Polarized interpolation and normal postulation for curves on Fano hypersurfaces

A general hypersurface $X$ of degree $\le n$ in projective space contains curves $C$ of any genus $g\ge 0$ and sufficiently large degree depnding on $g$ whose normal and conormal bundles have good postulation or natural cohomology in the sense that each twist has either $H^0=0$ or $H^1=0$. This implies a polarized version of the interpolation property for $C$ on $X$.