Decidability of singularities in the Ekedahl--Oort stratification

TL;DR

This paper investigates the regularity and singularities of Ekedahl--Oort strata in Shimura varieties, providing criteria, algorithms, and connections to stacks of G-zips, enhancing understanding of their geometric structure.

Contribution

It characterizes codimension one regularity of Ekedahl--Oort strata via Frobenius action and introduces an algorithm to detect singularities, linking to G-zip stacks for split types.

Findings

Regularity in codimension one is characterized by Frobenius action.

An algorithm for detecting codimension one singularities is developed.

Connections between singularities and G-zip stacks are established.

Abstract

For an abelian type Shimura variety and an odd prime $p$ of good reduction, we characterize the regularity in codimension one of Zariski closures of Ekedahl--Oort strata in terms of the Frobenius action on the root datum. We give an algorithm that detects codimension one singularities for arbitrary Ekedahl--Oort strata. When the Shimura datum is of split type, we relate the singularities of Ekedahl--Oort strata to a stack of $G$-zips over the complex numbers. We study the existence of generalized Hasse invariants on this stack.