Rational curves on hypersurfaces of a projective variety

TL;DR

This paper extends results on rational curves to hypersurfaces within smooth projective varieties, proving a vanishing cohomology condition and applying it to confirm Clemens' conjecture for certain Calabi-Yau threefolds.

Contribution

It generalizes previous findings to broader hypersurfaces and establishes a key cohomology vanishing result, supporting Clemens' conjecture in new cases.

Findings

Vanishing of H^1(N_{c_0/X_0}) for generic hypersurfaces

Application to Clemens' conjecture for Calabi-Yau threefolds

Extension of previous results to arbitrary smooth projective varieties

Abstract

In this paper, we extend our result in [3] to hypersurfaces of any smooth projective variety $Y$. Precisely we let $X_0$ be a generic hypersurface of $Y$ and $c_0:\mathbf P^1\to X_0$ be a generic birational morphism to its image, i.e. $c_0\in Hom_{bir}(\mathbf P^1, X_0)$ is generic, such that (1) $dim(X_0)\geq 3$, (2) $H^1( N_{c_0/Y})=0$. Then \begin{equation} H^1(N_{c_0/X_0})=0. \end{equation} As an application we prove that the Clemens' conjecture holds for Calabi-Yau complete intersections of dimension 3.