  We present explicit methods for computing the discriminant curves and the
associated Kodaira type fiber degeneracies of elliptically fibered Calabi-Yau
threefolds. These methods are applied to a specific three-family, SU(5) grand
unified theory of particle physics within the context of Heterotic M-Theory. It
is demonstrated that there is always a region of moduli space where a bulk
space five-brane is wrapped on a pure fiber in the Calabi-Yau threefold.
Restricting the discussion to the smooth parts of the discriminant curve, we
explore the properties of the N=2 BPS supermultiplets that arise on the
worldvolume of this five-brane due to the degeneration of the elliptic fiber.
The associated degenerating M membranes are shown to project to string
junctions in the base space. We use string junction techniques to explicitly
compute the light BPS hyper- and vector multiplet spectrum for each Kodaira
type fiber near the smooth part of the discriminant curve in the SU(5) GUT
theory.
