Postulation for 2-superfat points in the plane

We study the postulation of 0-dimensional schemes given by unions of 2-superfat points in general position in the plane, i.e., the union of local schemes defined by the intersection of two distinct double lines. We prove that such schemes have good postulation, i.e., they have the expected Hilbert function. We also show the good postulation of such schemes when we add a general 3-fat point. Finally, we use these results to answer a peculiar kind of interpolation problem.