Symbolic powers in weighted oriented graphs

TL;DR

This paper characterizes symbolic powers of edge ideals in weighted oriented graphs, showing their similarities to underlying graphs and providing explicit formulas and bounds for specific graph classes.

Contribution

It offers explicit descriptions of symbolic powers using strong vertex covers and compares their behavior to ordinary powers, including bounds on regularity.

Findings

Explicit description of symbolic powers using strong vertex covers.

Similarity in behavior between symbolic and ordinary powers of ideals.

Bounds on regularity of symbolic powers with equality conditions.

Abstract

Let $D$ be a weighted oriented graph with the underlying graph $G$ when vertices with non-trivial weights are sinks and $I(D), I(G) $ be the edge ideals corresponding to $D$ and $G,$ respectively. We give explicit description of the symbolic powers of $I(D)$ using the concept of strong vertex covers. We show that the ordinary and symbolic powers of $I(D)$ and $I(G)$ behave in a similar way. We provide a description for symbolic powers and Waldschmidt constant of $I(D)$ for certain classes of weighted oriented graphs. When $D$ is a weighted oriented odd cycle we compute $\reg (I(D)^{(s)}/I(D)^s)$ and prove $\reg I(D)^{(s)}\leq\reg I(D)^s$ and show that equality holds when there is only one vertex with non-trivial weight.