Nodal curves and postulation of generic fat points on P-2

We study the postulation of a general union Z of fat points of P-2, when mo st of the connected components of Z have multiplicity 2. This problem is re lated to the existence of "good" families of curves on P-2 with prescribed singularities, most of them being nodes, and to the cohomology of suitable line bundles on blowing ups of the plane.