Perpendicular categories, null cones and dense orbits

Let Q be a quiver, k an algebraically closed field and d a prehomogeneous dimension vector for Q with canonical decomposition . We denote by ZQ,d the set of common zeros of all semi-invariants in k[rep(Q,d)] of positive degree. We study the relations between the existence of a dense Gl(d)-orbit in the null cone ZQ,d and the position of the representations T1,…,Tr in the Auslander–Reiten quiver associated to Q.