On $p$-adic integral moduli schemes and local models for PEL type D

We construct flat integral moduli schemes of PEL type D and the corresponding flat orthogonal Rapoport–Zink spaces with parahoric level structure over a $p$-adic integer ring. The construction relies on proving a conjecture of Pappas–Rapoport: for an even orthogonal similitude group over a complete discretely valued field of residue characteristic $p>2$, and for arbitrary parahoric level, the associated spin local model is flat, normal, Cohen–Macaulay, with reduced special fiber. In the course of the proof, we also show that in the quasi-split but non-split case, the Rapoport–Zink (naive) local model is topologically flat, verifying a conjecture of Pappas–Rapoport–Smithling. In the maximal parahoric case, we also describe the Schubert varieties in the special fiber in moduli-theoretic terms. Finally, for a maximal parahoric case we construct an explicit regular semi-stable model by blowing up the spin local model along the unique closed Schubert cell in its special fiber.