Mathematics, Statistics, and Computer Science
Waldschmidt constant, monomial ideals, symbolic powers, graphs, hypergraphs, fractional chromatic number, linear programming, resurgence
Given a squarefree monomial ideal I ⊆ R = k[x1, . . . , xn], we show that α(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 α(I) in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of I. Moreover, expressing α(I) as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on α(I), thus verifying a conjecture of Cooper-Embree- Ha`-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 Pn with few components compared to n, and we find the Waldschmidt constant for the Stanley-Reisner ideal of a uniform matroid.
Source Publication Title
Journal of Algebraic Combinatorics
Bocci, C., Cooper, S., Guardo, E., Harbourne, B., Janssen, M., Nagel, U., Seceleanu, A., Van Tuyl, A., & Vu, T. (2016). Waldschmidt Constant for Squarefree Monomial Ideals. Journal of Algebraic Combinatorics, 1. https://doi.org/10.1007/s10801-016-0693-7