Semigroups, Cartier divisors and convex bodies

TL;DR

This paper explores the deep connections between convex geometry and algebraic geometry, specifically relating mixed volumes of convex bodies to intersection theory of Cartier divisors and b-divisors, establishing inequalities and providing an overview of the area.

Contribution

It presents a unified geometric and algebraic framework for understanding inequalities between mixed volumes and intersection indices, including new insights into Khovanskii--Teissier inequalities.

Findings

Classical geometric inequalities follow from algebraic inequalities.

Collected and summarized results from 45 years of research.

Presented new theorems with proofs based on geometric and algebraic arguments.

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.