Abstract: We consider Shimura varieties associated to a unitary group of signature (n-1,1). For these varieties, we construct p-adic integral models over odd primes p which ramify in the imaginary quadratic field with level subgroup at p given by the stabilizer of a vertex lattice in the hermitian space. Our models are given by a variation of the construction of the splitting models of Pappas-Rapoport and they have a simple moduli theoretic description. By an explicit calculation, we show that these splitting models are normal, flat, Cohen-Macaulay and with reduced special fiber. In fact, they have relatively simple singularities: we show that a single blow-up along a smooth codimension one subvariety of the special fiber produces a semi-stable model. This also implies the existence of semi-stable models of the corresponding Shimura varieties.

Abstract: We consider Shimura varieties associated to a unitary group of signature (n-s,s) where n is even. For these varieties, we construct smooth p-adic integral models for s=1 and regular p-adic integral models for s=2 and s=3 over odd primes p which ramify in the imaginary quadratic field with level subgroup at p given by the stabilizer of a π-modular lattice in the hermitian space. Our construction,  which has an explicit moduli-theoretic description, is given by an explicit resolution of a corresponding local model.

Abstract: We consider Shimura varieties associated to a unitary group of signature (n-2,2). We give regular p-adic integral models for these varieties over odd primes p which ramify in the imaginary quadratic field with level subgroup at p given by the stabilizer of a selfdual lattice in the hermitian space. Our construction is given by an explicit resolution of a corresponding local model.

Abstract: We consider Shimura varieties for orthogonal or spin groups acting on hermitian symmetric domains of type IV. We give regular p-adic integral models for these varieties over odd primes p at which the level subgroup is the connected stabilizer of a vertex lattice in the orthogonal space. Our construction is obtained by combining results of Kisin and the first author with an explicit presentation and resolution of a corresponding local model.

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.