Tacnodes and cusps

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.