  We show that F-theory compactifications with abelian gauge factors generally
exhibit a non-trivial global gauge group structure. The geometric origin of
this structure lies with the Shioda map of the Mordell--Weil generators. This
results in constraints on the U(1) charges of non-abelian matter consistent
with observations made throughout the literature. In particular, we find that
F-theory models featuring the Standard Model algebra actually realise the
precise gauge group [SU(3)xSU(2)xU(1)]/Z6. Furthermore, we explore the
relationship between the gauge group structure and geometric (un-)higgsing. In
an explicit class of models, we show that, depending on the global group
structure, an SU(2)xU(1) gauge theory can either unhiggs into an SU(2)xSU(2) or
an SU(3)xSU(2) theory. We also study implications of the charge constraints as
a criterion for the F-theory 'swampland'.
