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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
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##### 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
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