On orthogonal local models of Hodge type

TL;DR

This paper investigates the local geometric structure of orthogonal Shimura varieties at bad primes, providing explicit equations for local models and proving their reducedness and Cohen-Macaulay property.

Contribution

It introduces explicit equations for orthogonal local models using the Pappas-Zhu construction, revealing their hypersurface structure and singularity properties.

Findings

Local models are hypersurfaces in determinantal schemes.

The special fiber of the local model is reduced.

The local model is Cohen-Macaulay.

Abstract

We study local models that describe the singularities of Shimura varieties of non-PEL type for orthogonal groups at primes where the level subgroup is given by the stabilizer of a single lattice. In particular, we use the Pappas-Zhu construction and we give explicit equations that describe an open subset around the "worst" point of orthogonal local models given by a single lattice. These equations display the affine chart of the local model as a hypersurface in a determinantal scheme. Using this we prove that the special fiber of the local model is reduced and Cohen-Macaulay.