  Let S be a smooth algebraic surface satisfying the following property:
H^i(\oc_S(C))=0 (i=1,2) for any irreducible and reduced curve C of S. The aim
of this paper is to provide a characterization of special linear systems on S
which are singular along a set of double points in general position. As an
application, the dimension of such systems is evaluated in case S is an
Abelian, an Enriques, a K3 or an anticanonical rational surface.
