Six line configurations and string dualities

Abstract

We study the family of K3 surfaces of Picard rank sixteen associated with the double cover of the projective plane branched along the union of six lines, and the family of its Van Geemen-Sarti partners, i.e., K3 surfaces with special Nikulin involutions, such that quotienting by the involution and blowing up recovers the former. We prove that the family of Van Geemen-Sarti partners is a four-parameter family of K3 surfaces with $H\oplus E_7(-1)\oplus E_7(-1)$ lattice polarization. We describe explicit Weierstrass models on both families using even modular forms on the bounded symmetric domain of type $IV$. We also show that our construction provides a geometric interpretation, called geometric two-isogeny, for the F-theory/heterotic string duality in eight dimensions. As a result, we obtain novel F-theory models, dual to non-geometric heterotic string compactifications in eight dimensions with two non-vanishing Wilson line parameters.

42 pages; minor typos corrected in version 2

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• The Satake sextic in elliptic fibrations on K3
• Non-Geometric F-Theory-Heterotic Duality
• Rational points in the moduli space of genus two
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• Coble fourfold, $S_6$-invariant quartic threefolds, and Wiman-Edge sextics
• Machine learning for moduli space of genus two curves and an application to isogeny based cryptography
• Normal forms for Kummer surfaces
• On K3 surfaces of Picard rank 14
• Inverse period mappings of $K3$ surfaces and a construction of modular forms for a lattice with the Kneser conditions
• On the mixed-twist construction and monodromy of associated Picard-Fuchs systems
• On the duality of F-theory and the CHL string in seven dimensions
• Geometry of Prym varieties for special bielliptic curves of genus three and five
• On holomorphic conformal structures associated with lattice polarized K3 surfaces
• Eight-dimensional non-geometric heterotic strings and enhanced gauge groups
• New perspectives in the duality of M-theory, heterotic strings, and F-theory
• Sequence of families of lattice polarized $K3$ surfaces, modular forms and degrees of complex reflection groups
• K3 Surfaces and Orthogonal Modular Forms