I‑AIMS Institute for Artificial Intelligence
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Six line configurations and string dualities

Adrian Clingher, Andreas Malmendier, Tanush Shaska

I-AIMS paper 2018-3 · Published in Comm. Math. Phys. 371 (2019), 159–196 · 2019

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 ⊕ E₇(−1) ⊕ E₇(−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.

MSC classes: 11F03, 14J28, 14J81

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