Take the 2-minute tour ×  Suppose G is an algebraic group (over
a field, say; maybe even over ℂ) and H⊆G is a closed subgroup.
Does there necessarily exist an action of G on a scheme X and a
point x∈X such that H=Stab(x)?  Before you jump out of your seat
and say, "take X=G/H," let me point out that the question is
basically equivalent to "Is G/H a scheme?" If G/H is a scheme,
you can take X=G/H. On the other hand, if you have X and x∈X,
then the orbit of x (which is G/H) is open in its closure, so it
inherits a scheme structure (it's an open subscheme of a closed
subscheme of X).  I say "basically equivalent" because in my
argument, I assumed that the action of G on X is quasi-compact
and quasi-separated so that the closure of the orbit (i.e. the
scheme-theoretic closed image of G×{x}→X) makes sense. I'm also
using Chevalley's theorem to say the the image is open in its
closure, which requires that the action is locally finitely
presented. I suppose it's possible that there's a bizarre
example where this fai