F-theory models with $U(1)\times {\mathbb{Z}}_2,{\mathbb{Z}}_4$ and transitions in discrete gauge groups

We examine the proposal in the previous paper to resolve the puzzle in transitions in discrete gauge groups. We focus on a four-section geometry to test the proposal. We observed that a discrete ${\mathbb{Z}}_2$ gauge group enlarges and $U(1)$ also forms in F-theory along any bisection geometries locus in the four-section geometry built as the complete intersections of two quadrics in ${\mathbb{P}}^3$ fibered over any base. Furthermore, we demonstrate that giving vacuum expectation values to hypermultiplets breaks the enlarged $U(1)\times {\mathbb{Z}}_2$ gauge group down to a discrete ${\mathbb{Z}}_4$ gauge group via Higgsing. We thus confirmed that the proposal in the previous paper is consistent when a four-section splits into a pair of bisections in the four-section geometry. This analysis may be useful for understanding the Higgsing processes occurring in the transitions in discrete gauge groups in six-dimensional F-theory models. We also discuss the construction of a family of six-dimensional F-theory models in which $U(1)\times {\mathbb{Z}}_4$ forms.