A hierarchy of parametrizing varieties for representations

Abstract

The primary purpose is to introduce and explore projective varieties, ${\text{GRASS}}_{\mathbf{d}}(\Lambda )$, parametrizing the full collection of those modules over a finite dimensional algebra $\Lambda$ which have dimension vector $\mathbf{d}$. These varieties extend the smaller varieties previously studied by the author; namely, the projective varieties encoding those modules with dimension vector $\mathbf{d}$ which, in addition, have a preassigned top or radical layering. Each of the ${\text{GRASS}}_{\mathbf{d}}(\Lambda )$ is again partitioned by the action of a linear algebraic group, and covered by certain representation-theoretically defined affine subvarieties which are stable under the unipotent radical of the acting group. A special case of the pertinent theorem served as a cornerstone in the work on generic representations by Babson, Thomas, and the author. Moreover, applications are given to the study of degenerations.