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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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
