Let $G$ be a linearly reductive algebraic group, and let $X$ be an irreducible affine variety, over an algebraically closed field $\mathbb{K}$, with a regular action of $G$. By Rosenlicht's result, we know that there's a $G$-invariant open set $U\subseteq X$ such that the geometric quotient $\Phi:U\rightarrow U/\!/G$ exists. So, "generic" orbits are contained in $U$. But is there something more specific about these "generic" orbits known?
My question:
This $U$ contains orbits of maximal dimension, but is there more specific information for which $x\in X$, the orbit $\mathcal{O}(x)\subset U$?
I'm interested in the setting of quiver representations:
Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. Let $A$ be a finite-dimensional (associative and unital) algebra over $\mathbb{K}$. Assume there is a quiver $Q=(Q_0,Q_1)$, where $Q_0$ is the set of vertices and $Q_1$ is the set of arrows, associated to $A$, i.e., $A\cong\mathbb{K}Q/I$, with $0 \neq I\subset \mathbb{K}Q$ an admissible ideal. Further assume that the algebra $A$ is such that the quiver $Q$ is acyclic.
For a dimension vector $\beta$, the representation space
$$\DeclareMathOperator\rep{rep}\DeclareMathOperator\GL{GL}
\rep_{\beta}(Q,I):=\biggl\lbrace M\in\prod_{a\in Q_1}\operatorname{Mat}_{\beta(ha)\times\beta(ta)}(\mathbb{K}) \;\bigg|\; \text{$M(r) = 0$ for all $r \in I$}\biggr\rbrace
$$ parametrizes $\beta$-dimensional representations of $(Q,I)$.
The linear algebraic group
$$
\GL_{\beta}:=\prod_{x\in Q_0}\GL_{\beta(x)}(\mathbb{K})
$$
acts on $\rep_{\beta}(Q,I)$ by change of basis.
Now take an irreducible component $\mathcal{C}\subseteq\rep_{\beta}(Q,I)$ and restrict the action of $\GL_{\beta}$ to $\mathcal{C}$.
So my question now becomes:
Is there any information for which representations $V\in\mathcal{C}$, the orbit $\mathcal{O}(V)\subset U$, other than saying $V$ has to be in "general" position?
Best Answer
Let me add more details to my comment above. Let $S$ be a scheme. Let $\overline{X}$ be a proper $S$-scheme, and let $X\subset \overline{X}$ be a dense Zariski open subscheme.
A closed subset $R\subset X\times_S X$ is an algebraic equivalence relation if it contains the diagonal (i.e., it is reflexive), if it is invariant under the involution $(\text{pr}_2,\text{pr}_1)$ of $X\times_S X$ (i.e., it is symmetric), and if it is transitive, i.e., $R$ contains the image of the following composition, $$R\times_{\text{pr}_2,X,\text{pr}_1} R \hookrightarrow (X\times_S X) \times_{\text{pr}_2,X,\text{pr}_1} (X\times_S X) = X\times_S X\times_S X \xrightarrow{\text{pr}_1,\text{pr}_3} X\times_S X.$$ For instance, for a group scheme $G$ over $S$ with an $S$-action on $X$, the closure $Z$ of the image of the associated map is an algebraic equivalence relation, $$\Psi:G\times_S X \to X\times_S X, \ (g,x) \mapsto (g\cdot x,x).$$
Denote by $\overline{R}$ the closure of $R$ in $X\times_S \overline{X}$.
For the projection, $\text{pr}_1:\overline{R}\to X$, there is a maximal open subscheme $U$ of $X$ over which the projection is flat. If $X$ is reduced and Noetherian, then $U$ is a dense open subscheme by Grothendieck's generic flatness / generic freeness theorem. Thus, there is an induced $S$-morphism from $U$ to the relative Hilbert scheme, $$f_{\overline{R}}:U \to \operatorname{Hilb}_{\overline{X}/S}.$$ This is the "modern" take on the classical construction of a quotient of $R$ as a "rational map".
In fact, in terms of making $U$ as big as possible, it is usually better to work with the Chow scheme (due to Angeniol in characteristic $0$, and due to David Rydh in positive characteristic and mixed characteristic). The maximal open subscheme $V$ of $X$ over which $\overline{R}$ is a "good algebraic cycle", and thus defines a morphism from $V$ to the relative Chow scheme, always contains $U$ by the existence of the Hilbert–Chow morphism.
As I mentioned in my comment, this construction is in Kollár's book Rational curves on algebraic varieties, specifically in Section 4 of Chapter IV. I do not believe that he explicitly singles out the application to forming group quotients, since that is not his main concern (he wants to construct the maximal quotient by the family of rational curves, not group quotients).