Slightly mixed symbolic powers of matroids are locally glicci

TL;DR

The paper addresses whether symbolic powers of matroid-related monomial ideals are glicci by introducing slightly mixed symbolic powers $J^{(\ell)}:N$ and proving two key advances. First, for $J$ a $C$-matroidal ideal, every $J^{(\ell)}:N$ is Cohen–Macaulay, extending known results on symbolic powers. Second, using a novel lifting framework and two-step Gorenstein linkage, the authors establish that these ideals are locally glicci, thereby confirming a local version of the glicci conjecture for symbolic powers of $C$-matroidal ideals. The results leverage a general lifting technique and a careful inductive analysis of primary decompositions to connect combinatorial matroid structure with algebraic linkage properties, with potential implications for broader classes of monomial ideals and their liaison classes.

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.

Theorems & Definitions (25)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• Definition 3.1