Convex bodies and algebraic equations on affine varieties

Abstract

Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This is a far reaching generalization of usual theory of Newton polytopes (which is concerned with toric varieties). As applications we give new, simple and transparent proofs of some well-known theorems in both algebraic geometry (e.g. Hodge Index Theorem) and convex geometry (e.g. Alexandrov-Fenchel inequality). Our main tools are classical Hilbert theory on degree of subvarieties of a projective space (in algebraic geometry) and Brunn-Minkowski inequality (in convex geometric).

Preliminary version, may contain several typos, 44 pages

Cited by in corpus (21)

• Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians
• Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory
• Convex Bodies Associated to Linear Series
• A comparison of Newton-Okounkov polytopes of Schubert varieties
• Newton-Okounkov bodies of Bott-Samelson varieties and Grossberg-Karshon twisted cubes
• Newton-Okounkov bodies for Bott-Samelson varieties and string polytopes for generalized Demazure modules
• Integrable systems, toric degenerations and Okounkov bodies
• Okounkov bodies and toric degenerations
• Newton-Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases
• Newton-Okounkov bodies of flag varieties and combinatorial mutations
• Mixed volume and an extension of intersection theory of divisors
• Polyhedral realizations of crystal bases and convex-geometric Demazure operators
• Intersection theory of toric $b$-divisors in toric varieties
• Convex bodies appearing as Okounkov bodies of divisors
• A remark on asymptotic enumeration of highest weights in tensor powers of a representation
• On projective completions of affine varieties determined by 'degree-like' functions
• Newton-Okounkov polytopes of flag varieties and marked chain-order polytopes
• Folding procedure for Newton-Okounkov polytopes of Schubert varieties
• Iterative dissection of Okounkov bodies of graded linear series on ${\mathbb{CP}}^2$
• Okounkov bodies and restricted volumes along very general curves
• Semi-toric degenerations of Richardson varieties arising from cluster structures on flag varieties