{
  "id": "1603.08711",
  "version": "v1",
  "published": "2016-03-29T10:24:43.000Z",
  "updated": "2016-03-29T10:24:43.000Z",
  "title": "On twists of smooth plane curves",
  "authors": [
    "Eslam Badr",
    "Francesc Bars",
    "Elisa Lorenzo"
  ],
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Given a smooth curve defined over a field $k$ that admits a non-singular plane model over $\\overline{k}$, a fixed separable closure of $k$, it does not necessarily have a non-singular plane model defined over the field $k$. We determine under which conditions this happens and we show an example of such phenomenon. Now, even assuming that such a smooth plane model exists, we wonder about the existence of non-singular plane models over $k$ for its twists. We characterize twists possessing such models and use such characterization to improve, for the particular case of smooth plane curves, the algorithm to compute twists of non-hyperelliptic curves wrote recently down by the third author. We also show an example of a twist not admitting such non-singular plane model. As a consequence, we get explicit equations for a non-trivial Brauer-Severi surface. Finally, we obtain a theoretical result to compute all the twists of smooth plane curves with cyclic automorphism group having a $k$-model whose automorphism group is generated by a diagonal matrix. Some examples are also provided.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-29T10:24:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G30",
      "11D41",
      "14H37",
      "14H50",
      "14H45",
      "12F12"
    ],
    "keywords": [
      "smooth plane curves",
      "non-singular plane model",
      "non-hyperelliptic curves wrote",
      "smooth plane model",
      "non-trivial brauer-severi surface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160308711B"
    }
  }
}