On quotient varieties and the affine embedding of certain homogeneous spaces

Abstract

We recall that if an algebraic group G operates regularly on a variety V, by a quotient variety is meant a pair (V/G, r), where V/G is a variety and r: V-* V/G is a rational map, everywhere defined and surjective, such that two points of V have the same image under r if and only if they have the same orbit on V, and such that, for any xE V, any rational function on V that is G-invariant (i.e., constant on orbits) and defined at x is actually (under the natural injection of function fields Q(V/G) -*Q(V), Q denoting the universal domain) a rational function on VIG that is defined at rx (cf. [1, expose 8]). Q(V/G) must therefore consist precisely of all G-invariant elements of Q(V), so r is separable. A quotient variety need not exist (obvious necessary condition: all orbits on V must be closed), but when it exists it is clearly unique to within an isomorphism; in this case, for any open subset UC V/G, T-1 U/G exists and equals U. PROPOSITION 1. Let the algebraic group G operate regularly on the variety V, all defined over the field k. Suppose there exists a quotient variety r: V-* V/G. Suppose also that for each point p of V that is algebraic over k there exists an open affine subset of V/G containing the image under r of each of the conjugates of p over k (a vacuous condition if V/G. can. be embedded in a projective space or if V/G and r are known to be defined over a regular extension of k, in particular if k is algebraically closed). Then V/G and r could have been taken so as to be defined over k. The G-invariant elements of Q(V) are generated by those in k(V), in other words there exists a variety W and a generically surjective rational map V-)W, both defined over k, such that for any field K between k and Q, K(W) is the field of G-invariant elements of K(V) [3, Theorem 2]. We have here a field descent problem, and supposing V/G and r to be defined over the extension field K of k, there are two cases to consider: K a regular extension of k, and K algebraic over k. In view of the unicity to within isomorphism of the quotient variety, the criteria of Weil [6] take care of the first case. [Of course this can also be done directly; e.g., supposing k algebraically closed, if