  We systematically analyze a broad class of dual heterotic and F-theory models
that give four-dimensional supergravity theories, and compare the geometric
constraints on the two sides of the duality. Specifically, we give a complete
classification of models where the heterotic theory is compactified on a smooth
Calabi-Yau threefold that is elliptically fibered with a single section and
carries smooth irreducible vector bundles, and the dual F-theory model has a
corresponding threefold base that has the form of a P^1 bundle. We formulate
simple conditions for the geometry on the F-theory side to support an
elliptically fibered Calabi-Yau fourfold. We match these conditions with
conditions for the existence of stable vector bundles on the heterotic side,
and show that F-theory gives new insight into the conditions under which such
bundles can be constructed. In particular, we find that many allowed F-theory
models correspond to vector bundles on the heterotic side with exceptional
structure groups, and determine a topological condition that is only satisfied
for bundles of this type. We show that in many cases the F-theory geometry
imposes a constraint on the extent to which the gauge group can be enhanced,
corresponding to limits on the way in which the heterotic bundle can decompose.
We explicitly construct all (4962) F-theory threefold bases for dual
F-theory/heterotic constructions in the subset of models where the common
twofold base surface is toric, and give both toric and non-toric examples of
the general results.
