{
  "id": "1308.0745",
  "version": "v3",
  "published": "2013-08-03T21:41:58.000Z",
  "updated": "2015-06-16T15:26:57.000Z",
  "title": "On the varieties of the second row of the split Freudenthal-Tits Magic Square",
  "authors": [
    "Jeroen Schillewaert",
    "Hendrik Van Maldeghem"
  ],
  "comment": "New title and slightly rewritten abstract as well as some changes in the introduction",
  "categories": [
    "math.AG",
    "math.DG",
    "math.GR",
    "math.RA"
  ],
  "abstract": "Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e.~the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-di\\-men\\-sional projective spaces, the line Grassmannians in 14-dimensional projective spaces, and the exceptional varieties of type $\\mathsf{E}_{6}$ in 26-dimensional projective space. Our theorem can be regarded as a far-reaching generalization of Mazzocca and Melone's approach to finite quadric Veronesean varieties. This approach takes projective properties of complex Severi varieties as smooth varieties as axioms.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-06-13T15:05:39.000Z",
      "title": "Severi varieties over arbitrary fields",
      "abstract": "Our main aim is to provide a uniform geometric characterization of the Severi varieties over arbitrary fields, i.e. the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-dimensional projective spaces, the line Grassmannians in 14-dimensional projective spaces, and the 26-dimensional exceptional varieties of type $\\mathsf{E}_{6}$. Our theorem can be regarded as a counterpart over arbitrary fields of the classification of smooth complex algebraic Severi varieties. Our axioms are based on an old characterization of finite quadric Veronese varieties by Mazzocca and Melone, and our results can be seen as a far-reaching generalization of Mazzocca and Melone's approach that characterizes finite varieties by requiring just the essential algebraic-geometric properties. We allow just enough generalization to capture the Severi varieties and some related varieties, over an arbitrary field. The proofs just use projective geometry.",
      "comment": "Some sections have been slightly rewritten. Removed some computations. We have added in some references to related work by Nash, Ionescu and Russo, and Russo",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-06-16T15:26:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "17C40",
      "51E24"
    ],
    "keywords": [
      "arbitrary field",
      "smooth complex algebraic severi varieties",
      "projective spaces",
      "finite quadric veronese varieties",
      "essential algebraic-geometric properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.0745S"
    }
  }
}