  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.
