The Normality of Certain Varieties of Special Lattices

TL;DR

This paper investigates the geometric properties of lattice varieties, establishing their normality, describing their structure, and comparing them to function field cases, with implications for their canonical bundles.

Contribution

It constructs a flat cover of lattice varieties, proves their local complete intersection and normality, and compares their structure to the function field case using p-linear algebra.

Findings

Lattice varieties are normal and locally complete intersections.

A flat cover of lattice varieties is constructed and shown to be a complete intersection.

The canonical bundle of lattice varieties is described, highlighting differences from affine Grassmannian objects.

Abstract

We begin with a short exposition of the theory of lattice varieties. This includes a description of their orbit structure and smooth locus. We construct a flat cover of the lattice variety and show that it is a complete intersection. We show that the lattice variety is locally a complete intersection and nonsingular in codimension one and hence normal. We then prove a comparison theorem showing that this theory becomes parallel to the function field case if linear algebra is replaced by $p$-linear algebra. We then compute the Lie algebra of the special linear group over the truncated Witt vectors. We conclude by applying these results to show how to describe the canonical bundle on the lattice variety and we use the description to show that lattice varieties are not isomorphic to the analogous objects in the affine Grassmannian.