Negative deformations of toric singularities that are smooth in   codimension two
  Given a polyhedral cone sigma with smooth two-dimensional faces and, moreover, a lattice point R in the dual cone of sigma, we describe the part of the versal deformation of the associated toric variety TV(sigma) that is built from the deformation parameters of multidegree R.   Let Q the polyhedron obtained by intersecting sigma with the hyperplane R=1. Then the base space is (the germ of) an affine scheme that reflects certain possibilities of splitting Q into Minkowski summands. 
Klaus Altmann, Lars Kastner
math.AG