# On the bad reduction of certain U(2, 1) Shimura varieties

## TL;DR

This paper investigates the structure, singularities, and stratifications of three related Picard modular surfaces in characteristic p, focusing on their components and morphisms, especially around superspecial points.

## Contribution

It provides a detailed analysis of the components, singularities, and stratifications of Picard modular surfaces and introduces a foliation on the blow-up at superspecial points.

## Key findings

- Classification of components and singularities of the surfaces.
- Description of stratifications and their behavior under morphisms.
- Introduction of a foliation on the blow-up at superspecial points.

## Abstract

Let $E$ be a quadratic imaginary field and let $p$ be a prime which is inert in $E.$ We study three types of Picard modular surfaces in positive characteristic $p$ and the morphisms between them. The first Picard surface, denoted $S$, parametrizes triples $(A,\phi,\iota)$ comprised of an abelian threefold $A$ with an action $\iota$ of the ring of integers $\mathcal{O}_{E}$, and a principal polarization $\phi$. The second surface, $S_{0}(p)$, parametrizes, in addition, a suitably restricted choice of a subgroup $H\subset A[p]$ of rank $p^{2}$. The third Picard surface, $\widetilde{S}$, parametrizes triples $(A,\psi,\iota)$ similar to those parametrized by $S$, but where $\psi$ is a polarization of degree $p^{2}$. We study the components, singularities and naturally defined stratifications of these surfaces, and their behaviour under the morphisms. A particular role is played by a foliation we define on the blow-up of $S$ at its superspecial points.