Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory
type: Item
instance of: scholarly article
title: Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory
zbMATH Open document ID: 1270.14022
author: Kiumars Kaveh
author: A. G. Khovanskii
published in: Annals of Mathematics. Second Series
publication date: 2012-10-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/0904.3350
reviewed by: Ivan V. Arzhantsev
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 6093944
zbMATH Keywords: semigroup of integral points
zbMATH Keywords: convex body
zbMATH Keywords: graded algebra
zbMATH Keywords: Hilbert function
zbMATH Keywords: Hodge index theory
zbMATH Keywords: mixed volume
zbMATH Keywords: Alexandrov-Fenchel inequality
zbMATH Keywords: Bernstein-Kushnirenko theorem
zbMATH Keywords: Cartier divisor
zbMATH Keywords: linear series
Wikidata QID: Q56602745
MaRDI profile type: Publication
OpenAlex ID: W2964023259
arXiv ID: 0904.3350
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DOI: 10.4007/ANNALS.2012.176.2.5
review text: The main aim of the paper is to generalize the notion of the Newton polytope of a polynomial to several more abstract contexts. The authors begin with proving general results on semigroups of integral points. Let \(S\) be an additive semigroup in the lattice \(\mathbb{Z}^n\subset\mathbb{R}^n\). Consider the group \(G(S)\) consisting of all the linear combinations \(\sum_i k_ia_i\), where \(a_i\in S\) and \(k_i\in\mathbb{Z}\), and the closed convex cone \(\text{Con}(S)\) that is the closure of the set of all linear combinations \(\sum_i \lambda_ia_i\) for \(a_i\in S\) and \(\lambda_i\geq 0\). Let us define the regularization of \(S\) as the semigroup \(\text{Reg}(S)=G(S)\cap\text{Con}\). Theorem~1.6 states that, for every closed strongly convex cone \(\text{Con}\) inside \(\text{Con}(S)\) that intersects the boundary of \(\text{Con}(S)\) only at the origin, there exists a constant \(N>0\) such that each point in the group \(G(S)\) that lies in \(\text{Con}\) and whose distance from the origin is bigger than \(N\) belongs to \(S\).NEWLINENEWLINEAssume that the cone \(\text{Con}(S)\) is strognly convex and denote by \(L\) its linear span in \(\mathbb{R}^n\). Let \(\dim L=q+1\). Fix a rational \(q\)-dimensional subspace \(M_0\subset L\) intersecting \(\text{Con}(S)\) only at the origin. Let \(M_k\), \(k\in\mathbb{Z}_{\geq 0}\), be the family of \(q\)-dimensional affine subspaces parallel to \(M_0\) such that each \(M_k\) intersects the cone \(\text{Con}(S)\) as well as the group \(G(S)\). The Hilbert function \(H_S(k)\) of a semigroup \(S\) is defined as the number of points in \(S\cap M_k\). Let us define the Newton-Okounkov body of \(S\) as the intersection \(\Delta(S)=\text{Con}(S)\cap M_1\). The authors prove that the function \(H_S(k)\) grows like \(a_qk^q\), where the coefficient \(a_q\) is equal to the (normalized in the appropriate way) \(q\)-dimensional volume of \(\Delta(S)\). Also the growth of the Hilbert function corresponding to the semigroup generated by …
description: scientific article; zbMATH DE number 6093944