Orbit closures and rank schemes

Abstract

Let A be a finitely generated associative algebra over an algebraically closed field k , and consider the variety \mathrm{mod}_A^d(k) of A -module structures on k^d . In case A is of finite representation type, equations defining the closure \bar{\mathcal O}_M are known for M \in \mathrm{mod}_A^d(k) ; they are given by rank conditions on suitable matrices associated with M . We study the schemes \mathcal{C}_M defined by such rank conditions for modules over arbitrary A , 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 \bar{\mathcal O}_M for a representation M of a quiver of type \mathbb{A}_n , a result Lakshmibai and Magyar established for the equioriented quiver of type \mathbb{A}_n in [12].