  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 $U$.
