A duality theorem for the ic-resurgence of edge ideals

TL;DR

This paper establishes a duality formula linking the ic-resurgence of edge ideals of a clutter and its blocker, using linear programming and polyhedral geometry, with applications to matroids and graphs.

Contribution

It proves a duality theorem for the ic-resurgence of edge ideals and applies it to specific classes like matroids and non-bipartite graphs.

Findings

The ic-resurgence of an edge ideal and its blocker coincide.

Derived formulas for resurgence in uniform matroids.

Established a formula for the Waldschmidt constant in certain graphs.

Abstract

The aim of this work is to use linear programming and polyhedral geometry to prove a duality formula for the ic-resurgence of edge ideals. We show that the ic-resurgence of the edge ideal $I$ of a clutter $\mathcal{C}$ and the ic-resurgence of the edge ideal $I^\vee$ of the blocker $\mathcal{C}^\vee$ of $\mathcal{C}$ coincide. If $\mathcal{C}$ is the clutter of bases of certain uniform matroids, we recover a formula for the resurgence of $I$, and if $\mathcal{C}$ is a connected non-bipartite graph with a perfect matching, we show a formula for the Waldschmidt constant of $I^\vee$.