Ordinary and symbolic powers of edge ideals of weighted oriented graphs

TL;DR

This paper investigates the conditions under which the symbolic and ordinary powers of edge ideals of weighted oriented graphs, especially certain trees and lines, are equal, providing new insights into their algebraic properties.

Contribution

It establishes that for a specific class of weighted oriented trees, all symbolic and ordinary powers of their edge ideals coincide, and it characterizes when this occurs for oriented lines.

Findings

Symbolic and ordinary powers coincide for certain weighted oriented trees.

Necessary and sufficient conditions are provided for equality of powers in oriented lines.

The results deepen understanding of algebraic properties of edge ideals in weighted oriented graphs.

Abstract

Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we show that all the symbolic and ordinary powers of $I(\mathcal{D})$ coincide when $\mathcal{D}$ is a weighted oriented certain class of tree. Finally, we give necessary and sufficient conditions for the equality of ordinary and symbolic powers of naturally oriented lines.