Top-stable degenerations of finite dimensional representations II

Let Λ be a finite dimensional algebra over an algebraically closed field. We exhibit slices of the representation theory of Λ that are always classifiable in stringent geometric terms. Namely, we prove that, for any semisimple object T∈Λ-mod, the class of those Λ-modules with fixed dimension vector (say d) and top T which do not permit any proper top-stable degenerations possesses a fine moduli space. This moduli space, ModuliMaxdT, is a projective variety. Despite classifiability up to isomorphism, the targeted collections of modules are representation-theoretically rich: indeed, any projective variety arises as ModuliMaxdT for suitable choices of Λ, d, and T. In tandem, we give a structural characterization of the finite dimensional representations that have no proper top-stable degenerations.