  Let Z be a zero-dimensional subscheme of the projective plane consisting of
the union of r>5 double points, I its defining ideal sheaf. It is known that I
has the expected cohomology when the points are distinct and in general
position (Hirschowitz '85). We extend this result by allowing infinitely near
points, one of them having bigger multiplicity. As an application, new bounds
are given for the existence of plane curves with tacnodes and higher order
cusps.
