  We show that genus-one fibrations lacking a global section fit naturally into
the geometric moduli space of Weierstrass models. Elliptic fibrations with
multiple sections (nontrivial Mordell-Weil rank), which give rise in F-theory
to abelian U(1) fields, arise as a subspace of the set of genus-one fibrations
with multisections. Higgsing of certain matter multiplets charged under abelian
gauge fields in the corresponding supergravity theories break the U(1) gauge
symmetry to a discrete gauge symmetry group. We further show that in six
dimensions every U(1) gauge symmetry arising in an F-theory model can be found
by Higgsing an SU(2) gauge symmetry with adjoint matter, and that a similar
structure holds for F-theory geometries giving 4D supergravity theories.
