  Let X be a projective variety which is covered by a family of rational curves
of minimal degree. The classic bend-and-break argument of Mori asserts that if
x and y are two general points, then there are at most finitely many curves in
that family which contain both x and y. In this work we shed some light on the
question as to whether two sufficiently general points actually define a unique
curve.
  As an immediate corollary to the results of this paper, we give a
characterization of projective spaces which improves on the known
generalizations of Kobayashi-Ochiai's theorem.
