On resurgence via asymptotic resurgence

TL;DR

This paper explores the relationship between resurgence and asymptotic resurgence of ideals, establishing conditions for rationality and bounds, and extending criteria for when resurgence is less than the ideal's big height.

Contribution

It demonstrates that ideals with Noetherian symbolic Rees algebras have rational resurgence and provides bounds on asymptotic resurgence based on known containments.

Findings

Resurgence is rational if the symbolic Rees algebra is Noetherian.

Derived bounds on asymptotic resurgence from a single containment.

Extended criteria for resurgence being less than the big height.

Abstract

The resurgence and asymptotic resurgence of an ideal in a polynomial ring are two statistics which measure the relationship between its regular and symbolic powers. We address two aspects of resurgence which can be studied via asymptotic resurgence. First, we show that if an ideal has Noetherian symbolic Rees algebra then its resurgence is rational. Second, we derive two bounds on asymptotic resurgence given a single known containment between a symbolic and regular power. From these bounds we recover and extend criteria for the resurgence of an ideal to be strictly less than its big height recently derived by Grifo, Huneke, and Mukundan. We achieve the reduction to asymptotic resurgence by showing that if the asymptotic resurgence and resurgence are different, then resurgence is a maximum instead of a supremum.