Convex bodies and multiplicities of ideals

TL;DR

This paper introduces convex regions associated with graded sequences of ideals in local algebras, encoding multiplicity information and generalizing classical inequalities like Brunn-Minkowski.

Contribution

It develops a new geometric framework linking convex bodies to ideal multiplicities, extending existing theories and inequalities in algebraic geometry.

Findings

Convex regions encode Samuel multiplicities.

A new proof and generalization of Brunn-Minkowski inequality.

Application to a broad class of local algebras.

Abstract

We associate convex regions in R^n to m-primary graded sequences of subspaces, in particular m-primary graded sequences of ideals, in a large class of local algebras (including analytically irreducible local domains). These convex regions encode information about Samuel multiplicities. This is in the spirit of the theory of Grobner bases and Newton polyhedra on one hand, and the theory of Newton-Okounkov bodies for linear systems on the other hand. We use this to give a new proof, as well as a generalization of a Brunn-Minkowski inequality for multiplicities due to Teissier and Rees-Sharp.