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A duality theorem for the ic-resurgence of edge ideals

Rafael H. Villarreal

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 $I^\vee$.

A duality theorem for the ic-resurgence of edge ideals

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 of a clutter and the ic-resurgence of the edge ideal of the blocker of coincide. If is the clutter of bases of certain uniform matroids, we recover a formula for the resurgence of , and if is a connected non-bipartite graph with a perfect matching, we show a formula for the Waldschmidt constant of .
Paper Structure (5 sections, 18 theorems, 85 equations, 1 figure)

This paper contains 5 sections, 18 theorems, 85 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Duality formula for the ic-resurgence
  4. Examples
  5. Procedures

Key Result

Theorem 1.1

intclos For each $1\leq j\leq p$, let $\rho_j$ be the optimal value of the following linear program with variables $y_1,\ldots,y_{s+3}$. Then, $\rho_{ic}(I)=\max\{\rho_j\}_{j=1}^p$ and $\rho_j$ is attained at a rational vertex of the polyhedron $\mathcal{P}_j$ of feasible points of Eq. lp-ic-resurge

Figures (1)

  • Figure 1: Graph $G$ is the complement of a cycle of length $7$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • ...and 28 more