Compactification of a Drinfeld Period Domain over a Finite Field

TL;DR

This paper investigates a natural compactification of the Drinfeld period domain over finite fields, analyzing its geometric structure, singularities, and modular interpretation, and constructs a smooth desingularization.

Contribution

It introduces a new compactification of the Drinfeld period domain, studies its geometric properties, and provides a modular interpretation and a smooth desingularization.

Findings

The compactification is normal but singular along boundary strata of codimension ≥2.

The boundary consists of smaller rank period domains glued in a dual manner.

A smooth projective desingularization with a normal crossings boundary divisor is constructed.

Abstract

We study a certain compactification of the Drinfeld period domain over a finite field which arises naturally in the context of Drinfeld moduli spaces. Its boundary is a disjoint union of period domains of smaller rank, but these are glued together in a way that is dual to how they are glued in the compactification by projective space. This compactification is normal and singular along all boundary strata of codimension $\ge2$. We study its geometry from various angles including the projective coordinate ring with its Hilbert function, the cohomology of twisting sheaves, the dualizing sheaf, and give a modular interpretation for it. We construct a natural desingularization which is smooth projective and whose boundary is a divisor with normal crossings. We also study its quotients by certain finite groups.