Rational curves on general projective hypersurfaces

Author:
Gianluca Pacienza

Journal:
J. Algebraic Geom. **12** (2003), 245-267

DOI:
https://doi.org/10.1090/S1056-3911-02-00328-4

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.

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

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

Email:
pacienza@math.jussieu.fr, pacienza@math.ohio-state.edu

Received by editor(s):
October 2, 2000

Published electronically:
October 17, 2002