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

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, ϕ, ι) comprised of an abelian threefold A with an action ι of the ring of integers O_E, and a principal polarization ϕ. The second surface, S₀(p), parametrizes, in addition, a suitably restricted choice of a subgroup H⊂ A[p] of rank p². The third Picard surface, S~, parametrizes triples (A, ψ, ι) similar to those parametrized by S, but where ψ is a polarization of degree p². We study the components, singularities and naturally defined stratifications of these surfaces, and their behavior under the morphisms. A particular role is played by a foliation we define on the blowup of S at its superspecial points.