Hostname: page-component-f554764f5-fr72s Total loading time: 0 Render date: 2025-04-12T00:14:00.902Z Has data issue: false hasContentIssue false
  • English
  • Français

Explicit Noether–Lefschetz for arbitrary threefolds

Published online by Cambridge University Press:  01 September 2007

ANGELO FELICE LOPEZ  and
CATRIONA MACLEAN
ANGELO FELICE LOPEZ
Affiliation:
Dipartimento di Matematica, Università di Roma Tre, Largo San Leonardo Murialdo 1, 00146, Roma, Italy. email: [email protected]
CATRIONA MACLEAN
Affiliation:
UFR de Mathèmatiques, UMR 5582 CNRS/Universitè J. Fourier 100, rue des Maths, BP 74, 38402 St Martin d'Heres, France. email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the Noether–Lefschetz locus of a very ample line bundle L on an arbitrary smooth threefold Y. Building on results of Green, Voisin and Otwinowska, we give explicit bounds, depending only on the Castelnuovo–Mumford regularity properties of L, on the codimension of the components of the Noether–Lefschetz locus of |L|.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 2 , September 2007 , pp. 323 - 342
Copyright
Copyright © Cambridge Philosophical Society 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Carlson, J. and Griffiths, P.. Infinitesimal Variations of Hodge Structure and the Global Torelli Problem. Géométrie Algébrique, ed. Beauville, A. (Sijthoff–Noordhoff, 1980), 51–76.Google Scholar
[2]Angelini, F.. Ample divisors on the blow up of P3 at points. Manuscripta Math. 93 (1) (1997), 39–48.CrossRefGoogle Scholar
[3]Ballico, E.. Ample divisors on the blow up of Pn at points. Proc. Amer. Math. Soc. 127 (9) (1999), 2527–2528.CrossRefGoogle Scholar
[4]Ein, L.. The ramification divisors for branched coverings of Pnk. Math. Ann. 261 (4) (1982), 483–485.CrossRefGoogle Scholar
[5]Ein, L. and Lazarsfeld, R.. Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension. Invent. Math. 111 (1) (1993), 51–67.CrossRefGoogle Scholar
[6]Gherardelli, F.. Un teorema di Lefschetz sulle intersezioni complete. Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. 28 (8) (1960), 610–614.Google Scholar
[7]Gotzmann, G.. Eine Bedingung für die Flachheit und das Hilbert polynom eines Graduierten Ringes. Math. Z. 158 (1) (1978), 61–70.CrossRefGoogle Scholar
[8]Green, M. L.. A new proof of the explicit Noether–Lefschetz theorem. J. Differential Geom. 27 (1) (1988), 155–159.CrossRefGoogle Scholar
[9]Green, M. L.. Restrictions of Linear Series to Hyperplanes, and Some Results of Macaulay and Gotzmann. Algebraic curves and projective geometry (Trento, 1988), 76–86, Lecture Notes in Math. 1389 (Springer, 1989).CrossRefGoogle Scholar
[10]Griffiths, P.. Periods of integrals on algebraic manifolds, II. Local study of the period mapping. Amer. J. Math. 90 (1968), 805–865.CrossRefGoogle Scholar
[11]Griffiths, P.. On the periods of certain rational integrals I, II. Ann. of Math. 90 (2) (1969), 460–495; ibid. 90 (2) (1969), 496–541.CrossRefGoogle Scholar
[12]Griffiths, P.. Hermitian Differential Geometry, Chern Classes, and Positive Vector Bundles. Global Analysis, Papers in honor of K. Kodaira (Princeton University Press, 1969), 181–251.Google Scholar
[13]Joshi, K.. A Noether–Lefschetz theorem and applications. J. Algebraic Geom. 4 (1) (1995), 105–135.Google Scholar
[14]Moishezon, B. G.. Algebraic homology classes on algebraic varieties. Izv. Akad. Nauk SSSR 31 (1967), 209–251.Google Scholar
[15]Otwinowska, A.. Monodromie d'une famille d'hypersurfaces; application à l'étude du lieu de Noether-Lefschetz. Submitted for publication.Google Scholar
[16]Otwinowska, A.. Composantes de petite codimension du lieu de Noether-Lefschetz: un argument asymptotique en faveur de la conjecture de Hodge pour les hypersurfaces. J. Algebraic Geom. 12 (2) (2003), 307–320.CrossRefGoogle Scholar
[17]Otwinowska, A.. Sur les variètès de Hodge des hypersurfaces. Preprint arXiv.math.AG/401092.Google Scholar
[18]Severi, F.. Una proprietà delle forme algebriche prive di punti multipli. Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. 15 (5) (1906), 691–696.Google Scholar
[19]Sommese, A. J. and Van de Ven, A.. On the adjunction mapping. Math. Ann. 278 (1–4) (1987), 593–603.CrossRefGoogle Scholar
[20]Voisin, C.Une précision concernant le théorème de Noether. Math. Ann. 280 (4) (1988), 605–611.CrossRefGoogle Scholar
[21]Voisin, C.Théorie de Hodge et géométrie algébrique complexe. Cours Spécialisés, 10 (Société Mathématique de France, 2002).Google Scholar