zbMATH — the first resource for mathematics

The Noether-Lefschetz theorem for the divisor class group. (English) Zbl 1189.14010
The authors generalize the Noether-Lefschetz Theorem to normal three-folds. Let \(X\) be a normal projective three-fold and let \(V\subset H^0(X,\mathcal{O}_X(1))\) be a base point free linear system, where \(\mathcal{O}_X(1)\) is an ample line bundle. Let \(f:X\to \mathbb{P}^N\) be the induced morphism and assume that \((f_*K_X)(1)\) is globally generated. Then for a very general hyperplane section \(Y\in |V|\), the natural map \(\mathrm{Cl}(X)\to\mathrm{Cl}(Y)\) is an isomorphism. The usual Noether-Lefschetz theorem follows easily.
The method is to first prove a formal Noether-Lefschetz theorem like the one proved by the reviewer and the second author in the classical set up and deduce the theorem using similar methods as in the quoted unpublished article. These methods were also used by the authors to prove the Grothendieck-Lefschetz theorem for normal projective varieties [J. Algebr. Geom. 15, No. 3, 563–590 (2006; Zbl 1123.14004)].

14C22 Picard groups
14C20 Divisors, linear systems, invertible sheaves
13C20 Class groups
Full Text: DOI
[1] Green, M., A new proof of the explicit algebraic noether – lefschetz theorem, J. differential geom., 27, 1, 155-159, (1988) · Zbl 0674.14005
[2] Grothendieck, A., Cohomologie locale des faisceaux coherents et theoremes de Lefschetz locaux et globaux (SGA2), (1968), North-Holland
[3] Deligne, P.; Katz, N., Séminaire de Géométrie algébrique du bois – marie—1967-1969. groupes de monodromie en géométrie algébrique. II, Lecture notes in math., vol. 340, (1973), Springer-Verlag, [SGA7II]
[4] Fulton, W., Intersection theory, Ergeb. math. grenzgeb. (3), vol. 2, (1998), Springer-Verlag · Zbl 0885.14002
[5] Esnault, H.; Viehweg, E., Lectures on vanishing theorems, DMV seminar, Band 20, (1992), Birkhäuser
[6] Fulton, W.; Harris, J., Representation theory: A first course, Grad. texts in math., vol. 129, (1991), Springer-Verlag, (corrected fifth printing) · Zbl 0744.22001
[7] Hartshorne, R., Algebraic geometry, Grad. texts in math., vol. 52, (1977), Springer-Verlag · Zbl 0367.14001
[8] Hartshorne, R., Ample subvarieties of algebraic varieties, Lecture notes in math., vol. 156, (1970), Springer-Verlag · Zbl 0208.48901
[9] Joshi, K., A noether – lefschetz theorem and applications, J. algebraic geom., 4, 1, 105-135, (1995) · Zbl 0843.14004
[10] N. Mohan Kumar, V. Srinivas, The Noether-Lefschetz theorem, unpublished notes, 1990
[11] Ramanujam, C.P., Remarks on Kodaira’s vanishing theorem, (), 36, 41-51, (1972), reprinted · Zbl 0276.32018
[12] Ravindra, G.V.; Srinivas, V., The grothendieck – lefschetz theorem for normal projective varieties, J. algebraic geom., 15, 563-590, (2006) · Zbl 1123.14004
[13] Mohan Kumar, N.; Rao, A.P.; Ravindra, G.V., Generators for vector bundles on generic hypersurfaces, Math. res. lett., 14, 4, 649-655, (2007) · Zbl 1134.14011
[14] Ravindra, G.V., The noether – lefschetz theorem via vanishing of coherent cohomology, Canad. math. bull., 51, 2, 283-290, (2008) · Zbl 1151.14008
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.