{
  "id": "1604.02381",
  "version": "v1",
  "published": "2016-04-08T15:41:31.000Z",
  "updated": "2016-04-08T15:41:31.000Z",
  "title": "Morphisms of 1-motives defined by line bundles",
  "authors": [
    "Cristiana Bertolin",
    "Sylvain Brochard"
  ],
  "comment": "23 pages",
  "categories": [
    "math.AG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-08T15:41:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14K30",
      "14C20"
    ],
    "keywords": [
      "line bundle",
      "linear morphism",
      "group homomorphism",
      "normal base scheme",
      "cartier dual"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160402381B"
    }
  }
}