Picard group of hypersurfaces in toric 3-folds

TL;DR

This paper generalizes a criterion for generic hypersurfaces to have the same Picard number as their ambient toric variety, with specific results for K3 surfaces in Fano toric 3-folds.

Contribution

It extends the Picard number criterion to hypersurfaces in complete simplicial toric varieties, including K3 surfaces in Fano toric 3-folds.

Findings

Criterion applies to hypersurfaces in simplicial toric varieties

Generic K3 surfaces in Fano toric 3-folds satisfy the criterion

Provides a broader understanding of Picard groups in toric hypersurfaces

Abstract

We show that the usual sufficient criterion for a generic hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to hypersurfaces in complete simplicial toric varieties. This sufficient condition is always satisfied by generic K3 surfaces embedded in Fano toric 3-folds.