Six line configurations and string dualities

TL;DR

This paper explores special K3 surface families related to six-line configurations, providing explicit models and a geometric interpretation of string dualities, leading to new F-theory models in eight dimensions.

Contribution

It introduces explicit Weierstrass models for K3 families with specific lattice polarization and interprets string dualities geometrically through two-isogenies.

Findings

Explicit Weierstrass models for K3 families derived.

Geometric two-isogeny offers a new perspective on string dualities.

New F-theory models dual to non-geometric heterotic compactifications.

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.