A076912 Number of degree-n rational curves on a general quintic threefold.

5, 2875, 609250, 317206375, 242467530000, 229305888887625, 248249742118022000, 295091050570845659250, 375632160937476603550000, 503840510416985243645106250, 704288164978454686113382643750

Offset

0,1

References

J. Bertin and C. Peters, Variations of Hodge structure ..., pp. 151-232 of J. Bertin et al., eds., Introduction to Hodge Theory, Amer. Math. Soc. and Soc. Math. France, 2002; see p. 220.

D. A. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Amer. Math. Soc., 1999.

Ellingsrud, Geir and Stromme, Stein Arild, The number of twisted cubic curves on the general quintic threefold (preliminary version). In Essays on Mirror Manifolds, 181-222, Int. Press, Hong Kong, 1992.

Links

Table of n, a(n) for n=0..10.

V. Batyrev, Review of "Mirror Symmetry and Algebraic Geometry", by D. A. Cox and S. Katz, Bull. Amer. Math. Soc., 37 (No. 4, 2000), 473-476.

P. Candelas et al., A pair of Calabi-yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), 21-74.

Geir Ellingsrud and Stein Arild Stromme, The number of twisted cubic curves on the general quintic threefold, Math. Scand. 76 (1995), no. 1, 5-34.

Geir Ellingsrud and Stein Arild Stromme, Bott's formula and enumerative geometry, J. Amer. Math. Soc. 9 (1996), 175-193; arXiv:alg-geom/9411005, 1994.

Encyclopedia of Mathematics, Clemens' conjecture.

Steven R. Finch, Enumerative geometry, February 24, 2014. [Cached copy, with permission of the author]

Trygve Johnsen and Steven L. Kleiman, Rational curves of degree at most 9 on a general quintic threefold, arXiv:alg-geom/9510015, 1995-1996.

Trygve Johnsen and Steven L. Kleiman, Toward Clemens' Conjecture in degrees between 10 and 24, arXiv:alg-geom/9601024, 1996.

B. Mazur, Perturbations, deformations and variations (and "near-misses") in geometry, physics, and number theory, Bull. Amer. Math. Soc., 41 (2004), 307-336.

David R. Morrison, Mathematical Aspects of Mirror Symmetry, arXiv:alg-geom/9609021, 1996; in Complex Algebraic Geometry (J. Kollár, ed.), IAS/Park City Math. Series, vol. 3, 1997, pp. 265-340.

R. Pandharipande, Rational curves on hypersurfaces (after A. Givental), Séminaire Bourbaki, Vol. 1997/98. Astérisque No. 252 (1998), Exp. No. 848, 5, 307-340.

Wikipedia, Quintic threefold

Shing-Tung Yau and Steve Nadis, String Theory and the Geometry of the Universe's Hidden Dimensions, Notices Amer. Math. Soc., 58 (Sep 2011), 1067-1076.

Example

a(1) = 2875 = number of lines in the quintic.

Crossrefs

Coincides with A060041 for n <= 9, but not for n = 10.

Cf. A060345, A076913.

Sequence in context: A274527 A258980 A096934 * A060041 A199878 A076908

Adjacent sequences: A076909 A076910 A076911 * A076913 A076914 A076915

Keyword

nonn

Author

N. J. A. Sloane, Nov 28 2002

Extensions

a(10) = A060041(10) - 6 * 17601000 added by Andrey Zabolotskiy, Sep 10 2022 (see Encyclopedia of Mathematics, Clemens' conjecture)

Status

approved