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

There exist exactly 166 4-dimensional reflexive polytopes Delta such that the corresponding 4-dimensional Gorenstein toric Fano variety P-Delta. has at worst terminal singularities in codimension 3 and the anticanonical divisor of P-Delta is divisible by 2 in its Picard group. For every such a polytope Delta, one naturally obtains a family F(Delta) of Fano hypersurfaces X subset of P-Delta. with at worst conifold singularities. A generic 3-dimensional Fano hypersurface X is an element of F(Delta) 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 Phi associated to the dual reflexive polytope Delta*. We show that if r = 1 then F is a normalized regular solution of a modular D3-equation that appears in the Golyshev correspondence. We expect that the power series Phi can be used to compute the small quantum cohomology ring of all Fano 3-folds Y with the Picard number r >= 2 if Y admits a conifold degeneration X is an element of F(Delta).