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].