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. Moreover, by using the explicit description of this affine chart, we resolve the singularities of our local model. By combining results of Kisin and Pappas, this leads to the construction of regular p-adic integral models for the corresponding orthogonal Shimura varieties.
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