Singularities of a certain class of toric varieties

Abstract

Hibi considered the class of algebras $k[\mathcal{L}]=k[x{\alpha} \res \alpha \in \mathcal{L}]$ with straightening laws associated to a finite distributive lattice $\mathcal{L}$ in his paper \cite{hibi}. In that paper he proves that these algebras are normal and integral domains. This result along with the work of Sturmfels and Eisenbud \cite{ES} on binomial prime ideals implies that the affine varieties associated to the algebra $k[\mathcal{L}]$ are normal toric varieties. In the present work we will consider the toric variety $X(\mathcal{L})=\mathrm{spec}( k[\mathcal{L}])$, 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 $\widehat{T{x\tau}}$ where $x\tau$ is a singular point associated to a torus orbit $O_\tau$ for the action of the torus $T$, where $\tau$ is a face of the cone $\sigma$