{
  "id": "math/0503026",
  "version": "v1",
  "published": "2005-03-02T00:21:51.000Z",
  "updated": "2005-03-02T00:21:51.000Z",
  "title": "Cubic equations for the hyperelliptic locus",
  "authors": [
    "Samuel Grushevsky"
  ],
  "journal": "Asian J Math, vol 8 (2004) no. 1, special issue dedicated to Yum-Tong Siu on his 60th birthday",
  "categories": [
    "math.AG"
  ],
  "abstract": "We discuss the conjecture of Buchstaber and Krichever that their multi-dimensional vector addition formula for Baker-Akhiezer functions characterizes Jacobians among principally polarized abelian varieties, and prove that it is indeed a weak characterization, i.e. that it is true up to additional components, or true precisely under a general position assumption. We also show that this addition formula is equivalent to Gunning's multisecant formula for the Kummer variety. We then use Buchstaber-Krichever's computation of the coefficients in the addition formula to obtain cubic relations among theta functions, which (weakly) characterize the locus of hyperelliptic Jacobians among irreducible abelian varieties. In genus 3 our equations are equivalent to the vanishing of one theta-null, and thus are known classically by work of Mumford and Poor, but already for genus 4 they appear to be new.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-03-02T00:21:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyperelliptic locus",
      "cubic equations",
      "multi-dimensional vector addition formula",
      "baker-akhiezer functions characterizes jacobians",
      "general position assumption"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......3026G"
    }
  }
}