Orbit closures and rank schemes

  • Christine Riedtmann

    Universität Bern, Switzerland
  • Grzegorz Zwara

    Nicolaus Copernicus University, Torun, Poland


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

Cite this article

Christine Riedtmann, Grzegorz Zwara, Orbit closures and rank schemes. Comment. Math. Helv. 88 (2013), no. 1, pp. 55–84

DOI 10.4171/CMH/278