  We discuss the use of gauge fields to stabilize complex structure moduli in
Calabi-Yau three-fold compactifications of heterotic string and M-theory. The
requirement that the gauge fields in such models preserve supersymmetry leads
to a complicated landscape of vacua in complex structure moduli space. We
develop methods to systematically map out this multi-branched vacuum space, in
a computable and explicit manner. In analyzing the resulting vacua, it is found
that the associated Calabi-Yau three-folds are sometimes stabilized at a value
of complex structure resulting in a singular compactification manifold. We
describe how it is possible to resolve these singularities, in some cases,
while maintaining computational control over the moduli stabilization
mechanism. The discussion is illustrated throughout the paper with explicit
worked examples.
