Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semigroups, Cartier divisors and convex bodies

Askold Khovanskii

TL;DR

The paper builds a unifying framework connecting convex geometry and algebraic geometry through Newton–Okounkov bodies, showing that mixed volumes of convex bodies correspond to birationally invariant intersection indices of Cartier divisors and Shokurov b-divisors. It develops a robust semigroup/polynomial language to transfer inequalities like Brunn–Minkowski and Alexandrov–Fenchel from convex geometry to algebraic geometry, via NO bodies and the Deg self-intersection function. By leveraging BKK/NO theory, it derives explicit formulas for intersection indices, extends them to nef and psef elements in the Grothendieck group, and proves algebraic proofs of classical geometric inequalities, including Minkowski-type criteria for divisors. The work thus provides a comprehensive, proof-light survey that standardizes the interplay between convex bodies and intersection theory, with implications for birational invariants and divisor theory over arbitrary algebraically closed fields.

Abstract

The theory of Newton--Okounkov bodies provides direct relations and points out analogies between the theory of mixed volumes of convex bodies, on the one hand, and the intersection theories of Cartier divisors and of Shokurov $b$-divisors, on the other hand. The classical inequalities between mixed volumes of convex bodies correspond to inequalities between intersection indices of nef Cartier divisors on an irreducible projective variety and between the birationally invariant intersection indices of nef type Shokurov $b$-divisors on an irreducible algebraic variety. Such algebraic inequalities are known as Khovanskii--Teissier inequalities. Our proof of these inequalities is based on simple geometric inequalities on two dimensional convex bodies and on pure algebraic arguments. The classical geometric inequalities follow from the algebraic inequalities. We collected results from a few papers which were published during the last forty five years. Some theorems of the present paper were never stated, but all ideas needed for their proofs are contained in the published papers. So we avoid all heavy proofs. Our goal is to present an overview of the area. Notions of homogeneous polynomials on commutative semigroups and of their polarizations provide an adequate language for discussing the subject. We use this language through the paper.

Semigroups, Cartier divisors and convex bodies

TL;DR

The paper builds a unifying framework connecting convex geometry and algebraic geometry through Newton–Okounkov bodies, showing that mixed volumes of convex bodies correspond to birationally invariant intersection indices of Cartier divisors and Shokurov b-divisors. It develops a robust semigroup/polynomial language to transfer inequalities like Brunn–Minkowski and Alexandrov–Fenchel from convex geometry to algebraic geometry, via NO bodies and the Deg self-intersection function. By leveraging BKK/NO theory, it derives explicit formulas for intersection indices, extends them to nef and psef elements in the Grothendieck group, and proves algebraic proofs of classical geometric inequalities, including Minkowski-type criteria for divisors. The work thus provides a comprehensive, proof-light survey that standardizes the interplay between convex bodies and intersection theory, with implications for birational invariants and divisor theory over arbitrary algebraically closed fields.

Abstract

The theory of Newton--Okounkov bodies provides direct relations and points out analogies between the theory of mixed volumes of convex bodies, on the one hand, and the intersection theories of Cartier divisors and of Shokurov -divisors, on the other hand. The classical inequalities between mixed volumes of convex bodies correspond to inequalities between intersection indices of nef Cartier divisors on an irreducible projective variety and between the birationally invariant intersection indices of nef type Shokurov -divisors on an irreducible algebraic variety. Such algebraic inequalities are known as Khovanskii--Teissier inequalities. Our proof of these inequalities is based on simple geometric inequalities on two dimensional convex bodies and on pure algebraic arguments. The classical geometric inequalities follow from the algebraic inequalities. We collected results from a few papers which were published during the last forty five years. Some theorems of the present paper were never stated, but all ideas needed for their proofs are contained in the published papers. So we avoid all heavy proofs. Our goal is to present an overview of the area. Notions of homogeneous polynomials on commutative semigroups and of their polarizations provide an adequate language for discussing the subject. We use this language through the paper.

Paper Structure

This paper contains 29 sections, 66 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Polynomial functions on commutative semigroups
  3. Commutative semigroups and related objects
  4. Definition of polynomial function on a semigroup
  5. Polarization of homogeneous polynomials
  6. Extension of polynomials to the Grothendieck group
  7. Semigroups of subspaces, divisors, and Shokurov $b$-divisors
  8. Semigroup $K(X)$
  9. Grothendieck semigroup $G_s(X)$ and Grothendieck group $G(X)$
  10. Semigroups $K_\mathrm{reg}(X)$, $K_\mathrm{emb}(X)$, and divisors
  11. The Grothendieck group $G(X)$ and the group of Shokurov $b$-divisors on $X$
  12. Semigroups related to Newton polyhedra theory
  13. The Semigroup $\operatorname{Set}_n$
  14. The semigroup $K_n$
  15. Birationally invariant an intersection index
  16. ...and 14 more sections

Key Result

Lemma 2.4

For any homomorphism $\tau:S\to G$, where $G$ is a commutative group, there are unique homomorphisms $\tau_1:G_s(S)\to G$ and $\tau_2:G(S)\to G$ such that $\tau =\tau_1\circ \rho$ and $\tau=\tau_2\circ\pi$, where $\rho:S\to G_s(S)$ and $\pi: S\to G(S)$ are the maps defined above. If $G$ is an additi

Theorems & Definitions (121)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Definition 2.9
  • Lemma 2.10
  • ...and 111 more