Slightly mixed symbolic powers of matroids are locally glicci

TL;DR

This paper introduces a new class of ideals called slightly mixed symbolic powers of matroid ideals, proving they are Cohen–Macaulay and locally glicci, thus advancing understanding of their algebraic and geometric properties.

Contribution

It establishes that all slightly mixed symbolic powers of matroid ideals are Cohen–Macaulay and locally glicci, a novel result in combinatorial commutative algebra.

Findings

All slightly mixed symbolic powers are Cohen–Macaulay.

These ideals are locally glicci.

All symbolic powers of matroid ideals 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.