On singularities of lattice varieties

TL;DR

This paper investigates the singularities of lattice varieties derived from distributive lattices, establishing conditions for smoothness and relating the number of diamonds incident on lattice points to the codimension.

Contribution

It proves a lower bound on the number of diamonds incident on lattice points and demonstrates smoothness of lattice varieties from product of chain lattices.

Findings

Number of diamonds incident on a lattice point ≥ codimension

Lattice varieties from product of chain lattices are smooth

Provides criteria for singularities in lattice varieties

Abstract

Toric varieties associated with distributive lattices arise as a fibre of a flat degeneration of a Schubert variety in a minuscule. The singular locus of these varieties has been studied by various authors. In this article we prove that the number of diamonds incident on a lattice point $\a$ in a product of chain lattices is more than or equal to the codimension of the lattice. Using this we also show that the lattice varieties associated with product of chain lattices is smooth.