K3 surfaces without section as double covers of Halphen surfaces, and F-theory compactifications

We construct several examples of genus-one fibered K3 surfaces without a global section with type $I_{n}$ fibers, by considering double covers of a special class of rational elliptic surfaces lacking a global section, known as Halphen surfaces of index 2. The resulting K3 surfaces have bisection geometries. F-theory compactifications on these K3 genus-one fibrations without a section times a K3 yield models that have $SU(n)$ gauge symmetries with a discrete $\mathbb{Z}_2$ symmetry.