Elliptic fibrations on toric hypersurfaces and mirror symmetry derived from Fano polytopes
arXiv:2208.01465 · doi:10.1007/s10711-025-01014-5
Abstract
We determine the Néron-Severi lattices of hypersurfaces with large Picard number in toric three-folds derived from Fano polytopes. On each surface, we introduce a particular elliptic fibration. In the proof of the main theorem, we show that the Néron-Severi lattice of each 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 surfaces.
24 pages, v1 was revised thoroughly, the final version will be published in Geometriae Dedicata, the supplementary data can be found here