Elliptic fibrations on toric $K3$ hypersurfaces and mirror symmetry derived from Fano polytopes

TL;DR

This paper classifies the Néron-Severi lattices of certain K3 hypersurfaces in toric three-folds from Fano polytopes, introduces elliptic fibrations on them, and provides evidence for a mirror symmetry conjecture.

Contribution

It determines the Néron-Severi lattices of K3 hypersurfaces in toric Fano contexts and proves a case of the Dolgachev mirror symmetry conjecture.

Findings

Néron-Severi lattices are generated by fibers, sections, and singular fiber components.

Constructs specific elliptic fibrations on K3 hypersurfaces.

Provides proof supporting the Dolgachev conjecture for Fano polytopes.

Abstract

We determine the N\'eron-Severi lattices of $K3$ hypersurfaces with large Picard number in toric three-folds derived from Fano polytopes. On each $K3$ surface, we introduce a particular elliptic fibration. In the proof of the main theorem, we show that the N\'eron-Severi lattice of each $K3$ surface is generated by a general fibre, sections and appropriately selected components of the singular fibres of our elliptic fibration. Our argument gives a certain proof of the Dolgachev conjecture for Fano polytopes, which is a conjecture on mirror symmetry for $K3$ surfaces.