Over a field of characteristic zero the geometry of orbit closures for equioriented \(A_n\) quiver was first studied by S. Abeasis et al. [Math. Ann. 256, 401-418 (1981; Zbl 0477.14027)] where it was established that the orbit closures are normal, Cohen-Macaulay, and have rational singularities. This result was generalized to the case of a quiver \(A_n\) with an arbitrary orientation by G. Bobiński and G. Zwara [Manuscr. Math. 105, No. 1, 103-109 (2001; Zbl 1031.16012)].
In the paper under review orbit closures for the non-equioriented \(A_3\) quiver are investigated. Namely, a minimal free resolution of the defining ideal of an orbit closure is explicitly constructed, a description of a minimal set of generators of the defining ideal is obtained, a classification of orbits closures which are Gorenstein is established.