Regularity of symbolic powers of square-free monomial ideals

Abstract

We study the regularity of symbolic powers of square-free monomial ideals. We prove that if $I=I_{\Delta }$ is the Stanley-Reisner ideal of a simplicial complex $\Delta$, then $\reg(I^{(n)}) \leqslant δ(n-1) +b$ for all $n⩾1$, where $δ= \lim\limits_{n\to\infty} \reg(I^{(n)})/n$, and $b = \max\{\reg(I_Γ) \mid Γ\text{ is a subcomplex of } Δ\text{ with } \F(Γ) \subseteq \F(Δ)\}$. This bound is sharp for any $n$. When $I=I(G)$ is the edge ideal of a simple graph $G$, we obtain a general linear upper bound $\reg(I^{(n)}) \leqslant 2n + \ordmatch(G)-1$, where $\ordmatch(G)$ is the ordered matching number of $G$.