Smooth Toric Quiver Varieties

TL;DR

This paper classifies smooth toric quiver varieties with identity dimension vector and canonical weight, showing they are products of projective spaces or their blowups, linking quiver representations with toric geometry.

Contribution

It provides a complete classification of smooth toric quiver varieties under specific conditions, connecting quiver data with classical geometric objects.

Findings

Smooth toric quiver varieties are characterized as products of projective spaces or blowups.

The classification applies specifically to quivers with the identity dimension vector and canonical weight.

The work bridges quiver representation theory and toric geometry.

Abstract

We study smoothness of toric quiver varieties. When a quiver $Q$ is defined with the identity dimension vector, the corresponding quiver variety is also a toric variety. So it has both fan representation and quiver representation. We work only on quivers with canonical weight and we classify smooth such toric quiver varieties. We show that a variety corresponding to a quiver with the identity dimension vector and the canonical weight is smooth if and only if it is a product of projective spaces or their blowups.