Rational curves on general projective hypersurfaces

Details Rational curves on general projective hypersurfaces Rational curves on general projective hypersurfaces

2000-10-03

arxiv.org/abs/math/0010037v2

Mathematics

Algebraic Geometry

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

Scientific paper

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 $\delta >1$.

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