  On a general Fano hypersurface in projective space, we determine for
infinitely many $k$ the minimal degree $e$ of a rational curve through a
general collection of $k$ points. In the case of a hypersurface of index 1, our
results hold for all $k\geq 1$. In an appendix, M.C. Chang proves an
arithmetical result which implies that in the case of index $>1$, the density
of the set of curve degrees $e$ covered by our method is approximately
$\frac{(n-d)(d-\frac{5}{2})}{(n-2)d}$.
