  Let X be a smooth projective surface. Here we study the postulation of a
general union Z of fat points of X, when most of the connected components of Z
have multiplicity 2. This problem is related to the existence of "good"
families of curves on X, with prescribed singularities, most of them being
nodes, and to the cohomology of suitable line bundles on blowing ups of X. More
precise statements are obtained in the case X=P^2.
