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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reviewer and the second author in the classical set up and deduce the theorem using similar methods as in the quoted unpublished article.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)].

##### MSC:
 14C22 Picard groups 14C20 Divisors, linear systems, invertible sheaves 13C20 Class groups
##### Keywords:
divisor class group; Noether-Lefschetz theorem; three-folds
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##### References:
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