{
  "id": "2110.13687",
  "version": "v2",
  "published": "2021-10-26T13:25:06.000Z",
  "updated": "2022-04-18T13:11:28.000Z",
  "title": "Quartic del Pezzo surfaces with a Brauer group of order 4",
  "authors": [
    "Julian Lyczak",
    "Roman Sarapin"
  ],
  "comment": "14 pages. Comments welcome v2: Added a study of these surfaces with a conic fibrations",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We study arithmetic properties of del Pezzo surfaces of degree 4 for which the Brauer group has the largest possible order using different fibrations into curves. We show that if such a surface admits a conic fibration, then it always has a rational point. We also answer a question of V\\'arilly-Alvarado and Viray by showing that the Brauer groups these surfaces cannot be vertical with respect to any projection away from a plane. We conclude that the available techniques for proving existence of rational points or even Zariski density do not directly apply if there is no Brauer-Manin obstruction to the Hasse principle. In passing we pick up the first examples of quartic del Pezzo surfaces with a Brauer group of order 4 for which the failure of the Hasse principle is explained by a Brauer-Manin obstruction.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-04-18T13:11:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quartic del pezzo surfaces",
      "brauer group",
      "brauer-manin obstruction",
      "rational point",
      "hasse principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}