{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{P}$ be a [[projective space|ZT9A6JK8QF]] of dimension at least $3$.\nA line $g$ is *tangent* to a set $O$ of points at $P \\in O$ if $g \\cap O = \\set{P}$.\nAn *ovoid* of $\\mathcal{P}$ is a set $O$ of points such that:\n\n1. every line meets $O$ in at most two points;\n2. for every point $P \\in O$, the union of the tangent lines to $O$ at $P$ is exactly a hyperplane through $P$.\n\n## Remarks\n\nThe hyperplane in clause 2 is the tangent hyperplane of $O$ at $P$.\nIn a projective plane the same two conditions define an [[oval|7B03TB64N9]].\n\nExample: in $\\mathrm{PG}(3, q)$ over the [[finite field|0ZXZ82FTX2]] $\\F_q$, an elliptic [[quadric|Q3Q4FF23VG]] is an ovoid.\nIt contains no line, so a line meets it in at most two points.\nAt each point, the tangent lines are the lines through that point in its tangent plane.\nIt has $q^2 + 1$ points.\nIn real projective $3$-space, a sphere is an ovoid.\n\nNon-example: a hyperbolic quadric $x_0 x_1 = x_2 x_3$ contains whole lines, such as $x_0 = x_2 = 0$, so clause 1 fails.\n\nIn $\\mathrm{PG}(3, q)$ every ovoid has $q^2 + 1$ points.\nFor $q = 2^{2e+1}$ with $e \\ge 1$, Tits found ovoids that are not quadrics.\nIn finite projective spaces of dimension at least $4$ there are no ovoids.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "HNTE5FPS14",
    "kind": "definition",
    "msc": [
      "51E20",
      "51E21"
    ],
    "name": "Ovoid",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Ovoid_(projective_geometry)"
      }
    ],
    "tag": "9CD4SGERVP",
    "uses": [
      "ZT9A6JK8QF",
      "Q3Q4FF23VG",
      "7B03TB64N9",
      "0ZXZ82FTX2"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/9CD4SGERVP/",
  "links": [
    {
      "field": "defines",
      "target": "HNTE5FPS14"
    },
    {
      "field": "uses",
      "target": "ZT9A6JK8QF"
    },
    {
      "field": "uses",
      "target": "Q3Q4FF23VG"
    },
    {
      "field": "uses",
      "target": "7B03TB64N9"
    },
    {
      "field": "uses",
      "target": "0ZXZ82FTX2"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/9CD4SGERVP.md",
  "msc": [
    "51E20",
    "51E21"
  ],
  "name": "Ovoid",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/9CD/9CD4SGERVP.md",
  "status": null,
  "tag": "9CD4SGERVP",
  "uri": "m2://9CD4SGERVP",
  "url": "https://tamnd.github.io/math2/wiki/ovoid-definition/",
  "writing": {
    "tag": "9CD4SGERVP",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}