Orbit closures and rational surfaces

In this paper we study the Grassmannian of submodules of a given dimension inside the radical of a finitely generated projective module P for a finite dimensional algebra Λ over an algebraically closed field. The orbit of such a submodule C   under the action of AutΛ(P)AutΛ(P) on the Grassmannian encodes information on the top-stable degenerations of P/CP/C. The goal of this article is to begin the study of the global geometry of the closures of such orbits. In dimension one, this geometry is determined by the local rings of singular points. The smallest dimension for which the global geometry is not determined by local data is two, and this case is our main focus. We give several examples to illustrate the interplay between the geometry of the projective surfaces which arise and the corresponding posets of top-stable degenerations.