  We use the Kempf-Lascoux-Weyman geometric technique in order to determine the
minimal free resolutions of some orbit closures of quivers. As a consequence,
we obtain that for Dynkin quivers orbit closures of 1-step representations are
normal with rational singularities. For Dynkin quivers of type $A$, we describe
explicit minimal generators of the defining ideals of orbit closures of 1-step
representations. Using this, we provide an algorithm for type $A$ quivers for
describing an efficient set of generators of the defining ideal of the orbit
closure of any representation.
