Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers

Let $k$ be a field, let $G$ be a group scheme of finite type over $k$ (Group schemes of finite type over a field) and let $X$ be a $k$-scheme (Schemes and morphisms over a base). An action of $G$ on $X$ is a morphism $\alpha :G{\times }_{k}X\to X$ (Morphisms of schemes) such that the unit and associativity diagrams commute: $\alpha (e\times {\mathrm{id}}_X)={\mathrm{id}}_X$ and $\alpha ({\mathrm{id}}_G\times \alpha )=\alpha (m\times {\mathrm{id}}_X)$, where $m$ is the multiplication of $G$. A morphism $f:X\to Y$ of $k$-schemes on which $G$ acts is equivariant if $f{\alpha }_X={\alpha }_Y({\mathrm{id}}_G\times f)$. The action is determined by its values on $R$-points, giving an action of the abstract group $G(R)$ on $X(R)$ for every $k$-algebra $R$.

For $x\in X(k)$ (Field-valued points and local-ring points) the orbit map is ${\varrho }_x:G\to X$, ${\varrho }_x(g)=\alpha (g,x)$; for separated finite-type $X$, its image on $k$-points is the rational orbit $G(k)\cdot x$, whereas its underlying topological image is $\mathrm{\mid }{\varrho }_x\mathrm{\mid }(\mathrm{\mid }G\mathrm{\mid })\subseteq \mathrm{\mid }X\mathrm{\mid }$. The orbit set $X(k)\mathrm{/}G(k)$ is the set of all rational orbits. The reduced orbit subscheme $O_x$ means this locally closed image with reduced structure, when local closedness is established; the orbit lemma below constructs it for smooth $G$. For a nonsmooth group, the orbit map need not factor through this reduced subscheme: translation of ${\alpha }_p$ on ${\mathbb{A}}^1$ at $0$ has a one-point reduced orbit but a nonconstant infinitesimal orbit map. A factorization $G\to O_x$ must therefore be justified or explicitly assumed.

The scheme-theoretic stabilizer (isotropy group) is the fibre product $G_x:=G{\times }_X\mathrm{Spec}k$ formed with ${\varrho }_x$ and the $k$-point $x$ (Scheme-theoretic fibre, Fibre product of schemes): for separated finite-type $X$ it is a closed subscheme of $G$, and for every $k$-algebra $R$ its $R$-points are $G_x(R)=\{g\in G(R):\alpha (g,x_R)=x_R\}$. The pair $R=G{\times }_{k}X\rightrightarrows X$ with $s(g,z)=z$ and $t(g,z)=\alpha (g,z)$ is the action groupoid of the action, a pre-relation on $X$ (Quotient sheaves and representable quotients for pre-relations and group actions). When $X$ is separated and of finite type over $k$, $k$-points are closed, so $G_x$ is a closed subgroup scheme of $G$ (Morphisms and closed subgroup schemes of group schemes).

The reduced orbit $O_x$ need not equal the scheme-theoretic image (Scheme-theoretic image): the latter is closed and is normally the orbit closure, whereas the orbit is only locally closed. Fibres and the kernel pair of ${\varrho }_x:G\to X$ are always defined; a kernel pair over $O_x$ requires a factorization through the reduced orbit (Immersion of schemes).