{
  "id": "2011.14466",
  "version": "v1",
  "published": "2020-11-29T23:15:29.000Z",
  "updated": "2020-11-29T23:15:29.000Z",
  "title": "The Batyrev-Tschinkel conjecture for a non-normal cubic surface and its symmetric square",
  "authors": [
    "Nils Gubela",
    "Julian Lyczak"
  ],
  "comment": "19 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We complete the study of points of bounded height on irreducible non-normal cubic surfaces by doing the point count on the cubic surface $W$ given by $t_0^2 t_2 = t_1^2 t_3$ over any number field. We show that the order of growth agrees with a conjecture by Batyrev and Manin and that the constant reflects the geometry of the variety as predicted by a conjecture of Batyrev and Tschinkel. We then provide the point count for its symmetric square $\\mathrm{Sym}^2 W$. Although we can explain the main term of the counting function, the Batyrev--Manin conjecture is only satisfied after removing a thin set. Finally we interpret the main term of the count on $\\mathrm{Sym}^2(\\mathbb P^2 \\times \\mathbb P^1)$ done by Le Rudulier using these conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-29T23:15:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symmetric square",
      "batyrev-tschinkel conjecture",
      "main term",
      "point count",
      "irreducible non-normal cubic surfaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}