Representation varieties of algebras with nodes

TL;DR

This paper investigates how node splitting in quivers affects the geometry of their representation varieties, establishing correspondences that preserve properties like normality and rational singularities, and providing explicit equations for these varieties.

Contribution

It introduces a method to relate representation varieties of different algebras via node splitting, extending understanding beyond hereditary cases and including explicit ideal generators.

Findings

Node splitting induces correspondences preserving geometric properties.

Many non-hereditary algebras have normal, rational singularity components.

Explicit generators for defining ideals of irreducible components are provided.

Abstract

We study the behavior of representation varieties of quivers with relations under the operation of node splitting. We show how splitting a node gives a correspondence between certain closed subvarieties of representation varieties for different algebras, which preserves properties like normality or having rational singularities. Furthermore, we describe how the defining equations of such closed subvarieties change under the correspondence. By working in the "relative setting" (splitting one node at a time), we demonstrate that there are many non-hereditary algebras whose irreducible components of representation varieties are all normal with rational singularities. We also obtain explicit generators of the prime defining ideals of these irreducible components. This class contains all radical square zero algebras, but also many others, as illustrated by examples throughout the paper. We also show the above is true when replacing irreducible components by orbit closures, for a more restrictive class of algebras. Lastly, we provide applications to decompositions of moduli spaces of semistable representations of certain algebras.