Birational classification of moduli spaces of representations of quivers

Let alpha be a Schur root; let h = hef(r) (alpha(v)) and let p = 1 similar to [alpha/h,alpha/h]. Then a moduli space of representations of dimension vector alpha is birational to p h by h matrices up to simultaneous conjugacy. Therefore, if h = 1, 2, 3 or 4, then such a moduli space is a rational variety and if h divides 420 it is a stably rational variety.