Deformations of toric pairs and the smoothing of Batyrev Calabi-Yau threefolds

Let $Δ$ be a reflexive four-polytope with primitive edges. We construct deformations of the toric fourfold defined by its face fan that smooth the singularities of a generic anticanonical Calabi-Yau hypersurface over a chosen two-dimensional face. The construction applies when the face cone is smoot