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
Keywords
Examples
a(1) = 2875 = number of lines in the quintic.
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
- 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.
Extensions
a(10) = A060041(10) - 6 * 17601000 added by Andrey Zabolotskiy, Sep 10 2022 (see Encyclopedia of Mathematics, Clemens' conjecture)