Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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
Full-text PDF

Abstract | 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.


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

  • [Ba-Mu] 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] J. BERTIN, J.-P. DEMAILLY, L. ILLUSIE, C. PETERS. Introduction à la théorie de Hodge. Panorama et synthèses, publications SMF (1996).
  • [C-D-K] 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. 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] M. GREEN. Components of maximal dimension in the Noether-Lefschetz locus. J. Differential Geometry 29, p.295-302 (1989).
  • [IVHS I] J. CARLSON, M. GREEN, P. GRIFFITHS, J. HARRIS. Infinitesimal variations of Hodge structures I. Compositio. Math. 50, p.109-205 (1983).
  • [L] A. F. LOPEZ. Noether-Lefschetz theory and the Picard group of projective surfaces. Mem. Amer. Math. Soc. 89 (1991).
  • [G-H] P. GRIFFITHS, J. HARRIS. On the Noether-Lefschetz theorem and some remarks on codimension two cycles. Compositio. Math. 50, p.207-265 (1983).
  • [M] F. S. MACAULAY. Some properties of enumeration in the theory of modular systems. Proc. Lond. Math. Soc 26, p.531-555 (1927).
  • [O] A. OTWINOWSKA. Sur la fonction de Hilbert des algèbres graduées de dimension 0. Soumis (2000).
  • [V 1] C. VOISIN. Une précision concernant le théorème de Noether. Math. Ann. 280, p.605-611 (1989).
  • [V 2] 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

DOI: https://doi.org/10.1090/S1056-3911-02-00349-1
Received by editor(s): October 31, 2000
Published electronically: November 14, 2002

American Mathematical Society