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On K3 surfaces of Picard rank 14

Adrian Clingher, Andreas Malmendier

TL;DR

This work classifies complex K3 surfaces with finite automorphism groups that are polarized by rank-14 2-elementary lattices, identifying the three possibilities $P_{14}$, $P'_{14}$, and $P''_{14}$ and providing explicit quartic birational models for each. It describes their coarse moduli spaces via modular invariants, constructs explicit Jacobian elliptic fibrations, and determines the dual graphs of smooth rational curves, including duals under the van Geemen-Sarti-Nikulin construction. A detailed analysis of the P-, P'-, and P''-polarized families is given, along with Vinberg’s $P''_{14}$-polarized model and the associated double-sextic dual surfaces $\mathcal{Y}$, which are shown to be intimately connected through Nikulin duality. The results illuminate the interplay between lattice polarizations, elliptic fibrations, and modular moduli in high-Picard-rank K3 surfaces, with implications for related string-geometry constructions and explicit geometric realizations via quartic and Vinberg-type models.

Abstract

We study complex algebraic K3 surfaces with finite automorphism groups and polarized by rank-fourteen, 2-elementary lattices. Three such lattices exist, namely $H \oplus E_8(-1) \oplus A_1(-1)^{\oplus 4}$, $H \oplus E_8(-1) \oplus D_4(-1)$, and $H \oplus D_8(-1) \oplus D_4(-1)$. As part of our study, we provide birational models for these surfaces as quartic projective hypersurfaces and describe the associated coarse moduli spaces in terms of suitable modular invariants. Additionally, we explore the connection between these families and dual K3 families related via the Nikulin construction.

On K3 surfaces of Picard rank 14

TL;DR

This work classifies complex K3 surfaces with finite automorphism groups that are polarized by rank-14 2-elementary lattices, identifying the three possibilities , , and and providing explicit quartic birational models for each. It describes their coarse moduli spaces via modular invariants, constructs explicit Jacobian elliptic fibrations, and determines the dual graphs of smooth rational curves, including duals under the van Geemen-Sarti-Nikulin construction. A detailed analysis of the P-, P'-, and P''-polarized families is given, along with Vinberg’s -polarized model and the associated double-sextic dual surfaces , which are shown to be intimately connected through Nikulin duality. The results illuminate the interplay between lattice polarizations, elliptic fibrations, and modular moduli in high-Picard-rank K3 surfaces, with implications for related string-geometry constructions and explicit geometric realizations via quartic and Vinberg-type models.

Abstract

We study complex algebraic K3 surfaces with finite automorphism groups and polarized by rank-fourteen, 2-elementary lattices. Three such lattices exist, namely , , and . As part of our study, we provide birational models for these surfaces as quartic projective hypersurfaces and describe the associated coarse moduli spaces in terms of suitable modular invariants. Additionally, we explore the connection between these families and dual K3 families related via the Nikulin construction.

Paper Structure

This paper contains 38 sections, 39 theorems, 204 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction and summary of results
  2. The P-polarized K3 surfaces
  3. The P'-polarized K3 surfaces
  4. The P"-polarized K3 surfaces
  5. Motivation and general overview
  6. Acknowledgments
  7. Lattice theoretic considerations for certain K3 surfaces
  8. Jacobian elliptic fibrations
  9. Coarse moduli spaces
  10. The case of P-polarized K3 surfaces
  11. The case of P'-polarized K3 surfaces
  12. The case of H + D8 + E8-polarized K3 surfaces
  13. Projective models for certain K3 surfaces
  14. Projective model for P-polarized K3 surfaces
  15. Fibration (1)
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $(\alpha, \beta, \gamma, \delta , \varepsilon, \zeta, \eta, \iota, \kappa, \lambda) \in \mathbb{C}^{10}$. Consider the projective surface in $\mathbb{P}^3=\mathbb{P}(\mathbf{X}, \mathbf{Y}, \mathbf{Z}, \mathbf{W})$ defined by the homogeneous quartic equation Assuming general parameters, the surface $\mathcal{X}$ obtained as the minimal resolution of (quartic1) is a K3 surface endowed with a c

Figures (17)

  • Figure 1: Rational curves on $\mathcal{X}$ with $\mathrm{NS}(\mathcal{X})=P_{14}$
  • Figure 2: The alternate fibration on $\mathcal{X}$
  • Figure 3: The standard fibration on $\mathcal{X}$
  • Figure 4: The base-fiber dual fibration on $\mathcal{X}$ (using $L_3$)
  • Figure 5: The base-fiber dual fibration on $\mathcal{X}$ (using $L_5$)
  • ...and 12 more figures

Theorems & Definitions (76)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 66 more