Title: On the Picard group of low-codimension subvarieties

Authors: Enrique Arrondo and Jorge Caravantes

Issue: Volume 58 (2009), Issue 3, 1023-1050

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 (up 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.