Convex bodies and asymptotic invariants for powers of monomial ideals

TL;DR

This paper develops a framework connecting convex bodies to monomial ideals through asymptotic Newton polyhedra, enabling new bounds on invariants like the Waldschmidt constant via linear optimization.

Contribution

It introduces a method to construct asymptotic Newton polyhedra from monomial ideal decompositions, generalizing powers and providing computable bounds on invariants.

Findings

Established a lower bound on the Waldschmidt constant using convex bodies.

Defined the naive Waldschmidt constant as a new invariant.

Linked asymptotic invariants to solutions of linear optimization problems.

Abstract

Continuing a well established tradition of associating convex bodies to monomial ideals, we initiate a program to construct asymptotic Newton polyhedra from decompositions of monomial ideals. This is achieved by forming a graded family of ideals based on a given decomposition. We term these graded families powers since they generalize the notions of ordinary and symbolic powers. Asymptotic invariants for these graded families are expressed as solutions to linear optimization problems on the respective convex bodies. This allows to establish a lower bound on the Waldschmidt constant of a monomial ideal by means of a more easily computable invariant, which we introduce under the name of naive Waldschmidt constant.