Graded unipotent groups and Grosshans theory

Let U be a unipotent group which is graded in the sense that it has an extension H by the multiplicative group of the complex numbers such that all the weights of the adjoint action on the Lie algebra of U are strictly positive. We study embeddings of H in a general linear group G which possess Grosshans-like properties. More precisely, suppose H acts on a projective variety X and its action extends to an action of G which is linear with respect to an ample line bundle on X. Then, provided that we are willing to twist the linearisation of the action of H by a suitable (rational) character of H, we find that the H-invariants form a finitely generated algebra and hence define a projective variety X==H; moreover the natural morphism from the semistable locus in X to X==H is surjective, and semistable points in X are identified in X==H if and only if the closures of their H-orbits meet in the semistable locus. A similar result applies when we replace X by its product with the projective line; this gives us a projective completion of a geometric quotient of a U-invariant open subset of X by the action of the unipotent group U.