Morphisms of 1-motives defined by line bundles

TL;DR

This paper investigates line bundles on 1-motives over a normal base scheme, establishing a connection between Picard groups and morphisms to the dual, and proving the Theorem of the Cube for 1-motives.

Contribution

It introduces two independent methods to construct linear morphisms from 1-motives to their duals using line bundles, and proves their equivalence.

Findings

Computed a devissage of the Picard group of 1-motives.

Established a group homomorphism from Picard group to Hom(M, M*).

Proved the Theorem of the Cube for 1-motives.

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 d\'evissage of the Picard group of a 1-motive $M$ according to the weight filtration of $M$. This d\'evissage allows us to associate, to each line bundle $L$ on $M$, a linear morphism $\varphi_{L}: M \rightarrow M^*$ from $M$ to its Cartier dual. This yields a group homomorphism $\Phi : Pic(M) / Pic(S) \to Hom(M,M^*)$. We also prove the Theorem of the Cube for 1-motives, which furnishes another construction of the group homomorphism $\Phi : Pic(M) / Pic(S) \to Hom(M,M^*)$. Finally we prove that these two independent constructions of linear morphisms $M \to M^*$ using line bundles on $M$ coincide. However, the first construction, involving the d\'evissage 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.