Schubert varieties, toric varieties, and ladder determinantal varieties

We construct certain normal toric varieties (associated to finite distributive lattices) which are degenerations of the Grassmannians. We also determine the singular loci for certain normal toric varieties, namely the ones which are certain ladder determinantal varieties. As a consequence, we prove a refined version of the conjecture of Laksmibai & Sandhya [Criterion for smoothness of Schubert varieties in SL(n)/B, Proc. Ind. Acad. Sci., 100 (1990), 45-52] on the components of the singular locus, for certain Schubert varieties in the flag variety.