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arXiv · alg-geom/9510015

Rational curves of degree at most 9 on a general quintic threefold

Trygve Johnsen·Steven L. Kleiman
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Abstract

We prove the following form of the Clemens conjecture in low degree. Let $d\le9$, and let $F$ be a general quintic threefold in $\IP^4$. Then (1)~the Hilbert scheme of rational, smooth and irreducible curves of degree $d$ on $F$ is finite, nonempty, and reduced; moreover, each curve is embedded in $F$ with normal bundle $Ø(-1)\oplusØ(-1)$, and in $\IP^4$ with maximal rank. (2)~On $F$, there are no rational, singular, reduced and irreducible curves of degree $d$, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3)~On $F$, there are no connected, reduced and reducible curves of degree $d$ with rational components.

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Trygve Johnsen, Steven L. Kleiman. 1996-03-01. Rational curves of degree at most 9 on a general quintic threefold. https://arxiv.org/abs/alg-geom/9510015

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