Resurgence number of graded families of ideals

TL;DR

This paper introduces and studies the resurgence and asymptotic resurgence numbers for pairs of graded families of ideals in Noetherian rings, extending classical notions and exploring their properties and computations.

Contribution

It generalizes the concepts of resurgence and asymptotic resurgence to pairs of graded families, analyzing their finiteness, rationality, and behavior under integral closure.

Findings

Resurgence and asymptotic resurgence can be finite and rational under certain conditions.

These invariants can be computed via Rees valuations or as limits of sequences.

Properties of resurgence extend to pairs of families depending on Noetherian and finite generation conditions.

Abstract

We define the resurgence and asymptotic resurgence numbers associated to a pair of graded families of ideals in a Noetherian ring. These notions generalize the well-studied resurgence and asymptotic resurgence of an ideal in a polynomial ring. We examine when these invariant are finite and rational. We investigate situations where these invariant can be computed via Rees valuations or realized as actual limits of well-defined sequences. We study how the asymptotic resurgence changes when a family is replaced by its integral closure. Many examples are given to illustrate that whether or not known properties of resurgence and asymptotic resurgence of an ideal would extend to that of a pair of graded families of ideals generally depends on the Noetherian property and finite generation of the Rees algebras of these families.