The Grothendieck-Lefschetz theorem for normal projective varieties

Abstract

We prove that for a normal projective variety X X in characteristic 0 0 , and a base-point free ample line bundle L L on it, the restriction map of divisor class groups Cl ⁡ ( X ) → Cl ⁡ ( Y ) \operatorname {Cl} (X)\to \operatorname {Cl}(Y) is an isomorphism for a general member Y ∈ | L | Y\in |L| provided that dim ⁡ X ≥ 4 \dim {X}\geq 4 . This is a generalization of the Grothendieck-Lefschetz theorem, for divisor class groups of singular varieties.