The Structure of Symbolic Powers of Matroids

We describe the structure of the symbolic powers I⁽⁾ 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⁽⁾; (b) describe the (non--standard graded) symbolic Rees algebra Rₛ(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 Rₛ(I); (d) provide algebraic applications, including formulas for the symbolic defects of I, the initial degree of I⁽⁾, 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⁽⁾ for some ≥ 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⁽⁾ for some ≥ 3.