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

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. Using toric mirror symmetry, we define a r-dimensional generalized hypergeometric power series associated to the dual reflexive polytope. We show that if r =1 then this series is a normalized regular solution of a modular D3-equation that appears in the Golyshev correspondence. We expect that the multidimensional power series can be used to compute the small quantum cohomology ring of all Fano 3-folds Y with the Picard number r >1 if Y admit a conifold degeneration X.