  Let k be a field of positive characteristic. We construct, for each dominant
coweight \lambda of the standard maximal torus in the special linear group, a
closed subvariety D(\lambda) of the multigraded Hilbert scheme of an affine
space over k, such that the k-valued points of D(\lambda) can be interpreted as
lattices in k((z))^n endowed with infinitesimal structure. Moreover, for any
\lambda we construct a universal homeomorphism from D(\lambda) to a Demazure
resolution of the Schubert variety associated with \lambda in the affine
Grassmannian. Lattices in D(\lambda) have non-trivial infinitesimal structure
if and only if they lie over the boundary of the big cell.
