Commentarii Mathematici Helvetici

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Volume 88, Issue 1, 2013, pp. 55–84
DOI: 10.4171/CMH/278

Published online: 2013-01-07

Orbit closures and rank schemes

Christine Riedtmann and Grzegorz Zwara[1]

(1) Nicolaus Copernicus University, Torun, Poland

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].

Keywords: Modules, representations of quivers, orbit closures, subschemes, types of singularities

Riedtmann Christine, Zwara Grzegorz: Orbit closures and rank schemes. Comment. Math. Helv. 88 (2013), 55-84. doi: 10.4171/CMH/278