A homological criterion for the containment between symbolic and ordinary powers of some ideals of points in P2P2

We establish a criterion for the (failure of) the containment I(m)⊆IrI(m)⊆Ir for 3-generated ideals I   defining reduced sets of points in P2P2. Our criterion arises from studying the minimal free resolutions of the powers of I  , specifically the minimal free resolutions for ImIm and IrIr. We apply this criterion to two point configurations that have recently arisen as counterexamples to a question of B. Harbourne and C. Huneke: the family of Fermat configurations and the Klein configuration.