and $a\in Q_1$. Two points in ${\mathrm{Hom}}$ are in the same $GL$-orbit if and only if they define isomorphic representations of $Q$. The closures of the $GL$-orbits are called [[****]{}quiver loci]{}.[^2]

-   [@G] The action of $GL(Q,d)$ on ${\mathrm{Hom}}(Q,d)$ has finitely many orbits for all dimension vectors $d:Q_0\rightarrow{{\mathbb N}}$, if and only if $Q$ is a [[****]{}Dynkin quiver]{}, i.e. if the undirected graph underlying $Q$ is a Dynkin diagram of type $A_{n\geq 1}$, $D_{n\geq 4}$, $E_6$, $E_7$, or $E_8$.

-   [@LM; @BZ] For $Q$ of type $A,D$, the quiver loci have rational singularities. (To our knowledge the $E$ cases are still open.)

-   [@R] If $Q$ is a Dynkin quiver, each quiver locus $\Omega \subseteq {\mathrm{Hom}}$ is the image of a birational linear Kempf collapsing, i.e. there exists a parabolic subgroup $P \leq GL$ and a $P$-invariant linear subspace $Z\leq {\mathrm{Hom}}$ such that $GL \times^P Z {\mathop{\twoheadrightarrow}}GL\cdot Z = \Omega$ is birational.

In [@R], Reineke constructs each $Z$ explicitly using the [*Auslander-Reiten quiver*]{} of $Q$. We recapitulate this construction precisely in Section \[sec:finitequivers\].

Modulo the construction of a certain ordering, we can state the resulting quiver formulae here. It is well-known [@G] that the indecomposable representations of a Dynkin quiver 