On the equations and classification of toric quiver varieties

TL;DR

This paper studies toric quiver varieties, showing their embeddings are generated by degree at most 3 and outlining a finite classification method for fixed dimensions.

Contribution

It establishes bounds on the generators of the ideal and provides a classification procedure for toric quiver varieties of fixed dimension.

Findings

Homogeneous ideal generated by elements of degree at most 3 for acyclic quivers.

Finiteness of d-dimensional toric quiver varieties up to isomorphism.

Outline of a classification procedure for these varieties.

Abstract

Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight there is an associated quasiprojective toric variety together with a canonical embedding into projective space. It is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most $3$. In each fixed dimension $d$ up to isomorphism there are only finitely many $d$-dimensional toric quiver varieties. A procedure for their classification is outlined.