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

Tomonao Matsumura, Atsuhira Nagano

TL;DR

This work determines the Néron–Severi and transcendental lattices of K3 hypersurfaces in toric threefolds derived from 3‑dimensional Fano polytopes, using carefully chosen Jacobian elliptic fibrations. It introduces the evident lattices $E_k$ generated by a fibre, sections, and fibre components, and proves ${ m NS}(S_k)=E_k$ for all $k=6, ldots,18$, with ${ m Tr}(S_k) extstyleuildrel rianglear oar Uoxplus L_k$, thereby yielding a full Dolgachev mirror pair for these polytopes via ${ m Tr}(S_k) extstyleuildrel rianglear o Uoxplus { m NS}(S_{P^ times})$. The analysis combines explicit elliptic fibrations, period mappings, and discriminant form computations, and yields detailed Mordell–Weil group data across the 18 Fano polytopes. The results establish mirror symmetry for toric K3 hypersurfaces attached to Fano polytopes and provide concrete lattice and MW structures that illuminate the arithmetic of these highly Picard‑rank surfaces. Overall, the paper advances the Dolgachev program in this toric setting by turning lattice‑theoretic insight into explicit geometric and arithmetic results.

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.

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

TL;DR

This work determines the Néron–Severi and transcendental lattices of K3 hypersurfaces in toric threefolds derived from 3‑dimensional Fano polytopes, using carefully chosen Jacobian elliptic fibrations. It introduces the evident lattices generated by a fibre, sections, and fibre components, and proves for all , with , thereby yielding a full Dolgachev mirror pair for these polytopes via . The analysis combines explicit elliptic fibrations, period mappings, and discriminant form computations, and yields detailed Mordell–Weil group data across the 18 Fano polytopes. The results establish mirror symmetry for toric K3 hypersurfaces attached to Fano polytopes and provide concrete lattice and MW structures that illuminate the arithmetic of these highly Picard‑rank surfaces. Overall, the paper advances the Dolgachev program in this toric setting by turning lattice‑theoretic insight into explicit geometric and arithmetic results.

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.
Paper Structure (16 sections, 21 theorems, 28 equations)

This paper contains 16 sections, 21 theorems, 28 equations.

Table of Contents

  1. Mirror pairs of toric $K3$ hypersurfaces from Fano polytopes
  2. Preliminaries and notations
  3. Dolgachev conjecture for toric $K3$ hypersurfaces
  4. $K3$ hypersurfaces with explicit parameters
  5. Jacobian elliptic fibrations and evident lattices
  6. Period mappings and Picard numbers
  7. Lattice structures
  8. Mordell-Weil groups for our elliptic fibrations
  9. $E_k$ as Néron-Severi lattice
  10. Cases of $k=10,14,15$
  11. Cases of $k=7,8,11,18$
  12. Case of $k= 16$
  13. Cases of $k= 9, 13, 17$
  14. Case of $k=6$
  15. Case of $k=12$
  16. ...and 1 more sections

Key Result

Theorem 1

Let $P_k$ ($k\in \{1,\ldots,18\}$) be the Fano polytope in Table 1. Then, the intersection matrix of the transcendental lattice ${\rm Tr}(S_{P_k})$ is isometric to $U\oplus L_k$, where $U$ is the hyperbolic lattice of rank two and $L_k$ is the lattice in Table 1.

Theorems & Definitions (35)

  • Theorem 1
  • Definition 1.1
  • Conjecture 1.1
  • Proposition 1.1
  • Proposition 1.2
  • Remark 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Remark 2.1
  • ...and 25 more