Polarized interpolation and normal postulation for curves on Fano hypersurfaces - Ran, Ziv (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 514A general hypersurface $X$ of degree at most $n$ in projective $(n+1)$-space contains curves $C$ of any genus $g≥ 0$ and sufficiently large degree depending on $g$ whose normal and conormal bundles in $X$ have good postulation or natural cohomology in a 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 $X$.