  In this work we review a systematic, algorithmic construction of dual
heterotic/F-theory geometries corresponding to 4-dimensional, N = 1
supersymmetric compactifications. We look in detail at a class of well-defined
Calabi-Yau fourfolds for which the standard formulation of the duality map
appears to fail, leading to dual heterotic geometry which appears naively
incompatible with the spectral cover construction of vector bundles. In the
simplest class of examples the F-theory background consists of a generically
singular elliptically fibered Calabi-Yau fourfold with E7 symmetry. The vector
bundles arising in the corresponding heterotic theory appear to violate an
integrality condition of an SU(2) spectral cover. A possible resolution of this
puzzle is explored by studying the most general form of the integrality
condition. This leads to the geometric challenge of determining the Picard
group of surfaces of general type. We take an important first step in this
direction by computing the Hodge numbers of an explicit spectral surface and
bounding the Picard number.
