A076912 - OEIS

%I #34 Sep 11 2022 10:11:02

%S 5,2875,609250,317206375,242467530000,229305888887625,

%T 248249742118022000,295091050570845659250,375632160937476603550000,

%U 503840510416985243645106250,704288164978454686113382643750

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

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

%D 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.

%H V. Batyrev, <a href="http://www.ams.org/journal-getitem?pii=S0273-0979-00-00875-2">Review of "Mirror Symmetry and Algebraic Geometry"</a>, by D. A. Cox and S. Katz, Bull. Amer. Math. Soc., 37 (No. 4, 2000), 473-476.

%H P. Candelas et al., <a href="http://dx.doi.org/10.1016/0550-3213(91)90292-6">A pair of Calabi-yau manifolds as an exactly soluble superconformal theory</a>, Nuclear Phys. B 359 (1991), 21-74.

%H Geir Ellingsrud and Stein Arild Stromme, <a href="http://dx.doi.org/10.7146/math.scand.a-12522">The number of twisted cubic curves on the general quintic threefold</a>, Math. Scand. 76 (1995), no. 1, 5-34.

%H Geir Ellingsrud and Stein Arild Stromme, <a href="http://arxiv.org/abs/alg-geom/9411005">Bott's formula and enumerative geometry</a>, J. Amer. Math. Soc. 9 (1996), 175-193; arXiv:alg-geom/9411005, 1994.

%H Encyclopedia of Mathematics, <a href="https://encyclopediaofmath.org/wiki/Clemens%27_conjecture">Clemens' conjecture</a>.

%H Steven R. Finch, <a href="/A013587/a013587.pdf">Enumerative geometry</a>, February 24, 2014. [Cached copy, with permission of the author]

%H Trygve Johnsen and Steven L. Kleiman, <a href="http://arxiv.org/abs/alg-geom/9510015">Rational curves of degree at most 9 on a general quintic threefold</a>, arXiv:alg-geom/9510015, 1995-1996.

%H Trygve Johnsen and Steven L. Kleiman, <a href="http://arxiv.org/abs/alg-geom/9601024">Toward Clemens' Conjecture in degrees between 10 and 24</a>, arXiv:alg-geom/9601024, 1996.

%H B. Mazur, <a href="http://dx.doi.org/10.1090/S0273-0979-04-01024-9">Perturbations, deformations and variations (and "near-misses") in geometry, physics</a>, and number theory, Bull. Amer. Math. Soc., 41 (2004), 307-336.

%H David R. Morrison, <a href="http://arXiv.org/abs/alg-geom/9609021">Mathematical Aspects of Mirror Symmetry</a>, arXiv:alg-geom/9609021, 1996; in Complex Algebraic Geometry (J. Kollár, ed.), IAS/Park City Math. Series, vol. 3, 1997, pp. 265-340.

%H R. Pandharipande, <a href="http://www.numdam.org/book-part/SB_1997-1998__40__307_0/">Rational curves on hypersurfaces (after A. Givental)</a>, Séminaire Bourbaki, Vol. 1997/98. Astérisque No. 252 (1998), Exp. No. 848, 5, 307-340.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Quintic_threefold">Quintic threefold</a>

%H Shing-Tung Yau and Steve Nadis, <a href="http://www.ams.org/notices/201108/rtx110801067p.pdf">String Theory and the Geometry of the Universe's Hidden Dimensions</a>, Notices Amer. Math. Soc., 58 (Sep 2011), 1067-1076.

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

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

%Y Cf. A060345, A076913.

%K nonn

%O 0,1

%A _N. J. A. Sloane_, Nov 28 2002

%E a(10) = A060041(10) - 6 * 17601000 added by _Andrey Zabolotskiy_, Sep 10 2022 (see Encyclopedia of Mathematics, Clemens' conjecture)