The Waldschmidt constant for squarefree monomial ideals

TL;DR

This paper links the Waldschmidt constant of squarefree monomial ideals to linear programming and fractional graph theory, providing new bounds and explicit calculations for specific classes of ideals.

Contribution

It expresses the Waldschmidt constant as a linear program and relates it to fractional chromatic numbers, proving a conjecture and enabling explicit computations.

Findings

Derived a linear programming formulation for the Waldschmidt constant.

Established a Chudnovsky-like lower bound for the constant.

Computed the Waldschmidt constant for specific classes of ideals, such as unions of linear subspaces and matroids.

Abstract

Given a squarefree monomial ideal $I \subseteq R =k[x_1,\ldots,x_n]$, we show that $\widehat\alpha(I)$, the Waldschmidt constant of $I$, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of $I$. By applying results from fractional graph theory, we can then express $\widehat\alpha(I)$ in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of $I$. Moreover, expressing $\widehat\alpha(I)$ as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on $\widehat\alpha(I)$, thus verifying a conjecture of Cooper-Embree-H\`a-Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of $\mathbb{P}^n$ with few components compared to $n$, and we find the Waldschmidt constant for the Stanley-Reisner ideal of a uniform matroid.