  The present paper concerns the invariants of generically nef vector bundles
on ruled surfaces. By Mehta - Ramanathan Restriction Theorem and by Miyaoka
characterization of semistable vector bundles on a curve, the generic nefness
can be considered as a weak form of semistability. We establish a Bogomolov
type inequality for generically nef vector bundles with nef general fiber
restriction on ruled surfaces with no negative section. This gives an
affermative answer in this case to a problem posed by Th. Peternell.
  Concerning ruled surfaces with a negative section, we prove a a similar
result for generically nef vector bundles, with nef and balanced general fiber
restriction and with a numerical condition on first Chern class, which is
satisfied, for instance, if in its class there is a reduced divisor.
  Finally, we use such results to bound the invariants of curve fibrations,
which factorize through finite covers of ruled surfaces.
