Rational curves on hypersurfaces

TL;DR

This paper proves new results on the properties of rational curves on generic hypersurfaces, including vanishing cohomology of the normal sheaf, and addresses conjectures related to rational coverings and classifications of such curves.

Contribution

It extends previous results to all generic hypersurfaces, proving vanishing of higher cohomology and solving two Voisin conjectures about rational curves.

Findings

Normal sheaf of a generic rational map has vanishing higher cohomology

Provides a solution to Voisin's conjecture on rational coverings of hypersurfaces

Classifies rational curves on hypersurfaces of general type

Abstract

In this paper, we prove three related results; (1) Extension of our result in [10] to all generic hypersurfaces. More precisely, the normal sheaf of a generic rational map $c_0$ to a generic hypersurface $X_0$ of $\mathbf P^n, n\geq 4$ has a vanishing higher cohomology, \begin{equation} H^1(N_{c_0/X_0})=0. \end{equation} As applications we give (2) A solution to a Voisin's conjecture [9] on a covering of a generic hypersurface by rational curves (3) A classification of rational curves on hypersurfaces of general type--a solution to another Voisin's conjecture [9].