Singular loci of Grassmann-Hibi toric varieties

For the toric variety X associated to the Bruhat poset of Schubert varieties in the Grassmannian, we describe the singular locus in terms of the faces of the associated polyhedral cone. We also determine the tangent cones at the maximal singularities of X. These turn out to be again toric varieties. In the case of X being associated to the Bruhat poset of Schubert varieties in the Grassmannian of 2-planes in a n-dimensional vector space (over the base field), we also prove a certain product formula relating the multiplicities at certain singular points.