Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Noether-Lefschetz locus for surfaces


Author: Sung-Ock Kim
Journal: Trans. Amer. Math. Soc. 324 (1991), 369-384
MSC: Primary 14J10; Secondary 14J25, 14M10
MathSciNet review: 1043861
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Abstract: We generalize M. Green's Explicit Noether-Lefschetz Theorem to the family of smooth complete intersection surfaces in the higher dimensional projective spaces. Moreover, we give a new proof of the Density Theorem due to C. Ciliberto, J. Harris, and R. Miranda [5].


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  • [1] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932
  • [2] Raoul Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203–248. MR 0089473
  • [3] James Carlson, Mark Green, Phillip Griffiths, and Joe Harris, Infinitesimal variations of Hodge structure. I, Compositio Math. 50 (1983), no. 2-3, 109–205. MR 720288
  • [4] James A. Carlson and Phillip A. Griffiths, Infinitesimal variations of Hodge structure and the global Torelli problem, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 51–76. MR 605336
  • [5] Ciro Ciliberto, Joe Harris, and Rick Miranda, General components of the Noether-Lefschetz locus and their density in the space of all surfaces, Math. Ann. 282 (1988), no. 4, 667–680. MR 970227, 10.1007/BF01462891
  • [6] Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57 (French). MR 0498551
  • [7] Lawrence Ein, An analogue of Max Noether’s theorem, Duke Math. J. 52 (1985), no. 3, 689–706. MR 808098, 10.1215/S0012-7094-85-05235-4
  • [8] Mark L. Green, Koszul cohomology and the geometry of projective varieties. II, J. Differential Geom. 20 (1984), no. 1, 279–289. MR 772134
  • [9] Mark L. Green, The period map for hypersurface sections of high degree of an arbitrary variety, Compositio Math. 55 (1985), no. 2, 135–156. MR 795711
  • [10] Mark L. Green, A new proof of the explicit Noether-Lefschetz theorem, J. Differential Geom. 27 (1988), no. 1, 155–159. MR 918461
  • [11] Mark L. Green, Components of maximal dimension in the Noether-Lefschetz locus, J. Differential Geom. 29 (1989), no. 2, 295–302. MR 982176
  • [12] -, Griffiths' infinitesimal invariant and the Abel-Jacobi map, Preprint.
  • [13] -, Koszul cohomology and geometry, Preprint.
  • [14] P. Griffiths, Periods of integrals on algebraic manifolds. I, II, Amer. J. Math. 90 (1968), 568-626, 805-865.
  • [15] -, On the periods of certain rational integrals, Ann. of Math. (2) 90 (1969), 460-541.
  • [16] -, Periods of integrals on algebraic manifolds. III, Inst. Hautes Études Sci. Publ. Math. 63 (1970), 125-180.
  • [17] Phillip Griffiths and Joe Harris, On the Noether-Lefschetz theorem and some remarks on codimension-two cycles, Math. Ann. 271 (1985), no. 1, 31–51. MR 779603, 10.1007/BF01455794
  • [18] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
  • [19] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • [20] Robert Lazarsfeld, A sharp Castelnuovo bound for smooth surfaces, Duke Math. J. 55 (1987), no. 2, 423–429. MR 894589, 10.1215/S0012-7094-87-05523-2
  • [21] S. Lefschetz, On certain numerical invariants of algebraic varieties, Trans. Amer. Math. Soc. 22 (1921), 326-363.
  • [22] -, L'analysis situs et la geometrie algebrique, Gauthier-Villars, Paris, 1924.
  • [23] A. Lopez, On the Picard group of projective surfaces, Thesis, Brown Univ., Providence, R.I., 1988.
  • [24] David Mumford, Lectures on curves on an algebraic surface, With a section by G. M. Bergman. Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. MR 0209285
  • [25] M. Noether, Zur Grundlegung der Theorie der algebraischen Raumcurven, Berliner Abh., Berlin, 1882.
  • [26] C. Voisin, Une precision du théorème de Noether, Preprint.
  • [27] -, Composantes de petite codimension du lieu de Noether-Lefschetz, Preprint.

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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1043861-7
Article copyright: © Copyright 1991 American Mathematical Society