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

Demazure resolutions as varieties of lattices with infinitesimal structure