{
  "id": "1502.05959",
  "version": "v1",
  "published": "2015-02-20T18:09:11.000Z",
  "updated": "2015-02-20T18:09:11.000Z",
  "title": "On the existence of ordinary and almost ordinary Prym varieties",
  "authors": [
    "Ekin Ozman",
    "Rachel Pries"
  ],
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We study the relationship between the $p$-rank of a curve and the $p$-ranks of the Prym varieties of its cyclic covers in characteristic $p >0$. For arbitrary $p$, $g \\geq 3$ and $0 \\leq f \\leq g$, we generalize a result of Nakajima by proving that the Prym varieties of all unramified cyclic degree $\\ell \\not = p$ covers of a generic curve $X$ of genus $g$ and $p$-rank $f$ are ordinary. Furthermore, when $p \\geq 5$, we prove that there exists a curve of genus $g$ and $p$-rank $f$ having an unramified degree $\\ell=2$ cover whose Prym is almost ordinary. Using work of Raynaud, we use these two theorems to prove results about the (non)-intersection of the $\\ell$-torsion group scheme with the theta divisor of the Jacobian of a generic curve $X$ of genus $g$ and $p$-rank $f$. The proofs involve geometric results about the $p$-rank stratification of the moduli space of Prym varieties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-20T18:09:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G10",
      "14H10",
      "14H30",
      "14H40",
      "14K25",
      "11G20",
      "11M38",
      "14H42",
      "14K10",
      "14K15"
    ],
    "keywords": [
      "ordinary prym varieties",
      "generic curve",
      "torsion group scheme",
      "cyclic covers",
      "moduli space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150205959O"
    }
  }
}