Orbit closures and rank schemes

Let A be a finitely generated associative algebra over an algebraically closed field k, and consider the variety mod A.k/ of A-module structures on k d . In case A is of finite representation type, equations defining the closure x OM are known for M 2 mod A.k/; they are given by rank conditions on suitable matrices associated with M . We study the schemes CM defined by such rank conditions for modules over arbitraryA, comparing them with similar schemes defined for representations of quivers and obtaining results on singularities. One of our main theorems is a description of the ideal of x OM for a representation M of a quiver of type An, a result Lakshmibai and Magyar established for the equioriented quiver of type An in [12]. Mathematics Subject Classification (2010). Primary 14L30; Secondary 14B05, 16G20, 16G70.