In mathematics, a quintic threefold is a degree 5 dimension 3 hypersurface in 4-dimensional projective space. Non-singular quintic threefolds are Calabi–Yau manifolds.
The Hodge diamond of a non-singular quintic 3-fold is
Clemens (1984) conjectured that the number of rational curves of a given degree on a generic quintic threefold is finite. (Some smooth but non-generic quintic threefolds have infinite families of lines on them.) This was verified for degrees up to 7 by Katz (1986) who also calculated the number 609250 of degree 2 rational curves. Philip Candelas, Xenia C. de la Ossa, and Paul S. Green et al. (1991) conjectured a general formula for the number of rational curves of any degree, which was proved by Givental (1996) The number of rational curves of various degrees on a generic quintic threefold is given by
Since the generic quintic threefold is a Calabi–Yau threefold and the moduli space of rational curves of a given degree is a discrete, finite set (hence compact), these have well-defined Donaldson–Thomas invariants (the "virtual number of points"); at least for degree 1 and 2, these agree with the actual number of points.
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- Clemens, Herbert (1984), "Some results about Abel-Jacobi mappings", Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), Ann. of Math. Stud. 106, Princeton University Press, pp. 289–304, MR 756858
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- Givental, Alexander B. (1996), "Equivariant Gromov-Witten invariants", International Mathematics Research Notices 1996 (13): 613–663, doi:10.1155/S1073792896000414, ISSN 1073-7928, MR 1408320
- Katz, Sheldon (1986), "On the finiteness of rational curves on quintic threefolds", Compositio Mathematica 60 (2): 151–162, ISSN 0010-437X, MR 868135
- Pandharipande, Rahul (1998), "Rational curves on hypersurfaces (after A. Givental)", Astérisque (252): 307–340, ISSN 0303-1179, MR 1685628