Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Rational curves on general projective hypersurfaces

Author: Gianluca Pacienza
Journal: J. Algebraic Geom. 12 (2003), 245-267
Published electronically: October 17, 2002
MathSciNet review: 1949643
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Abstract | References | Additional Information

Abstract: In this article, we study the geometry of $k$-dimensional subvarieties with geometric genus zero of a general projective hypersurface $X_d\subset \mathbf{P}^n$ of degree $d=2n-2-k$, where $k$ is an integer such that $1\leq k\leq n-5$. As a corollary of our main result, we obtain that the only rational curves lying on the general hypersuface $X_{2n-3}\subset \mathbf{P}^n$, for $n\geq 6,$are the lines.

References [Enhancements On Off] (What's this?)

  • [B] R. Bott, Homogeneous vector bundles, Ann. of Math. 66 (1957), 203-248.
  • [C] H. Clemens, Curves in generic hypersurfaces, Ann. Sci. École Norm. Sup. 19 (1986), 629-636.
  • [CLR] L. Chiantini, A. F. Lopez and Z. Ran, Subvarieties of generic hypersurfaces in any variety, Math. Proc. Cambr. Phil. Soc. 130 (2001), 259-268.
  • [DM] O. Debarre and L. Manivel, Sur la variété des espaces linéaires contenus dans une intersection complète, Math. Ann. 312 (1998), 549-574.
  • [E1] L. Ein, Subvarieties of generic complete intersections, Invent. Math. 94 (1988), 163-169.
  • [E2] L. Ein, Subvarieties of generic complete intersections II, Math. Ann. 289 (1991), 465-471.
  • [G] M. Green, Koszul cohomology and the geometry of projective varieties II, J. of Diff. Geometry 20 (1984), 279-289.
  • [M1] D. Mumford, Lectures on curves on an algebraic surface, Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J. 1966.
  • [M2] D. Mumford, Rational equivalence of $0$-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195-204.
  • [V1] C. Voisin, Variations de structure de Hodge et zéro-cycles sur les surfaces générales, Math. Ann. 299 (1994), 77-103.
  • [V2] C. Voisin, On a conjecture of Clemens on rational curves on hypersurfaces, J. of Diff. Geometry 44 (1996), 200-214.
  • [V3] C. Voisin, A correction on ``A conjecture of Clemens on rational curves on hypersurfaces", J. of Diff. Geometry 49 (1998), 601-611.

Additional Information

Gianluca Pacienza
Affiliation: Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4, Place Jussieu, F-75252 Paris CEDEX 05 - FRANCE
Address at time of publication: Department of Mathematics, Ohio State University, 100 Mathematics Building, 231 West 18th Avenue, Columbus, Ohio 43210-1174

Received by editor(s): October 2, 2000
Published electronically: October 17, 2002

American Mathematical Society