerations and Okounkov bodies (2012, methodology): “The tool we use for this purpose—as initially proposed by Okounkov in [42,44] and further developed in [26,28,34]—is a valuation on the field C(X) of rational functions on the variety X .”
Vanishing sequences and Okounkov bodies (2013, background): “We refer to [2,15,19] for more information on this construction.”