On the resurgence and asymptotic resurgence of homogeneous ideals

TL;DR

This paper investigates the properties of resurgence and asymptotic resurgence of homogeneous ideals, especially in the context of squarefree monomial ideals and graph cover ideals, providing bounds, computations, and characterizations.

Contribution

It introduces improved bounds for asymptotic resurgence, analyzes how ideal operations affect resurgence, and characterizes bipartite graphs via resurgence measures.

Findings

Sharp bounds for resurgence and asymptotic resurgence of cover ideals.

Explicit computations for specific classes of graphs.

Characterization of bipartite graphs through resurgence values.

Abstract

Let $\mathbb{K}$ be a field and $R = \mathbb{K}[x_1, \ldots, x_n]$. We obtain an improved upper bound for asymptotic resurgence of squarefree monomial ideals in $R$. We study the effect on the resurgence when sum, product and intersection of ideals are taken. We obtain sharp upper and lower bounds for the resurgence and asymptotic resurgence of cover ideals of finite simple graphs in terms of associated combinatorial invariants. We also explicitly compute the resurgence and asymptotic resurgence of cover ideals of several classes of graphs. We characterize a graph being bipartite in terms of the resurgence and asymptotic resurgence of edge and cover ideals. We also compute explicitly the resurgence and asymptotic resurgence of edge ideals of some classes of graphs.