  We study F-Theory compactifications to four dimensions that exhibit discrete
gauge symmetries. Geometrically these arise by deforming elliptic fibrations
with two sections to a genus-one fibration with a bi-section. From a
four-dimensional field-theory perspective they are remnant symmetries from a
Higgsed U(1) gauge symmetry. We implement such symmetries in the presence of an
additional SU(5) symmetry and associated matter fields, giving a geometric
prescription for calculating the induced discrete charge for the matter curves
and showing the absence of Yukawa couplings that are forbidden by this charge.
We present a detailed map between the field theory and the geometry, including
an identification of the Higgs field and the massless states before and after
the Higgsing. Finally we show that the Higgsing of the U(1) induces a G-flux
which precisely accounts for the change in the Calabi-Yau Euler number so as to
leave the D3 tadpole invariant.
