  The paper concerns the affine varieties that are homogeneous with respect to
a (non-standard) graduation over the group Z^m. Among the other properties it
is shown that every such a variety can be embedded in its Zariski tangent space
at the origin, so that it is smooth if and only if it is isomorphic to an
affine space. The results directly apply to the study of Hilbert schemes of
subvarieties in P^n.
