Asymptotic resurgences for ideals of positive dimensional subschemes of projective space

TL;DR

This paper extends the concept of resurgence to higher dimensional subschemes in projective space, providing bounds and conjectures for symbolic powers of ideals, especially for unions of lines.

Contribution

It introduces asymptotic resurgence for higher dimensional subschemes and establishes bounds, advancing the understanding beyond zero-dimensional cases.

Findings

Derived upper and lower bounds for asymptotic resurgence

Applied bounds to ideals of unions of general lines in projective space

Proposed a Nagata type conjecture for symbolic powers of lines in P^3

Abstract

Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci-Harbourne defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings, with a focus on zero dimensional subschemes of projective space; the methods and results obtained there have much less to say about higher dimensional subschemes. Here we take the first steps toward extending this work to higher dimensional subschemes. We introduce new asymptotic versions of the resurgence and obtain upper and lower bounds on them for ideals of smooth subschemes, generalizing what is done by Bocci-Harbourne. We apply these bounds to ideals of unions of general lines in ${\bf P}^N$. We also pose a Nagata type conjecture for symbolic powers of ideals of lines in ${\bf P}^3$.