  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.
