  In the context of string dualities, fibration structures of Calabi-Yau
manifolds play a prominent role. In particular, elliptic and K3 fibered
Calabi-Yau fourfolds are important for dualities between string
compactifications with four flat space-time dimensions. A natural framework for
studying explicit examples of such fibrations is given by Calabi-Yau
hypersurfaces in toric varieties, because this class of varieties is
sufficiently large to provide examples with very different features while still
allowing a large degree of explicit control. In this paper, many examples for
elliptic K3 fibered Calabi-Yau fourfolds are found (not constructed) by
searching for reflexive subpolyhedra of reflexive polyhedra corresponding to
hypersurfaces in weighted projective spaces. Subpolyhedra not always give rise
to fibrations and the obstructions are studied. In addition, perturbative gauge
algebras for dual heterotic string theories are determined. In order to do so,
all elliptically fibered toric K3 surfaces are determined. Then, the
corresponding gauge algebras are calculated without specialization to
particular polyhedra for the elliptic fibers. Finally, the perturbative gauge
algebras for the fourfold fibrations are extracted from the generic fibers and
monodromy.
