Rocky Mountain Journal of Mathematics

First order deformations of pairs and non-existence of rational curves

Bin Wang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $X_0$ be a smooth hypersurface (assumed not to be generic) in projective space $\mathbf {P}^n$, $n\geq 4$, over complex numbers, and $C_0$ a smooth rational curve on $X_0$. We are interested in deformations of the pair $C_0$ and $X_0$. In this paper, we prove that, if the first order deformations of the pair exist along each deformation of the hypersurface $X_0$, then $\deg (C_0)$ cannot be in the range \[ \bigg ( m\frac {2\deg (X_0)+1}{\deg (X_0)+1}, \frac {2+m(n-2)}{2n-\deg (X_0)-1}\bigg ), \] where $m$ is any non negative integer less than \[ \dim (H^0(\mathcal {O}_{\mathbf {P}^n}(1))|_{C_0} )-1. \]

Article information

Source
Rocky Mountain J. Math., Volume 46, Number 2 (2016), 663-678.

Dates
First available in Project Euclid: 26 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1469537480

Digital Object Identifier
doi:10.1216/RMJ-2016-46-2-663

Mathematical Reviews number (MathSciNet)
MR3529086

Zentralblatt MATH identifier
1356.14032

Subjects
Primary: 14J70: Hypersurfaces 14N10: Enumerative problems (combinatorial problems) 14N25: Varieties of low degree

Keywords
Hypersurface rational curve normal bundle

Citation

Wang, Bin. First order deformations of pairs and non-existence of rational curves. Rocky Mountain J. Math. 46 (2016), no. 2, 663--678. doi:10.1216/RMJ-2016-46-2-663. https://projecteuclid.org/euclid.rmjm/1469537480


Export citation

References

  • H. Clemens, Curves in generic hypersurfaces, Ann. Sci. École Norm. 19 (1986), 629–636.
  • ––––, Some results on Abel-Jacobi mappings, in Topics in transcendental algebraic geometry, Princeton University Press, Princeton, 1984.
  • ––––, Homological equivalence, modulo algebraic equivalence, is not finitely generated, Publ. Math IHES 58 (1983), 19–38.
  • L. Chiantini, A.F. Lopez and Z. Ran, Subvarieties of generic hypersurfaces in any variety, Math. Proc. Camb. Phil. Soc. 130 (2001), 259–268.
  • L. Ein, Subvarieties of generic complete intersections, Invent. Math. 94 (1988), 163–169.
  • S. Katz, On the finiteness of rational curves on quintic threefolds, Comp. Math. 60 (1986), 151–162.
  • K. Kodaira, On stability of compact submanifolds of complex manifolds, Amer. J. Math. 85 (1963), 79–94.
  • G. Pacienza, Rational curves on general projective hypersurfaces, J. Alg. Geom. 12 (2003), 471–476.
  • C. Voisin, On a conjecture of Clemens on rational curves on hypersurfaces, J. Differ. Geom. 44 (1996), 200–213.
  • ––––, A correction: “On a conjecture of Clemens on rational curves on hypersurfaces,” J. Differ. Geom. 49 (1998), 601–611.
  • B. Wang, Obstructions to the deformations of curves to other hypersurfaces, arXiv:1110.0184v3, 2011.
  • G. Xu, Subvarieties of general hypersurfaces in projective space, J. Differ. Geom. 39 (1994), 139–172.