×

zbMATH — the first resource for mathematics

Noether-Lefschetz locus for surfaces. (English) Zbl 0739.14019
The Noether-Lefschetz theorem says that a general surface \(S\) of degree \(d\geq 4\) in \(\mathbb{C}\mathbb{P}^ 3\) contains only curves which are complete intersections of \(S\) with another surface. The Noether-Lefschetz locus \(\Sigma_ 3\) is given by the “non general” surfaces, that are smooth surfaces \(S\) for which \(\hbox{Pic}(S)\) is not generated by the hyperplane class. The problem is to study \(\Sigma_ 3\) and more generally \(\Sigma_ n\), defined in a similar way for surfaces in \(\mathbb{C}\mathbb{P}^ n\). M. Green [J. Differ. Geom. 27, No. 1, 155-159 (1988; Zbl 0674.14005) and 29, No. 2, 295-302 (1989; Zbl 0674.14003) and No. 3, 545– -555 (1989; Zbl 0692.14003)] and C. Ciliberto, J. Harris and R. Miranda [Math. Ann. 282, No. 4, 667-680 (1988; Zbl 0671.14017) and now also in Mem. Am. Math. Soc.] have obtained many interesting results in this area. In particular the components of maximal codimension in \(\Sigma_ 3\) are infinitely many and dense in the space of non singular surfaces (both in Zariski and classical topology). The paper contains, as a main result, a new, simpler proof of this result.

MSC:
14C22 Picard groups
14M10 Complete intersections
14J10 Families, moduli, classification: algebraic theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[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. · Zbl 0559.14017
[2] Raoul Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203 – 248. · Zbl 0094.35701
[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. Phillip Griffiths and Joe Harris, Infinitesimal variations of Hodge structure. II. An infinitesimal invariant of Hodge classes, Compositio Math. 50 (1983), no. 2-3, 207 – 265. Phillip A. Griffiths, Infinitesimal variations of Hodge structure. III. Determinantal varieties and the infinitesimal invariant of normal functions, Compositio Math. 50 (1983), no. 2-3, 267 – 324. · Zbl 0531.14006
[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. · Zbl 0479.14007
[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. · Zbl 0671.14017
[6] Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5 – 57 (French). Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5 – 77 (French).
[7] Lawrence Ein, An analogue of Max Noether’s theorem, Duke Math. J. 52 (1985), no. 3, 689 – 706. · Zbl 0589.14034
[8] Mark L. Green, Koszul cohomology and the geometry of projective varieties. II, J. Differential Geom. 20 (1984), no. 1, 279 – 289. · Zbl 0559.14009
[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. · Zbl 0588.14004
[10] Mark L. Green, A new proof of the explicit Noether-Lefschetz theorem, J. Differential Geom. 27 (1988), no. 1, 155 – 159. · Zbl 0674.14005
[11] Mark L. Green, Components of maximal dimension in the Noether-Lefschetz locus, J. Differential Geom. 29 (1989), no. 2, 295 – 302. · Zbl 0674.14003
[12] -, Griffiths’ infinitesimal invariant and the Abel-Jacobi map, Preprint. · Zbl 0692.14003
[13] -, Koszul cohomology and geometry, Preprint. · Zbl 0800.14004
[14] P. Griffiths, Periods of integrals on algebraic manifolds. I, II, Amer. J. Math. 90 (1968), 568-626, 805-865. · Zbl 0169.52303
[15] -, On the periods of certain rational integrals, Ann. of Math. (2) 90 (1969), 460-541. · Zbl 0215.08103
[16] -, Periods of integrals on algebraic manifolds. III, Inst. Hautes Études Sci. Publ. Math. 63 (1970), 125-180. · Zbl 0212.53503
[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. · Zbl 0552.14011
[18] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. · Zbl 0408.14001
[19] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001
[20] Robert Lazarsfeld, A sharp Castelnuovo bound for smooth surfaces, Duke Math. J. 55 (1987), no. 2, 423 – 429. · Zbl 0646.14005
[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. · Zbl 0187.42701
[25] M. Noether, Zur Grundlegung der Theorie der algebraischen Raumcurven, Berliner Abh., Berlin, 1882. · JFM 15.0684.01
[26] C. Voisin, Une precision du théorème de Noether, Preprint.
[27] -, Composantes de petite codimension du lieu de Noether-Lefschetz, Preprint. · Zbl 0724.14029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.