A homological criterion for the containment between symbolic and ordinary powers of some ideals of points in \(\mathbb{P}^2\) (Q2349337)

The study of the containment \(I^{(m)}\subset I^r\) for the homogeneous (radical) ideal of a set of points in \(\mathbb{P}^2\) is the subject of this paper (\(I^{(m)}\), the symbolic \(m\)-power of \(I\), is the saturation of the ideal \(I^m\)). The minimal free resolution of the ideals \(I^m\) is studied, and for ideals with three generators a criterion is given to check that containment, specifically to check if \(I^{(2)}\subset I^3\) (which was one of the first open problems posed about this question). Counterexamples when \(I^{(2)}\subset I^3\) does not hold are studied and proved using the criterion found (all sets of points are given by singular loci for special arrangements of lines).