Abstract. We prove that for a normal projective variety X in characteristic 0, and a base-point free ample line bundle L on it, the restriction map of divisor class groups Cl(X) → Cl(Y ) is an isomorphism for a general member Y ∈ |L| provided that dim X ≥ 4. This is a generalization of the Grothendieck-Lefschetz Theorem, for divisor class groups of singular varieties.We work over k, an algebraically closed field of characteristic 0. Let X be a smooth projective variety over k and Y a smooth complete intersection subvariety of X. The Grothendieck-Lefschetz theorem states that if dimension Y ≥ 3, the Picard groups of X and Y are isomorphic.In this paper, we wish to prove an analogous statement for singular varieties, with the Picard group replaced by the divisor class group.Let X be an irreducible projective variety which is regular in codimension 1 (for example, X may be irreducible and normal). Recall that for such X, the divisor class group Cl(X) is defined as the group of linear equivalence classes of Weil divisors on X (see [10], II, §6). If dim X = d, then Cl(X) coincides with the Chow group CH d−1 (X) as defined in Fulton's book [7]. If Y ⊂ X is an irreducible Cartier divisor, which is also regular in codimension 1, there is a well-defined restriction homomorphism 1where D is any irreducible Weil divisor in X distinct from Y , and [D ∩ Y ] denotes the Weil divisor on Y associated to the intersection scheme D ∩ Y . This may be viewed as a particular case of the refined Gysin homomorphism CH i (X) → CH i−1 (Y ) defined in [7], for i = dim X − 1. Now let X be an irreducible projective variety over k, regular in codimension 1, and let L be an ample line bundle over X, together with a linear subspace V ⊂ H 0 (X, L ) which gives a base point free ample linear system |V| on X. Let Y ∈ |V| be a general element of this linear system; by Bertini's theorem, we have Y sing = Y ∩ X sing . In this context, our main result is the following, which is an analogue of the GrothendieckLefschetz theorem. Theorem 1. In the above situation, for a dense Zariski open set of Y ∈ |V|, the restriction mapis an isomorphism, if dim X ≥ 4, and is injective, with finitely generated cokernel, if dim X = 3.

Let X be a normal projective threefold over a field of characteristic zero and |L| be a base-point free, ample linear system on X. Under suitable hypotheses on (X, |L|), we prove that for a very general member Y ∈ |L|, the restriction map on divisor class groups Cl(X) → Cl(Y ) is an isomorphism. In particular, we are able to recover the classical Noether-Lefschetz theorem, that a very general hypersurface X ⊂ P 3 C of degree 4 has Pic(X) ∼ = Z.We work over an algebraically closed field of characteristic zero, which shall be denoted by k. Let X be an irreducible normal projective 3-fold defined over k and O X (1) be an ample line bundle on X. Let V ⊂ H 0 (X, O X (1)) define a base-point free linear system |V |, so that the induced morphism f : X → P N := P(V ) is finite. If Y is a general member of this linear system on X, then Y is a normal projective surface, by Bertini's theorem. Since Y is an effective Cartier divisor in X, we have a natural homomorphism Cl(X) → Cl(Y ) on divisor class groups,

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We prove that any rank two arithmetically Cohen-Macaulay vector bundle on a general hypersurface of degree at least six in P 4 must be split.

We prove that any arithmetically Gorenstein curve on a smooth, general hypersurface X ⊂ P 4 of degree at least 6, is a complete intersection. This gives a characterisation of complete intersection curves on general type hypersurfaces in P 4 . We also verify that certain 1-cycles on a general quintic hypersurface are non-trivial elements of the Griffiths group.