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arXiv · 2608.20542

Regularity of Symbolic Powers of Co-Chordal Edge Ideals

Chwas Ahmed·Mohammed Rafiq Namiq
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Abstract

Let $G$ be a finite simple co-chordal graph with at least one edge, and let $I(G)$ be its edge ideal. Over an arbitrary field, we prove that the symbolic powers of $I(G)$ satisfy $\operatorname{reg} I(G)^{(s)}=2s$ for every $s\ge1$. Thus every symbolic power of $I(G)$ has a degree resolution. Using Takayama's formula and clique trees, we reduce the regularity problem to a topological one. Finite convex geometry then shows that nontrivial homology implies a suitable set of leaves, and a weighted counting argument gives the required regularity formula. Finally, we show that symbolic powers of co-chordal edge ideals are not necessarily componentwise linear.

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Chwas Ahmed, Mohammed Rafiq Namiq. 2026-08-20. Regularity of Symbolic Powers of Co-Chordal Edge Ideals. https://arxiv.org/abs/2608.20542

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