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 dévissage of the Picard group of a 1-motive $M$ according to the weight filtration of $M$. This dévissage allows us to associate, to each line bundle $L$ on $M$, a linear morphism ${\phi }_L:M\to M^{\ast }$ from $M$ to its Cartier dual. This yields a group homomorphism $\Phi :Pic(M)/Pic(S)\to Hom(M,M^{\ast })$. We also prove the Theorem of the Cube for 1-motives, which furnishes another construction of the group homomorphism $\Phi :Pic(M)/Pic(S)\to Hom(M,M^{\ast })$. Finally we prove that these two independent constructions of linear morphisms $M\to M^{\ast }$ using line bundles on $M$ coincide. However, the first construction, involving the dévissage of $Pic(M)$, is more explicit and geometric and it furnishes the motivic origin of some linear morphisms between 1-motives. The second construction, involving the Theorem of the Cube, is more abstract but perhaps also more enlightening.

29 pages. To appear in IMRN. Final version, including the remarks of the referee. Modified the numberings of the sections to match the published version

References in corpus (1)

• Foncteur de Picard d'un champ algébrique

Cited by in corpus (2)

• Mumford-Tate groups of 1-motives and Weil pairing
• The monodromy pairing for logarithmic 1-motifs