Singularities of zero sets of semi-invariants for quivers

Abstract

Let $Q$ be a quiver with dimension vector $\alpha$ prehomogeneous under the action of the product of general linear groups $\text{GL}\left(\alpha \right)$ on the representation variety $\text{Rep}\left(Q,\alpha \right)$. We study geometric properties of zero sets of semi-invariants of this space. It is known that for large numbers $N$, the nullcone in $\text{Rep}\left(Q,N\cdot \alpha \right)$ 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.