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On two-elementary K3 surfaces with finite automorphism group

Adrian Clingher, Andreas Malmendier, Flora Poon

TL;DR

This work classifies and geometrically realizes two-elementary, finite-automorphism K3 surfaces that carry a Jacobian elliptic fibration with a two-torsion section. By leveraging $L$-polarized lattice theory, van Geemen–Sarti and Nikulin involutions, and explicit quartic and double-sextic models, the authors construct and connect rank-$11$, rank-$12$, and rank-$13$ families to a rank-$10$ base, yielding detailed Jacobian fibrations and their frame structures. They compute the dual graphs of all smooth rational curves, determine polarizing divisors, and embed reducible fibers into these dual graphs, providing concrete geometric realizations via pencils and Inose-type quartics. The results extend the Inose/quartic approach to intermediate Picard ranks, illuminating how lattice data, elliptic fibrations, and explicit projective models interact in low-rank, $2$-elementary K3 surfaces, and offering comprehensive data on fibrations, automorphisms, and curve configurations.

Abstract

We study complex algebraic K3 surfaces of Picard ranks 11,12, and 13 of finite automorphism group that admit a Jacobian elliptic fibration with a section of order two. We prove that the K3 surfaces admit a birational model isomorphic to a projective quartic hypersurface and construct geometrically the frames of all supported Jacobian elliptic fibrations. We determine the dual graphs of all smooth rational curves for these K3 surfaces, the polarizing divisors, and the embedding of the reducible fibers in each frame into the corresponding dual graph.

On two-elementary K3 surfaces with finite automorphism group

TL;DR

This work classifies and geometrically realizes two-elementary, finite-automorphism K3 surfaces that carry a Jacobian elliptic fibration with a two-torsion section. By leveraging -polarized lattice theory, van Geemen–Sarti and Nikulin involutions, and explicit quartic and double-sextic models, the authors construct and connect rank-, rank-, and rank- families to a rank- base, yielding detailed Jacobian fibrations and their frame structures. They compute the dual graphs of all smooth rational curves, determine polarizing divisors, and embed reducible fibers into these dual graphs, providing concrete geometric realizations via pencils and Inose-type quartics. The results extend the Inose/quartic approach to intermediate Picard ranks, illuminating how lattice data, elliptic fibrations, and explicit projective models interact in low-rank, -elementary K3 surfaces, and offering comprehensive data on fibrations, automorphisms, and curve configurations.

Abstract

We study complex algebraic K3 surfaces of Picard ranks 11,12, and 13 of finite automorphism group that admit a Jacobian elliptic fibration with a section of order two. We prove that the K3 surfaces admit a birational model isomorphic to a projective quartic hypersurface and construct geometrically the frames of all supported Jacobian elliptic fibrations. We determine the dual graphs of all smooth rational curves for these K3 surfaces, the polarizing divisors, and the embedding of the reducible fibers in each frame into the corresponding dual graph.
Paper Structure (15 sections, 18 theorems, 70 equations, 13 figures, 1 table)

This paper contains 15 sections, 18 theorems, 70 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Elliptic fibrations of L-polarized K3 surfaces
  4. K3 surfaces with van Geemen-Sarti involution
  5. K3 surfaces with a different rank 10 polarization
  6. Double Sextics model
  7. Projective quartic model
  8. Construction of elliptic fibrations of families of Picard rank 11, 12 and 13
  9. Family of Picard rank 11
  10. Family of Picard rank 12
  11. Family of Picard rank 13
  12. Dual graphs of rational curves
  13. Family of Picard rank 11
  14. Family of Picard rank 12
  15. Family of Picard rank 13

Key Result

Theorem 2.1

Let $\mathcal{X}$ be a K3 surface satisfying Conditions cond_a-cond_c in Section sec_intro. Then, up to isometry, the lattice $L$ belongs to the list given in Table tab:extension. Table tab:extension also includes the frames of all possible Jacobian elliptic fibrations supported on $\mathcal{X}$.

Figures (13)

  • Figure 1: Rational curves on $\mathcal{X}$ with Néron-Severi lattice $H \oplus N$
  • Figure 2: Dual graph of all smooth rational curves with 1-fold (thin) and some 2-fold (thick) edges for $\mathrm{NS}(\mathcal{X}) = H \oplus D_4(-1) \oplus A_1(-1)^{\oplus 5}$
  • Figure 3: Dual graph of all smooth rational curves with 1-fold (thin) and a few 2-fold (thick) edges for $\mathrm{NS}(\mathcal{X}) = H \oplus D_6(-1) \oplus A_1(-1)^{\oplus 4}$
  • Figure 4: Dual graph of all smooth rational curves with 1-fold (thin) and some 2-fold (thick) edges for $\mathrm{NS}(\mathcal{X}) = H \oplus D_8(-1) \oplus A_1(-1)^{\oplus 3}$
  • Figure 5: Fibration with reducible fibers ${\color{magenta} \widetilde{A}_1}^{\oplus 9}$ and sections
  • ...and 8 more figures

Theorems & Definitions (49)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 39 more