Singularities of zero sets of semi-invariants for quivers

TL;DR

This paper investigates the geometric properties of zero sets of semi-invariants in quiver representation spaces, establishing conditions for reducedness and rational singularities, especially in the case of Dynkin quivers.

Contribution

It proves that nullcones become reduced for large N and provides criteria for zero sets to have rational singularities using Bernstein-Sato polynomials.

Findings

Nullcones become reduced for large N.

Zero sets can have rational singularities under certain conditions.

Codimension 1 orbit closures in Dynkin quivers have rational singularities.

Abstract

Let $Q$ be a quiver with dimension vector $\alpha$ prehomogeneous under the action of the product of general linear groups $\operatorname{GL}(\alpha)$ on the representation variety $\operatorname{Rep}(Q,\alpha)$. We study geometric properties of zero sets of semi-invariants of this space. It is known that for large numbers $N$, the nullcone in $\operatorname{Rep}(Q,N\cdot \alpha)$ becomes a complete intersection. First, we show that it also becomes reduced. Then, using Bernstein-Sato polynomials, we discuss some criteria for zero sets to have rational singularities. In particular, we show that for Dynkin quivers codimension $1$ orbit closures have rational singularities.