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

2022/08/02 by Tomonao Matsumura, Matsumura, Tomonao, Atsuhira Nagano +1
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2208.01465

openalex publication_date 2022/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We determine the Néron-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 Néron-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.

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