# Orbits in the open set given by Rosenlicht's result

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?

• The construction that I know depends on an auxiliary $G$-invariant dense open immersion $i:X\hookrightarrow \overline{X}$. Then the maximal $G$-invariant open subset on which $G$-orbits in $\overline{X}$ are flat is contained in Rosenlicht's open subset. Jan 11 at 11:54
• The reference that I know is in Koll'ar's book, "Rational curves on algebraic varieties." I do not have the book with me (I usually do, but not now). So I cannot point to a specific page number, but it is in the section where he discusses quotients by algebraic equivalence relations (such as the equivalence relation of lying in a common orbit for a group action). Jan 11 at 18:55
• The short answer it "no, not in this generality". The very purpose of GIT is to exhibit open sets $U$ for which membership is "decidable". See the preface of Mumford's book. So you have more chances of an affirmative answer if you stick to open subsets produced by GIT. Feb 2 at 11:59

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.