{
  "id": "2109.08137",
  "version": "v1",
  "published": "2021-09-16T17:51:48.000Z",
  "updated": "2021-09-16T17:51:48.000Z",
  "title": "G-torsors and universal torsors over nonsplit del Pezzo surfaces",
  "authors": [
    "Ulrich Derenthal",
    "Norbert Hoffmann"
  ],
  "comment": "9 pages",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of $S_L$ in Pic $S_L$, or a form of it containing the N\\'eron-Severi torus. Let $\\mathcal{G}$ be the G-torsor over $S_L$ obtained by extension of structure group from a universal torsor $\\mathcal{T}$ over $S_L$. We prove that $\\mathcal{G}$ does not descend to S unless $\\mathcal{T}$ does. This is in contrast to a result of Friedman and Morgan that such $\\mathcal{G}$ always descend to singular del Pezzo surfaces over $\\mathbb{C}$ from their desingularizations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-16T17:51:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J26",
      "14L30",
      "11E57",
      "14G27"
    ],
    "keywords": [
      "nonsplit del pezzo surfaces",
      "universal torsor",
      "smooth del pezzo surface",
      "singular del pezzo surfaces",
      "structure group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}