A homological criterion for the containment between symbolic and ordinary powers for some ideals of points in $\mathbb{P}^2$

TL;DR

This paper develops a homological criterion based on minimal free resolutions to determine when symbolic powers of certain point ideals in projective plane are contained within their ordinary powers, with applications to notable point configurations.

Contribution

It introduces a new homological criterion for containment problems of symbolic and ordinary powers of ideals of points in 2^2, based on minimal free resolutions, and applies it to specific configurations.

Findings

Criterion successfully identifies containment failures.

Applied to Fermat and Klein configurations.

Provides new insights into symbolic vs. ordinary power containment.

Abstract

We establish a criterion for the (failure of) the containment $I^{(m)}\subset I^r$ for 3-generated ideals $I$ defining reduced sets of points in $\mathbb{P}^2$. Our criterion arises from studying the minimal free resolutions of the powers of $I$, specifically the minimal free resolutions for $I^m$ and $I^r$. We apply this criterion to two point configurations that have recently arisen as counterexamples to a question of B. Harbourne and C. Huneke: the Fermat configuration and the Klein configuration.