Quiver representations with an irreducible null cone

Let d be a prehomogeneous dimension vector for a quiver Q. There is an action of the product Gl(d) of linear groups on the vector space rep(Q,d) of representations of Q with dimension vector d, and there is a representation T with a dense Gl(d)-orbit in rep(Q,d). We give a construction for a dense subset FQ,d of the variety ZQ,d of common zeros of all semi-invariants in k[rep(Q,d)] of positive degree, and we show that this set is stable for big dimension vectors, i.e. FQ,N⋅d={X⊕TN−1:X∈FQ,d}. Moreover, we show that the existence of a dense orbit in ZQ,d depends on a quiver Q⊥ such that the category of representations of Q⊥ is equivalent to the right perpendicular category T⊥.