The Finiteness of Smooth Curves of Degree $\le 11$ and Genus $\le 3$ on a General Complete Intersection of a Quadric and a Quartic in $\mathbb{P}^5$

Year 2022, Volume: 5 Issue: 3, 181 - 191, 23.09.2022

Edoardo Ballico

https://doi.org/10.33401/fujma.1069957

Abstract

Let $W\subset \mathbb{P}^5$ be a general complete intersection of a quadric hypersurface and a quartic hypersurface. In this paper, we prove that $W$ contains only finitely many smooth curves
$C\subset \mathbb{P}^5$ such that $d:= \deg ({C}) \le 11$, $g:= p_a({C}) \le 3$ and $h^1(\mathcal{O} _C(1)) =0$.

Keywords

Calabi-Yau threefold, Curves, Curves in a Calabi-Yau threefold

References

• [1] E. Cotterill, Rational curves of degree $11$ on a general quintic $3$-fold , Quart. J. Math., 63 (2012), 539-568.
• [2] D'Almeida, Courbes rationnelles de degr\'e $11$ sur une hypersurface quintique g\'en\'erale de $\mathbb {P ^4$ , Bull. Sci. Math., 136 (2012), 899-903.
• [3] T. Johnsen, S. Kleiman, Rational curves of degree at most $9$ on a general quintic threefold, Comm. Algebra, 24 (1996), 2721-2753.
• [4] T. Johnsen, S. Kleiman, Toward Clemens' conjecture in degrees between $10$ and $24$ , Serdica Math. J., 23 (1997), 131-142.
• [5] T. Johnsen, A. L. Knutsen, Rational curves in Calabi-Yau threefolds, Special issue in honor of Steven L. Kleiman. Comm. Algebra, 31 (8) (2003), 3917-3953.
• [6] S. Katz, On the finiteness of rational curves on quintic threefolds, Compositio Math., 60 (2) (1986), 151-162.
• [7] C. Voisin, On some problems of Kobayashi and Lang, in Current developments in Mathematics, pp. 53-125, Int. Press, Somerville, MA, 2003.
• [8] K. Oguiso, Two remarks on Calabi-Yau threefolds, J. Reine Angew. Math., 452 (1994), 153-161.
• [9] A. L. Knutsen, On isolated smooth curves of low genera in Calabi-Yau complete intersection threefolds , Trans. Amer. Math. Soc., 384 (10) (2012), 5243-5284.
• [10] E. Cotterill, Rational curves of degree $16$ on a general heptic fourfold, J. Pure Appl. Algebra, 218 (2014), 121-129.
• [11] G. Hana, T. Johnsen, Rational curves on a general heptic fourfold, Bull. Belg. Math. Soc., Simon Stevin, 16 (2009), 861-885.
• [12] C. Voisin, On a conjecture of Clemens on rational curves on hypersurfaces, J. Diff. Geometry, 44 (1) (1996), 200-213.
• [13] N. Mohan Kumar, A. P. Rao, G. V. Ravindra, On codimension two subvarieties in hypersurfaces, Motives and algebraic cycles, 167-174, Fields Inst. Commun., 56, Amer. Math. Soc., Providence, RI, 2009.
• [14] P. Candelas, X. de la Ossa, P. Green, L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B, 359 (1991), 21-74.
• [15] M. Kontsevich, Enumeration of rational curves via torus actions, in The Moduli Space of Curves, pp. 335-368, Progress in Math. 29, Birkh\"{a user, Basel, CH, 19958.
• [16] M. S. Narasimhan, S. Ramanan, Deformations of the moduli space of vector bundles over an algebraic curve, Ann. of Math., 101 (1975), 391-417.
• [17] M. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc., 7 (3) (1957), 414-452; reprinted in: Michael Atiyah Collected Works, Oxford, 1 (1988), 105-143.
• [18] R. Hartshorne, Algebraic Geometry , Springer-Verlag, Berlin--Heidelberg--New York, 1977.
• [19] L. Gruson, R. Lazarsfeld, Ch. Peskine, On a theorem of Castelnuovo and the equations defining space curves, Invent. Math., 72 (1983), 491-506.
• [20] R. Hartshorne, A. Hirschowitz, Smoothing Algebraic Space Curves, Algebraic Geometry, Sitges 1983, 98-131, Lecture Notes in Math., 1124, Springer, Berlin, 1985.
• [21] M. Green, R. Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math., 83 (1) (1986), 73-90.
• [22] D. Eisenbud, J. Harris, Finite projective schemes in linearly general position, J. Algebraic Geom., 1 (1) (1992), 15-30.
• [23] A. Bernardi, A. Gimigliano, M. Ida, Computing symmetric rank for symmetric tensors, J. Symbolic Comput., 46 (2011) 34-53.
• [24] Ph. Ellia, Ch. Peskine, Groupes de points de ${\bf {P ^2$: caractere et position uniforme, in: Algebraic geometry (L' Aquila, 1988), 111-116, Lecture Notes in Math., 1417, Springer, Berlin, 1990.
• [25] D. Perrin, Courbes passant par $m$ points g\'{e n\'{e raux de $\mathbb {P ^3$ , Bull. Soc. Math., France, M\'{e moire 28/29 (1987).
• [26] P. Jahnke, T. Peternell, I. Radloff, Some Recent Developments in the Classification Theory of Higher Dimensional Manifolds, Global Aspects of Complex Geometry, 311-357, Springer, Berlin, 2006.