Resolutions of defining ideals of orbit closures for quivers of type $A_3$

TL;DR

This paper constructs explicit minimal free resolutions of the defining ideals of orbit closures for quivers of type A3, revealing geometric properties and Gorenstein conditions of these orbit closures.

Contribution

It provides explicit minimal free resolutions for the defining ideals of orbit closures in type A3 quivers, a novel explicit construction in this context.

Findings

Explicit minimal free resolutions constructed

Characterization of Gorenstein orbit closures

Insights into geometric properties of orbit closures

Abstract

We investigate the properties of coordinate rings of orbit closures for quivers of type $A_3$ by considering the desingularization given by Reineke. We construct explicit minimal free resolutions of the defining ideals of the orbit closures thus giving us a minimal set of generators for the defining ideal. The resolution allows us to read off some geometric properties of the orbit closure. In addition, we give a characterization for the orbit closure to be Gorenstein.