Bounding symbolic powers via asymptotic multiplier ideals

TL;DR

This paper revisits and clarifies bounds on symbolic powers using asymptotic multiplier ideals, confirming previous improvements and suggesting further enhancements are unlikely with current techniques.

Contribution

It demonstrates that the original Ein-Lazarsfeld-Smith argument already achieves the known improvements and discusses the limitations of further improvements using similar methods.

Findings

The original bound by Ein-Lazarsfeld-Smith is as strong as the improved versions.

Examples indicate that further improvements with the same approach are unlikely.

The paper provides an exposition with some new examples and remarks.

Abstract

We revisit a bound on symbolic powers found by Ein-Lazarsfeld-Smith and subsequently improved by Takagi-Yoshida. We show that the original argument of Ein-Lazarsfeld-Smith actually gives the same improvement. On the other hand, we show by examples that any further improvement based on the same technique appears unlikely. This is primarily an exposition; only some examples and remarks might be new.