Rational curves on hypersurfaces

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 ${\mathbf{P}}^n,n\ge 4$ 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].

arXiv commentarXiv admin note: substantial text overlap with arXiv:1202.2831

References in corpus (1)

• Rational curves on generic quintic threefolds

Cited by in corpus (1)

• Rational curves on hypersurfaces of a projective variety