Singularities of toric varieties associated with finite distributive lattices

Wagner, David G.

Journal of Algebraic Combinatorics, Tome 5 (1996) no. 2, pp. 149-165.

Résumé

Summary: With each finite lattice $L$ we associate a projectively embedded scheme $V( L)$; as Hibi has shown, the lattice $D$ is distributive if and only if $V( D)$ is irreducible, in which case it is a toric variety. We first apply Birkhoff"s structure theorem for finite distributive lattices to show that the orbit decomposition of $V( D)$ gives a lattice isomorphic to the lattice of contractions of the bounded poset of join-irreducibles [^$( P)] \hat P$ of $D$. Then we describe the singular locus of $V( D)$ by applying some general theory of toric varieties to the fan dual to the order polytope of $P: V( D)$ is nonsingular along an orbit closure if and only if each fibre of the corresponding contraction is a tree. Finally, we examine the local rings and associated graded rings of orbit closures in $V( D)$. This leads to a second (self-contained) proof that the singular locus is as described, and a similar combinatorial criterion for the normal link of an orbit closure to be irreducible.

Détail

Keywords: toric variety, distributive lattice, singular locus, associated graded ring