The Waldschmidt constant for squarefree monomial ideals
Academic Article

Overview

Research

Identity

Additional Document Info

View All

Overview

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.