Morphisms of 1-motives defined by line bundles

Abstract

Let S be a normal base scheme. The aim of this paper is to study the line bundles on 1-motives defined over S. We first compute a devissage of the Picard group of a 1-motive M according to the weight filtration of M. This devissage allows us to associate, to each line bundle Ⅎ on M, a linear morphism φℲ : M → M∗ from M to its Cartier dual. This yields a group homomorphism Φ : Pic(M)/Pic(S) → Hom(M,M∗). We also prove the Theorem of the Cube for 1-motives, which furnishes another construction of the group homomorphism Φ : Pic(M)/Pic(S) → Hom(M,M∗). Finally, we prove that these two independent constructions of linear morphisms M → M∗ using line bundles on M coincide. However, the 1st construction, involving the devissage of Pic(M), is more explicit and geometric and it furnishes the motivic origin of some linear morphisms between 1-motives. The 2nd construction, involving the Theorem of the Cube, is more abstract but also more enlightening.