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. . Srinivas was partially supported by a Swarnajayanthi Fellowship of the D.S.T. 1 The terminology is from the non-singular case, where one

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... is considering restriction of line bundles. 563 564 G. V. RAVINDRA AND V. SRINIVAS This may be viewed as a particular case of the refined Gysin homomorphism 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 Grothendieck-Lefschetz theorem. Theorem 1. In the above situation, for a dense Zariski open set of Y ∈ |V|, the restriction map is an isomorphism, if dim X ≥ 4, and is injective, with finitely generated cokernel, if dim X = 3. Our proof is purely algebraic, in the style of the proof of the Grothendieck-Lefschetz theorem given in [11] , Chapter IV. The above result has an application in the theory of Deligne's 1-motives (see [4] ), which is discussed in §4 below; for this, it is of interest to have such an algebraic proof. In an appendix, we also sketch a different, transcendental proof of the theorem, when k = C, due to N. Fakhruddin, using results from stratified Morse theory, and properties of the weight filtration on cohomology. A version of the 5-lemma now implies that ker ψ 3 → ker ψ 4 and Coker ψ 3 → Coker ψ 4 are isomorphisms, as desired. Corollary 6.6. There is an exact sequence Proof. This follows from the proposition, and Lemma 6.1, since the explicit descriptions of K and Q imply that K ⊂ NS (X), and NS(Y ) Q.