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Six Line Configurations and String Dualities


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.

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Clingher, A., Malmendier, A. & Shaska, T. Six Line Configurations and String Dualities. Commun. Math. Phys. 371, 159–196 (2019).

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