{
  "id": "1308.1970",
  "version": "v2",
  "published": "2013-08-08T20:37:15.000Z",
  "updated": "2013-10-02T22:00:27.000Z",
  "title": "Index conditions and cup-product maps on abelian varieties",
  "authors": [
    "Nathan Grieve"
  ],
  "comment": "Revised Theorem 2.3",
  "categories": [
    "math.AG"
  ],
  "abstract": "We study questions surrounding cup-product maps which arise from pairs of non-degenerate line bundles on an abelian variety. Important to our work is Mumford's index theorem which we use to prove that non-degenerate line bundles exhibit positivity analogous to that of ample line bundles. As an application we determine the asymptotic behaviour of families of cup-product maps and prove that vector bundles associated to these families are asymptotically globally generated. To illustrate our results we provide several examples. For instance, we construct families of cup-product problems which result in a zero map on a one dimensional locus. We also prove that the hypothesis of our results can be satisfied, in all possible instances, by a particular class of simple abelian varieties. Finally, we discuss the extent to which Mumford's theta groups are applicable in our more general setting.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-02T22:00:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14K05"
    ],
    "keywords": [
      "abelian variety",
      "index conditions",
      "non-degenerate line bundles",
      "study questions surrounding cup-product maps",
      "simple abelian varieties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.1970G"
    }
  }
}