Comparing Powers of Edge Ideals

TL;DR

This paper investigates the relationships between symbolic and ordinary powers of edge ideals of graphs, providing structural insights and solutions for containment problems, especially for odd cycles, and exploring the symbolic defect sequence.

Contribution

It introduces a method to describe symbolic powers of edge ideals via an ideal addition, and offers solutions to containment problems and partial symbolic defect computations for odd cycle graphs.

Findings

Structural description of symbolic powers as sums of ordinary powers and an ideal J.

Solutions to containment questions for odd cycle edge ideals.

Partial computation of symbolic defect sequence for these ideals.

Abstract

Given a nontrivial homogeneous ideal $I\subseteq k[x_1,x_2,\ldots,x_d]$, a problem of great recent interest has been the comparison of the $r$th ordinary power of $I$ and the $m$th symbolic power $I^{(m)}$. This comparison has been undertaken directly via an exploration of which exponents $m$ and $r$ guarantee the subset containment $I^{(m)}\subseteq I^r$ and asymptotically via a computation of the resurgence $\rho(I)$, a number for which any $m/r > \rho(I)$ guarantees $I^{(m)}\subseteq I^r$. Recently, a third quantity, the symbolic defect, was introduced; as $I^t\subseteq I^{(t)}$, the symbolic defect is the minimal number of generators required to add to $I^t$ in order to get $I^{(t)}$. We consider these various means of comparison when $I$ is the edge ideal of certain graphs by describing an ideal $J$ for which $I^{(t)} = I^t + J$. When $I$ is the edge ideal of an odd cycle, our description of the structure of $I^{(t)}$ yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.