On the minimal free resolution for fat point schemes of multiplicity at most 3 in P2P2

Let ZZ be a fat point scheme in P2P2 supported on general points. Here we prove that if the multiplicities are at most 3 and the length of ZZ is sufficiently high then the number of generators of the homogeneous ideal IZIZ in each degree is as small as numerically possible. Since it is known that ZZ has maximal Hilbert function, this implies that ZZ has the expected minimal free resolution.