  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 an 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 the associated Yukawa couplings.
