Composantes de petite codimension du lieu de Noether-Lefschetz: Un argument asymptotique en faveur de la conjecture de Hodge pour les hypersurfaces
Author:
Ania Otwinowska
Journal:
J. Algebraic Geom. 12 (2003), 307-320
DOI:
https://doi.org/10.1090/S1056-3911-02-00349-1
Published electronically:
November 14, 2002
MathSciNet review:
1949646
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References |
Additional Information
Abstract:
This paper gives an asymptotic description of the Noether-Lefschetz locus for smooth projective hypersurfaces in $\mathbb {P}^{2n+1}_{\mathbb {C}}$ of large degree. I prove that successive small codimensional components of this locus correspond to surfaces containing a small degree subvariety of dimension $n$. This result generalises the work of Green and Voisin for surfaces in $\mathbb {P}^3_{\mathbb {C}}$ containing a line and a conic.
Résumé Cet article donne une description asymptotique du lieu de Noether-Lefschetz pour les hypersurfaces lisses de grand degré dans $\mathbb {P}^{2n+1}_{\mathbb {C}}$: les composantes succéssives de plus petite codimension de ce lieu sont constituées par les hypersurfaces contenant une sous-variété de dimension $n$ et de petit degré. Ce résultat généralise les travaus de Green et Voisin sur les surfaces de $\mathbb {P}^3_{\mathbb {C}}$ contenant une droite et une conique.
[Ba-Mu]bamu D. Bayer, D. Mumford. What can be computed in algebraic geometry? Computational algebraic geometry and commutative algebra (Cortona 1991), 1-48, Sympos. Math. XXXIV, Cambridge Univ. Press, Cambridge (1996).
[B-D-I-P]ps J. Bertin, J.-P. Demailly, L. Illusie, C. Peters. Introduction à la théorie de Hodge. Panorama et synthèses, publications SMF (1996).
[C-D-K]CDK E. Cattani, P. Deligne, A. Kaplan. On the locus of Hodge classes. Journal of the AMS, Vol. 8, n.2 (1995).
[C-H-M]C-H-M C. Cilberto, J. Harris, R. Miranda. General Components of the Noether-Lefschetz locus and their Density in the space of moduli. Math. Annalen 282, p.667-680 (1988).
[G]NLgreen M. Green. Components of maximal dimension in the Noether-Lefschetz locus. J. Differential Geometry 29, p.295-302 (1989).
[IVHS I]IVHS1 J. Carlson, M. Green, P. Griffiths, J. Harris. Infinitesimal variations of Hodge structures I. Compositio. Math. 50, p.109-205 (1983).
[L]lopez A. F. Lopez. Noether-Lefschetz theory and the Picard group of projective surfaces. Mem. Amer. Math. Soc. 89 (1991).
[G-H]hodgevariationnel P. Griffiths, J. Harris. On the Noether-Lefschetz theorem and some remarks on codimension two cycles. Compositio. Math. 50, p.207-265 (1983).
[M]macaulay F. S. Macaulay. Some properties of enumeration in the theory of modular systems. Proc. Lond. Math. Soc 26, p.531-555 (1927).
[O]O2 A. Otwinowska. Sur la fonction de Hilbert des algèbres graduées de dimension 0. Soumis (2000).
[V 1]NLclaire C. Voisin. Une précision concernant le théorème de Noether. Math. Ann. 280, p.605-611 (1989).
[V 2]NLclaire2 C. Voisin. Composantes de petite codimension du lieu de Noether-Lefschetz. Comment. Math. Helvetici 64, p.515-526 (1989).
[Ba-Mu]bamu D. Bayer, D. Mumford. What can be computed in algebraic geometry? Computational algebraic geometry and commutative algebra (Cortona 1991), 1-48, Sympos. Math. XXXIV, Cambridge Univ. Press, Cambridge (1996).
[B-D-I-P]ps J. Bertin, J.-P. Demailly, L. Illusie, C. Peters. Introduction à la théorie de Hodge. Panorama et synthèses, publications SMF (1996).
[C-D-K]CDK E. Cattani, P. Deligne, A. Kaplan. On the locus of Hodge classes. Journal of the AMS, Vol. 8, n.2 (1995).
[C-H-M]C-H-M C. Cilberto, J. Harris, R. Miranda. General Components of the Noether-Lefschetz locus and their Density in the space of moduli. Math. Annalen 282, p.667-680 (1988).
[G]NLgreen M. Green. Components of maximal dimension in the Noether-Lefschetz locus. J. Differential Geometry 29, p.295-302 (1989).
[IVHS I]IVHS1 J. Carlson, M. Green, P. Griffiths, J. Harris. Infinitesimal variations of Hodge structures I. Compositio. Math. 50, p.109-205 (1983).
[L]lopez A. F. Lopez. Noether-Lefschetz theory and the Picard group of projective surfaces. Mem. Amer. Math. Soc. 89 (1991).
[G-H]hodgevariationnel P. Griffiths, J. Harris. On the Noether-Lefschetz theorem and some remarks on codimension two cycles. Compositio. Math. 50, p.207-265 (1983).
[M]macaulay F. S. Macaulay. Some properties of enumeration in the theory of modular systems. Proc. Lond. Math. Soc 26, p.531-555 (1927).
[O]O2 A. Otwinowska. Sur la fonction de Hilbert des algèbres graduées de dimension 0. Soumis (2000).
[V 1]NLclaire C. Voisin. Une précision concernant le théorème de Noether. Math. Ann. 280, p.605-611 (1989).
[V 2]NLclaire2 C. Voisin. Composantes de petite codimension du lieu de Noether-Lefschetz. Comment. Math. Helvetici 64, p.515-526 (1989).
Additional Information
Ania Otwinowska
Affiliation:
Mathematical Institute, University of Warwick, Coventry CV4 7AL, England
Address at time of publication:
Université Paris-Sud, Bât 425, 91405 Orsay Cedex, France
Email:
ania@maths.warwick.ac.uk
Received by editor(s):
October 31, 2000
Published electronically:
November 14, 2002
