  Let X be a projective variety which is covered by rational curves, for
instance a Fano manifold over the complex numbers. In this setup,
characterization and classification problems lead to the natural question:
"Given two points on X, how many minimal degree rational curve are there which
contain those points?". A recent answer to this question led to a number of new
results in classification theory. As an infinitesimal analogue, we ask "How
many minimal degree rational curves exist which contain a prescribed tangent
vector?"
  In this paper, we give sufficient conditions which guarantee that every
tangent vector at a general point of X is contained in at most one rational
curve minimal degree. As an immediate application, we obtain irreducibility
criteria for the space of minimal rational curves.
