We analyze three aspects of N = 1 heterotic string compactifications
on elliptically fibered Calabi-Yau threefolds:
stability of vector bundles, five-brane instanton transitions and chiral matter.
First we show that relative Fourier-Mukai transformation preserves absolute stability.
This is relevant for vector bundles whose spectral cover is reducible. Then we
derive an explicit formula for the number of moduli which occur in (vertical)
five-brane instanton transitions provided a certain vanishing argument
applies. Such transitions increase the holonomy of the heterotic vector
bundle and cause gauge changing phase transitions.
In a M-theory description the transitions are associated with collisions of bulk
five-branes with one of the boundary fixed planes.
In F-theory they correspond to three-brane instanton transitions.
Our derivation relies on an index computation with data localized
along the curve which is related to the existence of chiral matter
in this class of heterotic vacua.
Finally, we show how to compute the number of chiral matter multiplets
for this class of vacua allowing to discuss associated Yukawa couplings.