Quintic threefold

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

			1
		0		0
	0		1		0
1		101		101		1
	0		1		0
		0		0
			1

Rational curves

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 virtual number of rational curves of any degree, which was proved by Givental (1996) (the fact that the virtual number equals the actual number relies on confirmation of Clemens' conjecture, currently known for degree at most 11 Cotterill (2012)). The number of rational curves of various degrees on a generic quintic threefold is given by

5, 2875, 609250, 317206375, 242467530000, ...(sequence A076912 in the OEIS).

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.

Examples

• The Fermat quintic threefold is given by ${\displaystyle V^{5}+W^{5}+X^{5}+Y^{5}+Z^{5}=0}$.
• Barth–Nieto quintic

References

• Candelas, Philip; de la Ossa, Xenia C.; Green, Paul S.; Parkes, Linda (1991), "A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory", Nuclear Physics B, 359 (1): 21–74, doi:10.1016/0550-3213(91)90292-6, MR 1115626
• 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
• Cotterill, Ethan (2012), "Rational curves of degree 11 on a general quintic 3-fold", The Quarterly Journal of Mathematics, 63 (3): 539–568, doi:10.1093/qmath/har001, MR 2967162
• Cox, David A.; Katz, Sheldon (1999), Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, 68, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1059-0, MR 1677117
• Givental, Alexander B. (1996), "Equivariant Gromov-Witten invariants", International Mathematics Research Notices, 1996 (13): 613–663, doi:10.1155/S1073792896000414, MR 1408320
• Katz, Sheldon (1986), "On the finiteness of rational curves on quintic threefolds", Compositio Mathematica, 60 (2): 151–162, MR 868135
• Pandharipande, Rahul (1998), "Rational curves on hypersurfaces (after A. Givental)", Astérisque, 1997/98 (252): 307–340, MR 1685628