{
  "id": "math/0702525",
  "version": "v1",
  "published": "2007-02-18T14:54:02.000Z",
  "updated": "2007-02-18T14:54:02.000Z",
  "title": "A conic bundle degenerating on the Kummer surface",
  "authors": [
    "Michele Bolognesi"
  ],
  "comment": "29 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $C$ be a genus 2 curve and $\\su$ the moduli space of semi-stable rank 2 vector bundles on $C$ with trivial determinant. In \\cite{bol:wed} we described the parameter space of non stable extension classes (invariant with respect to the hyperelliptic involution) of the canonical sheaf $\\omega$ of $C$ with $\\omega_C^{-1}$. In this paper we study the classifying rational map $\\phi: \\pr Ext^1(\\omega,\\omega^{-1})\\cong \\pr^4 \\dashrightarrow \\su\\cong \\pr^3$ that sends an extension class on the corresponding rank two vector bundle. Moreover we prove that, if we blow up $\\pr^4$ along a certain cubic surface $S$ and $\\su$ at the point $p$ corresponding to the bundle $\\OO \\oplus \\OO$, then the induced morphism $\\tilde{\\phi}: Bl_S \\ra Bl_p\\su$ defines a conic bundle that degenerates on the blow up (at $p$) of the Kummer surface naturally contained in $\\su$. Furthermore we construct the $\\pr^2$-bundle that contains the conic bundle and we discuss the stability and deformations of one of its components.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-02-18T14:54:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H60",
      "14J70"
    ],
    "keywords": [
      "kummer surface",
      "conic bundle degenerating",
      "vector bundle",
      "non stable extension classes",
      "trivial determinant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......2525B"
    }
  }
}