Polarized interpolation and normal postulation for curves on Fano hypersurfaces

TL;DR

This paper demonstrates that general Fano hypersurfaces contain curves with well-behaved normal bundles, satisfying a polarized interpolation property, which advances understanding of curve embeddings in algebraic geometry.

Contribution

It establishes the existence of curves with natural cohomology on Fano hypersurfaces and introduces a polarized interpolation property for these curves.

Findings

Curves of any genus and large degree exist on general Fano hypersurfaces.

Normal and conormal bundles of these curves exhibit good cohomological behavior.

The results imply a polarized version of the interpolation property for curves on hypersurfaces.

Abstract

A general hypersurface $X$ of degree $\leq n$ in projective space contains curves $C$ of any genus $g\geq 0$ 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 $X$.