  In this paper we study actions of reductive groups on affine spaces. We prove
that there is a fan structure on the space of characters of the group, which
parameterizes the possible invariant quotients. In the second half of the paper
we study various geometrical properties of the quotients: we compute their Chow
ring and the cohomology ring (in case we work over the complex numbers), both
with rational coefficients. Actually, the cycle map turns out to be an
isomorphism. Further, we construct families of such quotients over reduced base
schemes, and compute again the Chow and the cohomology rings. Finally, we prove
the vanishing of the higher cohomology groups of nef line bundles on the
geometric quotients that we obtain.
