Regularity of symbolic powers of square-free monomial ideals

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 $\operatorname{reg}(I^{(n)}) \leqslant \delta (n-1)+b$ for all $n \geqslant 1$, where $\delta = \operatorname{lim}_{n \to \infty} \operatorname{reg}(I^{(n)}) / n$, $b=\max \lbrace \operatorname{reg}(I_\Gamma ) \vert \Gamma$ is a subcomplex of $\Delta$ with $\mathcal{F}(\Gamma) \subseteq F(\Delta ) \rbrace$, and $F(\Gamma)$ and $F(\Delta)$ are the set of facets of $\Gamma$ and $\Delta$, respectively. 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 $\operatorname{reg}(I^{(n)}) \leqslant 2n+ \operatorname{ord-match} (G) - 1$, where $\operatorname{ord-match} (G)$ is the ordered matching number of $G$.

Keywords

Castelnuovo–Mumford regularity, symbolic power, edge ideal, matching

2010 Mathematics Subject Classification

05C90, 05E40, 05E45, 13D45

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The authors are supported by Project ICRTM.02_2021.02 of the International Centre for Research and Postgraduate Training in Mathematics (ICRTM), Institute of Mathematics, VAST.

Received 27 January 2022

Received revised 3 June 2022

Accepted 15 August 2022

Published 26 April 2023