Asymptotic Resurgence of Facet and Stanley-Reisner ideals of Matroids

Abstract

Matroid configurations -- introduced by Geramita, Harbourne, Migliore, and Nagel -- are projective varieties which generalize so-called \textit{star configurations} and whose defining ideals are obtained by appropriately specializing the Stanley-Reisner ideal of a matroid. Motivated by this connection, we study the asymptotic resurgence of the Stanley-Reisner ideals of matroids. A result of Villareal shows that it is equivalent to study the asymptotic resurgence of facet ideals. We prove a formula for the asymptotic resurgence of the facet ideal of a matroid in terms of the Waldschmidt constant of facet ideals of the contractions of the matroid. As a consequence, we show that asymptotic resurgence respects the weak order on matroids of the same rank. Therefore, the asymptotic resurgence of the facet ideal of a given matroid is bounded above by the asymptotic resurgence of the facet ideal of a so-called \textit{almost-uniform} matroid of the same rank, which we compute explicitly. Guardo, Harbourne, and Van Tuyl showed that the asymptotic resurgence of an ideal is bounded below by the ratio of the initial degree of the ideal by its Waldschmidt constant. We prove that this lower bound is an equality for facet ideals of many classes of matroids, including matroids of rank $k$ on a ground set of size $n\ge 2k$ whose dual is paving, perfect matroid designs, and sparse paving matroids arising from Steiner systems. For the latter two classes, we explicitly compute the asymptotic resurgence.