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We establish a criterion for the (failure of) the containment $I^{(m)}\subsetI^r$ for 3-generated ideals $I$ defining reduced sets of points in$\mathbb{P}^2$. Our criterion arises from studying the minimal free resolutionsof 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 recentlyarisen as counterexamples to a question of B. Harbourne and C. Huneke: theFermat configuration and the Klein configuration.

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