Rational curves of degree 11 on a general quintic threefold

Details Rational curves of degree 11 on a general quintic threefold Rational curves of degree 11 on a general quintic threefold

2007-11-17

arxiv.org/abs/0711.2758v2

Mathematics

Algebraic Geometry

Significant reorganization; the liaison-theoretic argument handling curves contained in complete intersections of three hyperc

Scientific paper

We prove that the incidence scheme of rational curves of degree 11 on quintic threefolds is irreducible. This implies a strong form of the Clemens conjecture in degree 11. Namely, on a general quintic threefold $F$ in $\mathbb{P}^4$, there are only finitely many smooth rational curves of degree 11, and each curve $C$ is embedded in $F$ with normal bundle $\mathcal{O}(-1) \oplus \mathcal{O}(-1)$. Moreover, in degree 11, there are no singular, reduced, and irreducible rational curves, nor any reduced, reducible, and connected curves with rational components on $F$.

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