(``Elementary'') proof of the Grothendieck-Lefschetz theorem for hypersurfaces (Q1327121)

Let \(Y \subset \mathbb{P}^ n_ k\) be an algebraic hypersurface. The Picard group of \(Y\) is isomorphic to \(\mathbb{Z}\) and is generated by the class of an hyperplane section of \(Y\). This result has been proved by Lefschetz using transcendental methods and Grothendieck using formal completion. We give a new proof in the framework of classical projective geometry.