Rational curves on general projective hypersurfaces
Abstract
Description
Let $k$ be an integer such that $1\leq k\leq n-5$, 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 $δ>1$.
Final version to appear in the Journal of Algebraic Geometry. Exposition improved, according to referee's suggestions. 26 pages, Latex
Final version to appear in the Journal of Algebraic Geometry. Exposition improved, according to referee's suggestions. 26 pages, Latex