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

TL;DR

This paper explores conifold degenerations of Fano 3-folds as hypersurfaces in toric varieties, linking toric geometry, mirror symmetry, and quantum cohomology to classify and compute invariants of these complex algebraic varieties.

Contribution

It introduces a new framework connecting toric degenerations of Fano 3-folds with hypergeometric series and modular equations, extending mirror symmetry techniques.

Findings

Identifies 166 reflexive polytopes with specific singularity properties.

Shows the hypergeometric series solves a modular D3-equation for Picard number 1.

Proposes a method to compute quantum cohomology for Fano 3-folds with higher Picard number.

Abstract

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.