Let \(Q\) be a tame connected quiver (i.e. the underlying graph \(|Q|\) of \(Q\) is a Dynkin or extended Dynkin diagram), and let \(d\) be a prehomogeneous dimension vector (i.e. the space of representations of \(Q\) with dimension vector \(d\) contains a Zariski open \(\text{GL}(d)\)-orbit). Denote by \(Z_{Q,d}\) the closed subscheme in the representation space of the common zeros of the basic semi-invariant polynomial functions \(f_1,\dots,f_s\) (the null-cone); recall that the zero sets \(Z(f_1),\dots,Z(f_s)\) are the codimension \(1\) irreducible components of the complement of the open orbit.
In an earlier paper, Ch. Riedtmann and G. Zwara [Comment. Math. Helv. 79, No. 2, 350-361 (2004; Zbl 1063.14052)] showed that \(Z_{Q,d}\) is a complete intersection if \(|Q|=A_n\) or \(\widetilde A_n\).
The main result of the present paper is that \(Z_{Q,d}\) is not too far from being a complete intersection for all tame prehomogeneous quiver settings. More precisely, it is proved that \(s-\text{codim}(Z_{Q,d})\leq\gamma(|Q|)\), where \(\gamma(|Q|)\in\{0,1,2,3,4\}\) is explicitly given for each (extended) Dynkin diagram, and the bound is sharp with the possible exception of the case of \(\widetilde E_8\). The proof uses the Auslander-Reiten theory for representations of tame quivers. (Also submitted to MR.)