Rational curves on general projective hypersurfaces

Abstract

Let $k$ be an integer such that $1\le k\le n-5$, and $X_{2n-2-k}\subset {\mathbf{P}}^n$ a general projective hypersurface of degree $d=2n-2-k$. In this paper we prove that the only $k$-dimensional subvariety $Y$ of $X_{2n-2-k}$ having geometric genus zero is the one covered by the lines. As an immediate corollary we obtain that, for $n>5$, the general $X_{2n-3}\subset {\mathbf{P}}^n$, contains no rational curves of degree $\delta >1$.

Final version to appear in the Journal of Algebraic Geometry. Exposition improved, according to referee's suggestions. 26 pages, Latex