On singularities of quiver moduli

TL;DR

This paper investigates the singularities of quiver moduli spaces, showing they can be embedded into bipartite quiver moduli spaces with similar singularities, and characterizes when these spaces are smooth based on the quiver type.

Contribution

It establishes a connection between arbitrary quiver moduli spaces and bipartite quiver moduli spaces, providing criteria for smoothness based on Dynkin classifications.

Findings

Moduli spaces of representations can be embedded into bipartite quiver moduli spaces.

Smoothness of all moduli spaces characterizes Dynkin or extended Dynkin quivers.

Singularities are preserved under the embedding.

Abstract

Any moduli space of representations of a quiver (possibly with oriented cycles) has an embedding as a dense open subvariety into a moduli space of representations of a bipartite quiver having the same type of singularities. A connected quiver is Dynkin or extended Dynkin if and only if all moduli spaces of its representations are smooth.