$SU(n) \times \mathbb{Z}_2$ in F-theory on K3 surfaces without section as double covers of Halphen surfaces

We investigate F-theory models with a discrete $\mathbb{Z}_2$ gauge symmetry and $SU(n)$ gauge symmetries. We utilize a class of rational elliptic surfaces lacking a global section, known as Halphen surfaces of index 2, to yield genus-one fibered K3 surfaces with a bisection, but lacking a global section. We consider F-theory compactifications on these K3 surfaces times a K3 surface to build such models. We construct Halphen surfaces of index 2 with type $I_n$ fibers, and we take double covers of these surfaces to obtain K3 surfaces without a section with two type $I_n$ fibers, and K3 surfaces without a section with a type $I_{2n}$ fiber. We study these models to advance the understanding of gauge groups that form in F-theory compactifications on the moduli of bisection geometries. Our results also show that the Halphen surfaces of index 2 can have type $I_n$ fibers up to $I_9$. We construct an example of such a surface and determine the complex structure of the Jacobian of this surface. This allows us to precisely determine the non-Abelian gauge groups that arise in F-theory compactifications on genus-one fibered K3 surfaces obtained as double covers of this Halphen surface of index 2, with a type $I_9$ fiber times a K3 surface. We also determine the $U(1)$ gauge symmetries for compactifications when K3 surfaces as double covers of Halphen surfaces with type $I_9$ fiber are ramified over a smooth fiber.