On resurgence via asymptotic resurgence

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. (C) 2021 Elsevier Inc. All rights reserved.