Rational Curves of Degree 10 on a General Quintic Threefold

We prove the "strong form" of the Clemens conjecture in degree 10. Namely, on a general quintic threefold F in ℙ 4 , there are only, finitely, many smooth rational curves of degree 10, and each curve C is embedded in F with normal bundle (−1) ⊕  (−1). Moreover, in degree 10, there are no s...

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Bibliographic Details
Published inCommunications in algebra Vol. 33; no. 6; pp. 1833 - 1872
Main Author Cotterill, Ethan
Format Journal Article
LanguageEnglish
Published Taylor & Francis Group 01.05.2005
Subjects
Curves of low genus
Gröbner bases
Polynomial ideals
Primary 14J30
Secondary 13P10
Special curves
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Summary:We prove the "strong form" of the Clemens conjecture in degree 10. Namely, on a general quintic threefold F in ℙ 4 , there are only, finitely, many smooth rational curves of degree 10, and each curve C is embedded in F with normal bundle (−1) ⊕  (−1). Moreover, in degree 10, there are no singular, reduced, and irreducible rational curves, nor any reduced, reducible, and connected curves with rational components on F.
ISSN:0092-7872
1532-4125
DOI:10.1081/AGB-200063325