{
  "id": "1912.08109",
  "version": "v1",
  "published": "2019-12-17T16:03:51.000Z",
  "updated": "2019-12-17T16:03:51.000Z",
  "title": "Special groups, versality and the Grothendieck-Serre conjecture",
  "authors": [
    "Zinovy Reichstein",
    "Dajano Tossici"
  ],
  "comment": "13 pages. Comments are welcome",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let $k$ be a base field and $G$ be an algebraic group over $k$. J.-P. Serre defined $G$ to be special if every $G$-torsor $X \\to Y$ is locally trivial in the Zariski topology for every $k$-scheme $Y$. In recent papers an a priori weaker condition was used: $G$ is called special if every $G$-torsor $X \\rightarrow \\operatorname{Spec}(K)$ is split for every field extension $K/k$. We show that these two definitions are equivalent. We also generalize this fact and propose a strengthened version of the Grothendieck-Serre conjecture based on the notion of essential dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-17T16:03:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "grothendieck-serre conjecture",
      "special groups",
      "priori weaker condition",
      "base field",
      "zariski topology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}