  We study aspects of heterotic/F-theory duality for compactifications with
Abelian gauge symmetries. We consider F-theory on general Calabi-Yau manifolds
with a rank one Mordell-Weil group of rational sections. By rigorously
performing the stable degeneration limit in a class of toric models, we derive
both the Calabi-Yau geometry as well as the spectral cover describing the
vector bundle in the heterotic dual theory. We carefully investigate the
spectral cover employing the group law on the elliptic curve in the heterotic
theory. We find in explicit examples that there are three different classes of
heterotic duals that have U(1) factors in their low energy effective theories:
split spectral covers describing bundles with S(U(m) x U(1)) structure group,
spectral covers containing torsional sections that seem to give rise to bundles
with SU(m) x Z_k structure group and bundles with purely non-Abelian structure
groups having a centralizer in E_8 containing a U(1) factor. In the former two
cases, it is required that the elliptic fibration on the heterotic side has a
non-trivial Mordell-Weil group. While the number of geometrically massless
U(1)'s is determined entirely by geometry on the F-theory side, on the
heterotic side the correct number of U(1)'s is found by taking into account a
Stuckelberg mechanism in the lower-dimensional effective theory. In geometry,
this corresponds to the condition that sections in the two half K3 surfaces
that arise in the stable degeneration limit of F-theory can be glued together
globally.
