Convex bodies and multiplicities of ideals

DOI10.1134/S0081543814060169zbMath1315.13013arXiv1302.2676MaRDI QIDQ2514596

Kiumars Kaveh, A. G. Khovanskii

Publication date: 3 February 2015

Published in: Proceedings of the Steklov Institute of Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1302.2676

zbMATH Keywords

multiplicitiesBrunn-Minkowski inequalityconvex geometryNewton polyhedraHilbert-Samuel functionSamuel multiplicities

Mathematics Subject Classification ID

Toric varieties, Newton polyhedra, Okounkov bodies (14M25) Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes (13F55) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series (13D40) Valuations and their generalizations for commutative rings (13A18) Deformations and infinitesimal methods in commutative ring theory (13D10)

Related Items (28)

The convex-set algebra and intersection theory on the toric Riemann-Zariski space ⋮ Interpolation of weighted extremal functions ⋮ Okounkov bodies and restricted volumes along very general curves ⋮ Uniform stable radius and Milnor number for non-degenerate isolated complete intersection singularities ⋮ A generalized FKG-inequality for compositions ⋮ Higher Lelong numbers and convex geometry ⋮ Examples of multiplicities and mixed multiplicities of filtrations ⋮ Multiplicity operators ⋮ Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties ⋮ The dual Minkowski problem for unbounded closed convex sets ⋮ Convexity of multiplicities of filtrations on local rings ⋮ Intersection theory of toric 𝑏-divisors in toric varieties ⋮ A Guided Tour to Normalized Volume ⋮ Existence of valuations with smallest normalized volume ⋮ Asymptotic multiplicities ⋮ Stability of valuations: higher rational rank ⋮ Durfee-type bound for some non-degenerate complete intersection singularities ⋮ Local Okounkov bodies and limits in prime characteristic ⋮ The normalized volume of a singularity is lower semicontinuous ⋮ On the theory of coconvex bodies ⋮ Convex bodies and multiplicities of ideals ⋮ The Minkowski equality of filtrations ⋮ Thresholds, valuations, and K-stability ⋮ Canonical decomposition of a difference of convex sets ⋮ Mixed multiplicities of filtrations ⋮ On the multiplicity of isolated roots of sparse polynomial systems ⋮ Mixed multiplicities of divisorial filtrations ⋮ Convex bodies and graded families of monomial ideals

Cites Work

• Fuchsian convex bodies: basics of Brunn-Minkowski theory
• Asymptotic multiplicities of graded families of ideals and linear series
• Singularities of differentiable maps, Volume 2. Monodromy and asymptotics of integrals. Transl. from the Russian by Hugh Porteous and revised by the authors and James Montaldi
• Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory
• Asymptotic multiplicities
• Sur une inégalité à la Minkowski pour les multiplicités
• Polyedres de Newton et nombres de Milnor
• Newton polyhedron, Hilbert polynomial, and sums of finite sets
• On multiplicities of graded sequences of ideals.
• Brunn-Minkowski inequality for multiplicities
• Multiplicities associated to graded families of ideals
• Local volumes of cartier divisors over normal algebraic varieties
• On the theory of coconvex bodies
• Convex bodies and multiplicities of ideals
• Inequalities related to certain couples of local rings
• Convex bodies associated to linear series
• On a Theorem of B. Teissier on Multiplicities of Ideals in Local Rings
• Multiplicities and log canonical threshold
• Completions of local morphisms and valuations.
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