{
  "id": "math/0512597",
  "version": "v1",
  "published": "2005-12-27T12:23:06.000Z",
  "updated": "2005-12-27T12:23:06.000Z",
  "title": "The action of the Frobenius map on rank 2 vector bundles over genus 2 curves in small characteristics",
  "authors": [
    "Laurent Ducrohet"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X$ be genus 2 curve defined over an algebraically closed field of characteristic $p$ and let $X\\_1$ be its $p$-twist. Let $M\\_X$ (resp. $M\\_{X\\_1}$) be the (coarse) moduli space of semi-stable rank 2 vector bundles with trivial determinant over $X$ (resp. $X\\_1$). The moduli space $M\\_X$ is isomorphic to the 3-dimensional projective space and is endowed with an action of the group $J[2]$ of order 2 line bundles over $X$. When $3\\leq p \\leq 7$, we show that the Verschiebung (i.e., the separable part of the action of Frobenius by pull-back) $V : M\\_{X\\_1} \\dashrightarrow M\\_X$ is completely determined by its restrictions to the lines that are invariant under the action of a non zero element of $J[2]$. Those lines correspond to elliptic curves that appear as Prym varieties and the Verschiebung restricts to the morphism induced by multiplication by $p$. Therefore, we are able to compute the explicit equations of the Verschiebung when the base field has characteristic 3, 5 or 7.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-12-27T12:23:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H60"
    ],
    "keywords": [
      "vector bundles",
      "frobenius map",
      "small characteristics",
      "moduli space",
      "non zero element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....12597D"
    }
  }
}