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Lemma 3. The open subscheme $X^{\text{ss}}_\rho$ of $X$ depends only on the image of $\rho$. Also, for every integer $n$, $X^{\text{ss}}_{\gamma_n\circ \rho}$ equals $X^{\text{ss}}_{\rho}$ as open subschemes of $X$.

Lemma 3. The open subscheme $X^{\text{ss}}_\rho$ of $X$ depends only on the image of $\rho$. Also, for every positive integer $n$, $X^{\text{ss}}_{\gamma_n\circ \rho}$ equals $X^{\text{ss}}_{\rho}$ as open subschemes of $X$.

Fundamental Theorem of Geometric Invariant Theory (Projective Case). [Theorem 1.10, p. 38, Geometric Invariant Theory, 3rd ed., Mumford, Fogarty, Kirwan, Ergebnis. Math. 34, Spring-Verlag, Berlin, 1994]
Among all $G$-invariant $k$-morphisms from $X^{\text{ss}}_\rho$ to $k$-schemes, there exists an initial such $k$-morphism, $$q:X^{\text{ss}}_\rho \to Y,$$ i.e., $q$ is a categorical quotient. This morphism is a uniform categorical quotient, i.e., it is a categorical quotient after finitely presented, flat base change of $Y$. This morphism is affine. This morphism is universally submersive, i.e., for every finitely presented morphism $Y'\to Y$ and for every subset $U\subset Y'$, $U$ is open if and only if the inverse image of $U$ in $Y'\times_{Y,q} X^{\text{ss}}_\rho$ is open. As a closed subscheme of $X^{\text{ss}}_\rho\times_{\text{Spec}\ k} X^{\text{ss}}_\rho$, the fiber product $R_q:=X^{\text{ss}}_\rho\times_{q,Y,q} X^{\text{ss}}_\rho$ equals $R_\rho$. The scheme $Y$ is projective, and there exists an ample invertible sheaf whose pullback under $q$ with its natural linearization is isomorphic to a positive tensor power of $\mathcal{L}$ with its induced linearization. Finally, there exists an open subscheme $Y_0$ of $Y$ whose inverse image equals the (properly) stable locus $X^{\text{s}}_{\rho,0}$, and the restriction $q_0:X^{\text{s}}_{\rho,0}\to Y_0$ is a uniform geometric quotient.

There is an important observation about the fiber product $R_q$.
Since $q$ is $G$-invariant, $R_q$ contains $R^{\text{pre}}_\rho$. Since $R_q$ is the fiber product of a morphism, this is an equivalence relation. Thus, $R_q$ contains $R_\rho$. The main content of equality and $R_q$ and $R_\rho$ is the following. For every geometric pair of points, $$(x_1,x_2):\text{Spec}\ \kappa \to X^{\text{ss}}_\rho\times_{\text{Spec} k}X^{\text{ss}}_\rho,$$ the images of $x_1$ and $x_2$ in $Y(\kappa)$ are equal if and only if the orbit closures of $x_1$ and $x_2$ in $X^{\text{ss}}_\rho\times_{\text{Spec}\ k}\text{Spec}\ \kappa$ have nonempty intersection. Moreover, every geometric fiber of $q$ contains a unique orbit that is closed in $X^{\text{ss}}_\rho$.

Fundamental Theorem of Geometric Invariant Theory (Projective Case). [Theorem 1.10, p. 38, Geometric Invariant Theory, 3rd ed., Mumford, Fogarty, Kirwan, Ergebnis. Math. 34, Spring-Verlag, Berlin, 1994]
Among all $G$-invariant $k$-morphisms from $X^{\text{ss}}_\rho$ to $k$-schemes, there exists an initial such $k$-morphism, $$q:X^{\text{ss}}_\rho \to Y,$$ i.e., $q$ is a categorical quotient. This morphism is a uniform categorical quotient, i.e., it is a categorical quotient after finitely presented, flat base change of $Y$. This morphism is affine. This morphism is universally submersive, i.e., for every finitely presented morphism $Y'\to Y$ and for every subset $U\subset Y'$, $U$ is open if and only if the inverse image of $U$ in $Y'\times_{Y,q} X^{\text{ss}}_\rho$ is open. As a closed subscheme of $X^{\text{ss}}_\rho\times_{\text{Spec}\ k} X^{\text{ss}}_\rho$, the fiber product $R_q:=X^{\text{ss}}_\rho\times_{q,Y,q} X^{\text{ss}}_\rho$ contains $R_\rho$, and they are equal as closed subsets. The scheme $Y$ is projective, and there exists an ample invertible sheaf whose pullback under $q$ with its natural linearization is isomorphic to a positive tensor power of $\mathcal{L}$ with its induced linearization. Finally, there exists an open subscheme $Y_0$ of $Y$ whose inverse image equals the (properly) stable locus $X^{\text{s}}_{\rho,0}$, and the restriction $q_0:X^{\text{s}}_{\rho,0}\to Y_0$ is a uniform geometric quotient.

There is an important observation about the fiber product $R_q$. Since $q$ is $G$-invariant, $R_q$ contains $R^{\text{pre}}_\rho$. Since $R_q$ is the fiber product of a morphism, this is an equivalence relation. Thus, $R_q$ contains $R_\rho$. The content of set-theoretic equality of $R_q$ and $R_\rho$ is the following. For every geometric pair of points, $$(x_1,x_2):\text{Spec}\ \kappa \to X^{\text{ss}}_\rho\times_{\text{Spec} k}X^{\text{ss}}_\rho,$$ the images of $x_1$ and $x_2$ in $Y(\kappa)$ are equal if and only if the orbit closures of $x_1$ and $x_2$ in $X^{\text{ss}}_\rho\times_{\text{Spec}\ k}\text{Spec}\ \kappa$ have nonempty intersection. Moreover, every geometric fiber of $q$ contains a unique orbit that is closed in $X^{\text{ss}}_\rho$.

Definition 4. The associated prerelation $R^{\text{pre}}_\rho$ is the minimal closed subscheme through which the following morphism factors, $$\Psi_\rho:T\times_{\text{Spec}\ k} X^{\text{ss}}_{\rho} \to X^{\text{ss}}_\rho \times_{\text{Spec}\ k} X^{\text{ss}}_\rho, \ \ (t,x)\mapsto (\rho_t(x),x).$$ The associated relation $R_\rho$ is the first iterate of this relation, i.e., the minimal closed subscheme through which the following morphism factors, $$(\text{pr}_1\circ \text{pr}_1,\text{pr}_2\circ \text{pr}_2):R^{\text{pre}}_\rho \times_{\text{pr}_2,X^{\text{ss}}_\rho,\text{pr}_1} R^{\text{pre}}_\rho \to X^{\text{ss}}_\rho\times_{\text{Spec}\ k} X^{\text{ss}}_{\rho}, \ \ ((x_1,x),(x,x_2))\mapsto (x_1,x_2).$$

There is an important observation about the fiber product $R_q$.
Since $q$ is $G$-invariant, $R_q$ contains $R^{\text{pre}}_\rho$. Since $R_q$ is the fiber product of a morphism, this is an equivalence relation. Thus, $R_q$ contains $R_\rho$. The main content of equality and $R_q$ and $R_\rho$ is the following. For every geometric pair of points, $$(x_1,x_2):\text{Spec}\ \kappa \to X^{\text{ss}}_\rho\times_{\text{Spec} \k}X^{\text{ss}}_\rho,$$ the images of $x_1$ and $x_2$ in $Y(\kappa)$ are equal if and only if the orbit closures of $x_1$ and $x_2$ in $X^{\text{ss}}_\rho\times_{\text{Spec}\ k}\text{Spec}\ \kappa$ have nonempty intersection. Mreover, every geometric fiber of $q$ contains a unique orbit that is closed in $X^{\text{ss}}_\rho$.

Definition 4. The associated prerelation $R^{\text{pre}}_\rho$ is the minimal closed subscheme through which the following morphism factors, $$\Psi_\rho:T\times_{\text{Spec}\ k} X^{\text{ss}}_{\rho} \to X^{\text{ss}}_\rho \times_{\text{Spec}\ k} X^{\text{ss}}_\rho, \ \ (t,x)\mapsto (\rho_t(x),x).$$ The associated relation $R_\rho$ is the first iterate of this relation, i.e., the minimal closed subscheme through which the following morphism factors, $$(\text{pr}_1\circ \text{pr}_1,\text{pr}_2\circ \text{pr}_2):R^{\text{pre}}_\rho \times_{\text{pr}_2,X^{\text{ss}}_\rho,\text{pr}_1} R^{\text{pre}}_\rho \to X^{\text{ss}}_\rho\times_{\text{Spec}\ k} X^{\text{ss}}_{\rho},$$ $$((x_1,x),(x,x_2))\mapsto (x_1,x_2).$$

There is an important observation about the fiber product $R_q$.
Since $q$ is $G$-invariant, $R_q$ contains $R^{\text{pre}}_\rho$. Since $R_q$ is the fiber product of a morphism, this is an equivalence relation. Thus, $R_q$ contains $R_\rho$. The main content of equality and $R_q$ and $R_\rho$ is the following. For every geometric pair of points, $$(x_1,x_2):\text{Spec}\ \kappa \to X^{\text{ss}}_\rho\times_{\text{Spec} k}X^{\text{ss}}_\rho,$$ the images of $x_1$ and $x_2$ in $Y(\kappa)$ are equal if and only if the orbit closures of $x_1$ and $x_2$ in $X^{\text{ss}}_\rho\times_{\text{Spec}\ k}\text{Spec}\ \kappa$ have nonempty intersection. Moreover, every geometric fiber of $q$ contains a unique orbit that is closed in $X^{\text{ss}}_\rho$.