Annals of Mathematics· 2012Q1
Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory
- 350citations
- Q1SCImago
- 2012year
Short summary
This paper introduces Newton-Okounkov bodies, generalizing Newton polytopes, for integral point semigroups, graded algebras, and linear series on varieties.
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Key points
- Introduces Newton-Okounkov bodies, generalizing Newton polytopes.
- Proves asymptotic approximation of semigroups by sublattices in convex cones.
- Shows polynomial growth for Hilbert functions and Brunn-Minkowski inequalities for growth coefficients in graded algebras.
- Generalizes Fujita approximation theorem for semigroups and graded algebras.
- Provides generalizations of Kushnirenko theorem and Hodge inequality for intersection theory.
AI-generated from the title and abstract; the full text is not read.
Abstract
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 the 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.
The authors' abstract, as published at the source. Annals of Mathematics, 2012 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics