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

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

1995-10-30

arxiv.org/abs/alg-geom/9510015v2

Mathematics

Algebraic Geometry

31 pages, minor revisions -- current 2/29/96, Plain Tex

Scientific paper

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 $\O(-1)\oplus\O(-1)$, 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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