  We study the partial resolutions of singularities related to Hilbert schemes
of points on an affine space. Consider a quotient of a vector space $V$ by an
action of a finite group $G$ of linear transforms. Under some additional
assumptions, we prove that the partial desingularization of Hilbert type is
smooth only if the action of $G$ is generated by complex reflections. This is
used to study the subvarieties of a Hilbert scheme of a complex torus. We show
that any subvariety of a generic deformation of a Hilbert scheme of a torus is
birational to a quotient of another torus by an action of a Weyl group of some
semisimple Lie algebra. In Appendix, we produce counterexamples to a false
theorem stated in our preprint math.AG/9801038.
