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arXiv · 0711.2758

Rational curves of degree 11 on a general quintic threefold

Ethan Cotterill
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Abstract

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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Ethan Cotterill. 2010-04-02. Rational curves of degree 11 on a general quintic threefold. https://arxiv.org/abs/0711.2758

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