The Structure of Symbolic Powers of Matroids

Abstract

We describe the structure of the symbolic powers $I^{(\ell)}$ of the Stanley-Reisner ideals, and cover ideals, $I$, of matroids. We (a) prove a structure theorem describing a minimal generating set for every $I^{(\ell)}$; (b) describe the (non--standard graded) symbolic Rees algebra $\mathcal{R}_s(I)$ of $I$ and show its minimal algebra generators have degree at most ht $I$; (c) provide an explicit, simple formula to compute the largest degree of a minimal algebra generator of $\mathcal{R}_s(I)$; (d) provide algebraic applications, including formulas for the symbolic defects of $I$, the initial degree of $I^{(\ell)}$, and the Waldschmidt constant of $I$; (e) provide a new algorithm allowing fast computations of very large symbolic powers of $I$. One of the by-products is a new characterization of matroids in terms of minimal generators of $I^{(\ell)}$ for some $\ell\geq 2$. In particular, it yields a new, simple characterization of matroids in terms of the minimal generators of $I^{(2)}$. This is the first characterization of matroids in terms of $I^{(2)}$, and it complements a celebrated theorem by Minh-Trung, Varbaro, and Terai-Trung which requires the investigation of homological properties of $I^{(\ell)}$ for some $\ell\geq 3$.