Conifold degenerations of Fano 3-folds as hypersurfaces in toric varieties

Understand

There exist exactly 166 4-dimensional reflexive polytopes such that the corresponding 4-dimensional Gorenstein toric Fano varieties have at worst terminal singularities in codimension 3 and their anticanonical divisor is divisible by 2.

• For every such a polytope, one naturally obtains a family of Fano hypersurfaces X with at worst conifold singularities.
• A generic 3-dimensional Fano hypersurface X can be interpreted as a flat conifold degeneration of some smooth Fano 3-folds Y whose classification up to deformation was obtained by Iskovskikh, Mori and Mukai.
• In this case, both Fano varieties X and Y have the same Picard number r.