  We study issues related to F-theory on Calabi-Yau fourfolds and its duality
to heterotic theory for Calabi-Yau threefolds. We discuss principally fourfolds
that are described by reflexive polyhedra and show how to read off some of the
data for the heterotic theory from the polyhedron. We give a procedure for
constructing examples with given gauge groups and describe some of these
examples in detail. Interesting features arise when the local pieces are fitted
into a global manifold. An important issue is how to compute the superpotential
explicitly. Witten has shown that the condition for a divisor to contribute to
the superpotential is that it have arithmetic genus 1. Divisors associated with
the short roots of non-simply laced gauge groups do not always satisfy this
condition while the divisors associated to all other roots do. For such a
`dissident' divisor we distinguish cases for which the arithmetic genus is
greater than unity corresponding to an X that is not general in moduli (in the
toric case this corresponds to the existence of non-toric parameters). In these
cases the `dissident' divisor D does not remain an effective divisor for
general complex structure. If however the arithmetic genus is less than or
equal to 0, then the divisor is general in moduli and there is a genuine
instability.
