Resurgence number and convex body associated to pairs of graded families of ideals

TL;DR

This paper explores how the asymptotic resurgence number of pairs of graded ideals can be understood through associated convex bodies, linking algebraic properties with combinatorial and geometric data.

Contribution

It introduces a method to analyze the asymptotic resurgence number using convex bodies, especially for monomial and invariant ideals, connecting algebraic and geometric perspectives.

Findings

Convex bodies encode asymptotic resurgence data.

Newton-Okounkov bodies relate to monomial ideals.

Rees packages help analyze invariant ideals.

Abstract

We discuss how to understand the asymptotic resurgence number of a pair of graded families of ideals from combinatorial data of their associated convex bodies. When the families consist of monomial ideals, the convex bodies being considered are the Newton-Okounkov bodies of the families. When ideals in the second family are classical invariant ideals, for instance, determinantal ideals or ideals of Pfaffians, these convex bodies are constructed from the associated Rees packages.