  We study F-theory compactifications with up to two Abelian gauge group
factors that are based on elliptically fibered Calabi-Yau 4-folds describable
as generic hypersurfaces. Special emphasis is put on elliptic fibrations based
on generic Bl^2 P^2[3]-fibrations. These exhibit a Mordell-Weil group of rank
two corresponding to two extra rational sections which give rise to two Abelian
gauge group factors. We show that an alternative description of the same
geometry as a complete intersection makes the existence of a holomorphic
zero-section manifest, on the basis of which we compute the U(1) generators and
a class of gauge fluxes. We analyse the fibre degenerations responsible for the
appearance of localised charged matter states, whose charges, interactions and
chiral index we compute geometrically. We implement an additional SU(5) gauge
group by constructing the four inequivalent toric tops giving rise to SU(5) x
U(1) x U(1) gauge symmetry and analyse the matter content. We demonstrate that
notorious non-flat points can be avoided in well-defined Calabi-Yau 4-folds.
These methods are applied to the remaining possible hypersurface fibrations
with one generic Abelian gauge factor. We analyse the local limit of our SU(5)
x U(1) x U(1) models and show that one of our models is not embeddable into E8
due to recombination of matter curves that cannot be described as a Higgsing of
E8. We argue that such recombination forms a general mechanism that opens up
new model building possibilities in F-theory.
