We use local mirror symmetry in type IIA string compactifications on
Calabi-Yau n+1 folds $X_{n+1}$ to construct vector bundles on (possibly
singular) elliptically fibered Calabi-Yau n-folds Z_n. The interpretation of
these data as valid classical solutions of the heterotic string compactified on
Z_n proves F-theory/heterotic duality at the classical level. Toric geometry is
used to establish a systematic dictionary that assigns to each given toric
n+1-fold $X_{n+1}$ a toric n fold Z_n together with a specific family of sheafs
on it. This allows for a systematic construction of phenomenologically
interesting d=4 N=1 heterotic vacua, e.g. on deformations of the tangent
bundle, with grand unified and SU(3)\times SU(2) gauge groups. As another
application we find non-perturbative gauge enhancements of the heterotic string
on singular Calabi-Yau manifolds and new non-perturbative dualities relating
heterotic compactifications on different manifolds.