A hierarchy of parametrizing varieties for representations

TL;DR

This paper introduces and studies projective varieties called $ ext{GRASS}_{f d}( ext{Λ})$ that parametrize modules over finite dimensional algebras with a fixed dimension vector, extending previous varieties and providing tools for degeneration analysis.

Contribution

It develops a hierarchy of parametrizing varieties for modules, generalizing earlier varieties with fixed top or radical layering, and explores their structure and applications.

Findings

Varieties are partitioned by linear algebraic group actions.

Varieties are covered by affine subvarieties stable under unipotent radicals.

Applications include the study of module degenerations.

Abstract

The primary purpose is to introduce and explore projective varieties, $\text{GRASS}_{\bf d}(\Lambda)$, parametrizing the full collection of those modules over a finite dimensional algebra $\Lambda$ which have dimension vector $\bf d$. These varieties extend the smaller varieties previously studied by the author; namely, the projective varieties encoding those modules with dimension vector $\bf d$ which, in addition, have a preassigned top or radical layering. Each of the $\text{GRASS}_{\bf d}(\Lambda)$ 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.