Smooth orbits are locally closed and their orbit maps are faithfully flat over every field

The orbit map ${\varrho }_x$ is quasi-compact, so by [F1] its scheme-theoretic image $Y=V(\mathcal{I})\hookrightarrow X$ exists. Let $Z_k:={\varrho }_x(\mathrm{\mid }G\mathrm{\mid })$. Its closure is the underlying space of $Y$: on an affine target chart with finitely many source charts, if a basic open $D(a)$ misses the image then every source algebra localized at $a$ is zero. A power of $a$ vanishes in each of the finitely many algebras, so a common power belongs to the kernel defining $Y$, and $D(a)$ misses $Y$. The reverse inclusion follows because the image lies in $Y$. On an affine chart $\mathrm{Spec}A\subseteq X$ with ${\varrho }_x^{-1}(\mathrm{Spec}A)$ covered by finitely many affine charts $\mathrm{Spec}{B}_j$, the ideal is $I=\ker (A\to {\prod }_j{B}_j)$, and for every field extension $K\mathrm{/}k$ one has $I{\otimes }_{k}K=\ker (A_K\to {\prod }_j(B_j{\otimes }_{k}K))$ because $k\to K$ is flat and tensor products are right exact and commute with finite products; hence $Y_K:=Y{\times }_{k}K$ is the scheme-theoretic image of ${\varrho }_{x,K}$ and $Y_K$ is the closure of $Z_K:={\varrho }_{x,K}(\mathrm{\mid }{G}_K\mathrm{\mid })$ in $X_K$.

For $K=\bar{k}$ the set $Z_K$ is constructible in $X_K$: cover the quasi-compact $X_K$ by finitely many affine charts $\mathrm{Spec}A$, cover each preimage by finitely many affine charts $\mathrm{Spec}{B}_j$, and each $A$ is Noetherian by [F12], and $B_j$ is a finite-type $A$-algebra because it is generated by finitely many elements over $K$. Hence [F2] gives finite presentation, so apply [F2] to $A\to B_j$ and to $b=1$; the full scheme-point image is the finite union of these constructible images.

The projection $\pi :Y_K\to Y$ is surjective and closed. It is the base change of $\mathrm{Spec}K\to \mathrm{Spec}k$, which is surjective; and $K\mathrm{/}k$ is algebraic, hence integral, so $Y_K\to Y$ is integral and, by [F3], the image of any closed $V(J)\subseteq Y_K$ is the closed set $V(J\cap Y)$; a surjective closed map is a quotient map.

The scheme $Y$ is geometrically reduced: by step 1.1 it suffices to note that each $B_j{\otimes }_{k}K$ is reduced, since $G_K$ has regular local rings by [F4], and that $A_K\mathrm{/}I{\otimes }_{k}K$ embeds into the product of the reduced rings $B_j{\otimes }_{k}K$.

Since $Z_K$ is constructible by step 1.2 and dense in $Y_K$ by step 1.1, it contains a dense open $U$ of $Y_K$: writing $Z_K$ as a finite union of locally closed subsets and intersecting with the finitely many irreducible components of the Noetherian space $Y_K$, one piece is dense in each component, and a locally closed subset dense in an irreducible space contains an open dense subset of it; remove from $Y_K$ the finitely many closed complements of those relative opens and all intersections of distinct components. The remaining subset is open and dense in $Y_K$ and contained in $Z_K$. Moreover every nonempty constructible subset of $Y_K$ contains a point closed in $Y_K$: a nonempty locally closed piece has a nonempty open subset of an affine chart, and by [F6] that open contains a $K$-point of the chart, which is closed in $Y_K$ because its residue field is $K$.

The image is saturated for $\pi$: for a point $y\in Y_K$ with $q=\pi (y)$ one has $G_K{\times }_{X_K,y}\mathrm{Spec}\kappa (y)\cong G{\times }_X\mathrm{Spec}\kappa (y)$, because the morphism $\mathrm{Spec}\kappa (y)\to X$ factors through $\mathrm{Spec}\kappa (q)\to X$; this in turn is isomorphic to $G_q{\times }_{\mathrm{Spec}\kappa (q)}\mathrm{Spec}\kappa (y)$ with $G_q=G{\times }_{X,q}\mathrm{Spec}\kappa (q)$, and the latter is nonempty whenever $G_q$ is, by [F7]. Since $G_q$ is nonempty exactly when $q$ lies in $Z_k={\varrho }_x(\mathrm{\mid }G\mathrm{\mid })$, this shows $y\in Z_K$ if and only if $\pi (y)\in Z_k$, so ${\pi }^{-1}(\pi (Z_K))=Z_K$.

Each translation by $g\in G(K)$ preserves $Y_K$: translating ${\varrho }_{x,K}$ is precomposing it with left translation on $G_K$, so its scheme-theoretic image is unchanged by [F1]. Set $O_K:=Z_K$ with the open subscheme structure it has in $Y_K$; this is legitimate because $Z_K$ is open in $Y_K$: every closed point $c$ of $Z_K$ lifts to a $K$-point $h$ of $G_K$ by [F6] applied to the nonempty finite-type fibre of ${\varrho }_{x,K}$ over $c$, and with a $K$-point $v$ of the nonempty open ${\varrho }_{x,K}^{-1}(U)$ one has $c={\varrho }_{x,K}(h)=(h{v}^{-1})\cdot {\varrho }_{x,K}(v)\in (h{v}^{-1})U$, so that ${\bigcup }_{g\in G(K)}gU$ is an open subset of $Z_K$ containing every closed point of $Z_K$; its constructible complement in $Z_K$ would otherwise contain a closed point of $Y_K$ by step 2.2, so $Z_K={\bigcup }_{g\in G(K)}gU$ is open in $Y_K$. By step 2.1 the open subscheme $O_K$ is reduced, and it is finite type over $K$ because it is locally of finite type as a locally closed subscheme of the finite-type $X_K$ and quasi-compact as the continuous image of the quasi-compact space $G_K$.

The morphism ${\varrho }_{x,K}:G_K\to O_K$ is surjective by construction, and ${\varrho }_x(\mathrm{\mid }G\mathrm{\mid })=\pi (Z_K)$ is open in $Y$: by step 1.3 and step 2.3 the set ${\pi }^{-1}(\pi (Z_K))=Z_K$ is open, and $\pi$ is a quotient map, so $\pi (Z_K)$ is open. Define $O_x:=\pi (Z_K)$ with the induced open subscheme structure in $Y$; it is finite type over $k$, since it is locally of finite type as a locally closed subscheme of the finite-type $X$ and quasi-compact as the image of the quasi-compact space $G$ under ${\varrho }_x$, and $(O_x{)}_K=O_K$ as open subschemes of $Y_K$.

Reduced-source factorization. For $F=k$ or $F=K$, write $O_F=O_x$ or $O_K$ respectively. The image $Y_F$ is geometrically reduced by step 2.1. Every translation by a point of $G(F)$ preserves the scheme-theoretic image, since translating ${\varrho }_{x,F}$ is the same as precomposing it with left translation on $G_F$. Moreover the action on $G_F{\times }_F{O}_F$ has underlying image in $O_F$: after extending a residue field further, any orbit point has a lift to $G$, and acting on that lift gives another lift. On affine charts let $A$ be a geometrically reduced algebra for $G_F$ and let $C$ be a reduced algebra for $O_F$. The map $A{\otimes }_{F}C\to {\prod }_{\mathfrak{p}\in \mathrm{Spec}C}(A{\otimes }_F\kappa (\mathfrak{p}))$ is injective: write a tensor with a finite independent list of coefficients in $A$, and coefficient comparison after scalar extension shows that each corresponding element of $C$ lies in all primes, hence is zero by [F5]. The target factors are reduced because $G_F$ is smooth by [F4], so $G_F{\times }_F{O}_F$ is reduced. Thus both this action source and $G_F$ are reduced. Pulling back a section of the ideal of $Y_F$ gives a function vanishing in every residue field of the source, hence zero by [F5]; both morphisms therefore factor through $Y_F$. Since their images lie in its open $O_F$, they then factor through $O_F$. Applied to $F=K$, this establishes the action and orbit map over $K$ without asserting that $O_K$ is open in $X_K$.

The subscheme $O_K$ is smooth over $K$: it is reduced by step 3.1 and finite type over the perfect field $K$; by [F9] its regular locus is a nonempty open subset, regularity equals smoothness over $K$, and the smooth locus $\mathrm{S}\mathrm{m}$ is a nonempty open subset stable under the $K$-automorphisms $g\in G(K)$. Since every $K$-point of $O_K$ is of the form $g\cdot x_K$ (the fibre over a $K$-point of $O_K$ is a nonempty finite-type $K$-scheme, hence has a $K$-point by [F6]), and since a nonempty open subset of a finite-type $K$-scheme contains a $K$-point by [F6], $\mathrm{S}\mathrm{m}$ meets $G(K)\cdot x_K$; then $x_K\in \mathrm{S}\mathrm{m}$ and all $K$-points of $O_K$ lie in $\mathrm{S}\mathrm{m}$. The closed complement $O_K\setminus \mathrm{S}\mathrm{m}$, if nonempty, would contain a closed $K$-point by [F6], contradicting the preceding conclusion. Thus $\mathrm{S}\mathrm{m}=O_K$.

The morphism ${\varrho }_{x,K}$ is faithfully flat: it is surjective by step 3.2; for flatness, apply [F8] on the finitely many disjoint integral open pieces of $O_K$ obtained by deleting the intersections of its irreducible components, obtaining a dense open $V\subseteq O_K$ over which ${\varrho }_{x,K}$ is flat. Every closed point $c$ of $O_K$ lies in some translate $gV$ by the argument of step 3.1, and over $gV$ the morphism ${\varrho }_{x,K}$ is conjugate by the isomorphisms $w\mapsto gw$ and $v\mapsto gv$ to the flat morphism over $V$, hence is flat there; the union of the translates contains all closed points, so its closed complement is empty by [F6] and it is all of $O_K$ and ${\varrho }_{x,K}$ is flat at every point.

The factorization over $k$. The same argument in step 4.1 with $F=k$ applies to $O_x$, the reduced open subscheme of $Y$ constructed in step 3.2. Its underlying image is $\mathrm{\mid }{\varrho }_x\mathrm{\mid }(\mathrm{\mid }G\mathrm{\mid })$, stable under the action by the field-lift argument, so $G{\times }_k{O}_x\to X$ and $G\to X$ factor first through $Y$ and then its open $O_x$. Hence $O_x$ is $G$-stable and ${\varrho }_x:G\to O_x$ is a morphism, with image $O_x$. Its base change is the orbit morphism $G_K\to O_K$ because $(O_x{)}_K=O_K$ by step 3.2.

Flatness and faithful flatness descend to $k$: the base changed morphism $({\varrho }_x{)}_K$ is ${\varrho }_{x,K}$, which is faithfully flat by step 4.3 and surjective by step 3.2, and $(O_x{)}_K=O_K$; flatness is checked on affine charts, where it descends along the faithfully flat ring map $k\to K$ by [F10].

Finally $O_x$ is smooth over $k$: by step 4.2 the base change $O_K=(O_x{)}_K$ is smooth over $K$; a finite-type $k$-algebra $A$ with $A{\otimes }_{k}K$ smooth, hence geometrically regular, over $K$ is geometrically regular over $k$ by [F10] and therefore smooth over $k$. Thus $O_x$ is a locally closed, $G$-stable, smooth finite-type $k$-subscheme of $X$, and ${\varrho }_x:G\to O_x$ is faithfully flat by step 5.2 and locally of finite presentation: on affine charts their ring map is of finite type, since its target is generated by finitely many elements over $k$, and its source is Noetherian by [F12]; [F2] then gives finite presentation.