Moduli spaces of graded representations of finite dimensional algebras

Let A be a basic finite dimensional algebra over an algebraically closed field, presented as a path algebra modulo relations; further, assume that A is graded by lengths of paths. The paper addresses the classifiability, via moduli spaces, of classes of graded A-modules with fixed dimension d and fixed top T. It is shown that such moduli spaces exist far more frequently than they do for ungraded modules. In the local case (i.e., when T is simple), the graded d-dimensional A-modules with top T always possess a fine moduli space which classifies these modules up to graded-isomorphism; moreover, this moduli space is a projective variety with a distinguished affine cover that can be constructed from quiver and relations of A. When T is not simple, existence of a coarse moduli space for the graded d-dimensional A-modules with top T forces these modules to be direct sums of local modules; under the latter condition, a finite collection of isomorphism invariants of the modules in question yields a partition into subclasses, each of which has a fine moduli space (again projective) parametrizing the corresponding graded-isomorphism classes. https://arxiv.org/pdf/1407.2659.pdf