Quiver Schemes for Nongeneric Stability and Cornering

TL;DR

This paper investigates the structure of quiver schemes under nongeneric stability conditions, revealing reduced points, embeddings, and connections to Quot schemes and Nakajima varieties.

Contribution

It proves that certain stability-zero Nakajima quiver schemes are reduced points and establishes embeddings induced by modules supported on stability-zero vertices.

Findings

Stability-zero Nakajima quiver schemes are reduced points.

Adding modules supported on stability-zero vertices induces closed embeddings.

Certain equivariant Quot schemes are isomorphic to Nakajima quiver varieties.

Abstract

We use the Le Bruyn--Procesi theorem to prove several results on quiver schemes for nongeneric stability conditions: We show that the stability-zero finite-type Nakajima quiver schemes are reduced points and give an example of a closely related nonreduced quiver scheme. In broader generality, we prove that adding modules supported on stability-zero vertices induces closed embeddings of quiver schemes, and show how in many cases the quiver scheme associated with the cornered algebra defines a limit to this system of embeddings. As an application, we show that there is an isomorphism between the underlying reduced schemes of certain equivariant Quot schemes and Nakajima quiver varieties.