On the Picard Group of Low-codimension Subvarieties

Abstract

We introduce a method to determine if n-dimensional smooth subvarieties of an ambient space of dimension at most 2n - 2 inherits the Picard group from the ambient space (as it happens when the ambient space is a projective space, according to results of Barth and Larsen). As an application, we give an affirmative answer (LIP to some mild natural numerical conditions) when the ambient space is a Grassmannian of lines (thus improving results of Barth, Van de Ven and Sommese) or a product of two projective spaces of the same dimension.