Elliptic fibrations on toric $K3$ hypersurfaces and mirror symmetry derived from Fano polytopes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916752847273984 |
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| author | Matsumura, Tomonao Nagano, Atsuhira |
| author_facet | Matsumura, Tomonao Nagano, Atsuhira |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_01465 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Elliptic fibrations on toric $K3$ hypersurfaces and mirror symmetry derived from Fano polytopes Matsumura, Tomonao Nagano, Atsuhira Algebraic Geometry 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. |
| title | Elliptic fibrations on toric $K3$ hypersurfaces and mirror symmetry derived from Fano polytopes |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2208.01465 |