Singularities of a certain class of toric varieties

Hibi considered the class of algebras k[ L ] = k[xalpha|alpha ∈ L ] with straightening laws associated to a finite distributive lattice L in his paper [2]. In that paper he proves that these algebras are normal and integral domains. This result along with the work of Sturmfels and Eisenbud [7] on binomial prime ideals implies that the affine varieties associated to the algebra k[ L ] are normal toric varieties. In the present work we will consider the toric variety X( L ) = spec(k[ L ]), we will give the combinatorial description of the cone sigma associated to it. The final result will be to give a standard monomial basis for the tangent cone Txt&d14; where xtau is a singular point associated to a torus orbit Otau for the action of the torus T, where tau is a face of the cone sigma...