  A microscopic model of a surface polymer network - membrane system is
introduced, with contact polymer surface interactions that can be either
repulsive or attractive and sliplinks of functionality four randomly
distributed over the supporting membrane surface anchoring the polymers to it.
For the supporting surface perturbed from a planar configuration and a small
relative number of surface sliplinks, we investigate an expansion of the free
energy in terms of the local curvatures of the surface and the surface density
of sliplinks, obtained through the application of the Balian - Bloch -
Duplantier multiple surface scattering method. As a result, the dependence of
the curvature elastic modulus, the Gaussian modulus as well as of the
spontaneous curvature of the "dressed" membrane, ~{\sl i.e.} polymer network
plus membrane matrix, is obtained on the mean polymer bulk end to end
separation and the surface density of sliplinks.
