Slightly mixed symbolic powers of matroids are locally glicci

Abstract

Let $\M$ be a matroid, and let $I_{\M}$ be either the Stanley--Reisner or the cover ideal of $\M$. In this paper we prove that for any matroid $\M$ on $[n]$, any $\ell\in \ZZ_+$, and any squarefree monomial $N\in R=\kk[x_1,\ldots,x_n]$, the ideal $I_{\M}^{(\ell)}:N$, which we call a ``slightly mixed symbolic power" of $I_{\M}$, is always Cohen--Macaulay and locally glicci. As a corollary, we obtain that all symbolic powers $I_{\M}^{(\ell)}$ are locally glicci.