The Waldschmidt constant for squarefree monomial ideals

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–Hà–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 compute the Waldschmidt constant for the Stanley–Reisner ideal of a uniform matroid.