Demazure resolutions as varieties of lattices with infinitesimal structure

TL;DR

This paper constructs a geometric framework linking Demazure resolutions to lattices with infinitesimal structure in positive characteristic, providing new insights into their algebraic and geometric properties.

Contribution

It introduces a novel construction of subvarieties of the multigraded Hilbert scheme representing lattices with infinitesimal structure and relates them to Demazure resolutions.

Findings

Established a universal homeomorphism between D(λ) and Demazure resolutions.

Characterized lattices with non-trivial infinitesimal structure as lying over the boundary of the big cell.

Connected lattice structures with geometric resolutions in the affine Grassmannian.

Abstract

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.