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Suppose G is an algebraic group with an action G×X→X on a scheme. Then many of the usual constructions you make when you talk about group actions on sets can be made scheme-theoretically. For example, if x∈X is a point (thought of as a map x:∗→X, where ∗ is Spec of a field or the base scheme), then the stabilizer Stab(x) is naturally a scheme because it is the fiber product

Stab(x) ----> G×X    (g,y)
  |            |       _
  |            |       |
  v   (x,x)    v       v
  ∗ --------> X×X   (gy,y)
  1. Does the orbit of a point have a natural scheme structure?
  2. Does the fixed locus (the set of points x∈X fixed by all of G) have a scheme structure?

For (1), if everything is sufficiently nice, then the morphism G×∗→X, given by g→g⋅x has a scheme-theortic closed image, and the actual image is constructible and invariant under the G-action, so the actual image is an open subset of its closure. Thus, the orbit gets the structure of an open subscheme of a closed subscheme of X. But this construction doesn't feel very natural.

For (2), you can obviously define the functor Fix(T)={t∈X(T)|t is fixed by every element of G(T)}. Is this functor always representable?

Edit: Given that Scott has given such an excellent (negative) answer to question (1) but not said anything about question (2), I've asked (2) as a separate question.

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2 Answers 2

up vote 8 down vote accepted

Question number 1: Let your base S be Spec k[x] (say k is an algebraically closed field), let X be Spec k[x,y], and let G be Ga,S, with action over the point s given by gs(xs) = (sgs) + xs. This action is transitive away from zero, so the orbit of the zero section is a plane with a slit. This is not an open subscheme of a closed subscheme of X, because it is not a scheme.

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This is a fantastic example! –  Anton Geraschenko Oct 29 '09 at 2:16

For (1), I think the moral of your remark is that the orbit is the scheme-theoretic image of G×∗→X. If you insist on viewing the 'actual' image as the orbit, well then your definition of what an 'orbit' is will only feel as natural as your definition of what the 'actual' image is.

For example, suppose we want to say something about 'actual' orbits such as 'this orbit is contained in the closure that orbit.' If I take orbit to mean scheme-theoretic image, then I can restate this as 'this orbit is contained in that orbit', which sounds more natural anyway.

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I have to disagree with this point of view. It just feels like an abuse of the word "orbit". Whatever the orbit is, the natural map from G to it should be a surjection, not just dominant. I think our only disagreement is about terminology. But I should also point out that in order to have a scheme-theoretic closed image, the map must be quasi-separated and quasi-compact. If you take G to be the discrete group ℤ and take X to be quasi-compact, then this condition will fail, so it's not even clear how to get a scheme structure on the closure of the orbit. –  Anton Geraschenko Oct 29 '09 at 2:12
Might I suggest "scheme-theoretic orbit"? –  Andrew Critch Oct 29 '09 at 5:35

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