Projective rational smoothness of varieties of representations for quivers of type $A$

Let ${\mathbf U}^+$ be the positive part of the quantized enveloping algebra ${\mathbf U}$of type $A_n$. The change of basis between canonical, and PBW-basis of ${\mathbf U}^+$ has a geometric interpretation in terms of local intersection cohomology of some affine algebraic varieties, namely the Zariski closures of orbits of representations of a quiver of type $A_n$. In this paper we study the local rational smoothness of these orbit closures and, in particular, the rational smoothness of their projectivization.