{
  "id": "2111.03546",
  "version": "v1",
  "published": "2021-11-05T15:08:32.000Z",
  "updated": "2021-11-05T15:08:32.000Z",
  "title": "Persistence of the Brauer-Manin obstruction on cubic surfaces",
  "authors": [
    "Carlos Rivera",
    "Bianca Viray"
  ],
  "comment": "5 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $X$ be a cubic surface over a global field $k$. We prove that a Brauer-Manin obstruction to the existence of $k$-points on $X$ will persist over every extension $L/k$ with degree relatively prime to $3$. In other words, a cubic surface has nonempty Brauer set over $k$ if and only if it has nonempty Brauer set over some extension $L/k$ with $3\\nmid[L:k]$. Therefore, the conjecture of Colliot-Th\\'el\\`ene and Sansuc on the sufficiency of the Brauer-Manin obstruction for cubic surfaces implies that $X$ has a $k$-rational point if and only if $X$ has a $0$-cycle of degree $1$. This latter statement is a special case of a conjecture of Cassels and Swinnerton-Dyer.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-11-05T15:08:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G05",
      "14J20",
      "14J26",
      "11D25",
      "11G35"
    ],
    "keywords": [
      "brauer-manin obstruction",
      "nonempty brauer set",
      "persistence",
      "cubic surfaces implies",
      "degree relatively prime"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}