{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "TA0NREQMPX"
    }
  ],
  "body": "\n## Summary\n\nLet $X \\subseteq \\mathbb{P}^n$ be a [[projective variety|A4H4CZ2V35]] of dimension $d$.\nIts degree is the number of points in which $X$ meets a general linear subspace of dimension $n - d$.\nAlgebraically, it is $d!$ times the leading coefficient of the [[Hilbert polynomial|RMH4FBCFDA]] of $X$.\nIt extends the degree of a polynomial to varieties and is the basic quantity in [[Bézout's theorem|KWNVF194D8]].\n\n## Explanations\n\n### L1 human\n\nA plane curve given by an equation of degree $e$ meets a general line in exactly $e$ points, so its degree is $e$.\nFor a surface in space one cuts with a general line instead, and for a curve in space with a general plane.\nThe twisted cubic, the image of $t \\mapsto (t, t^2, t^3)$, meets a general plane $a + bx + cy + dz = 0$ where the cubic $a + bt + ct^2 + dt^3$ vanishes, that is, in $3$ points.\nSo its degree is $3$.\nThe degree depends on how the variety sits in projective space, not only on the variety itself.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "14C17",
      "14N05"
    ],
    "name": "Degree of a projective variety",
    "same_as": {
      "wikidata": "Q5251838",
      "wikipedia": "Degree of an algebraic variety"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Degree_of_an_algebraic_variety"
      }
    ],
    "tag": "1XG9GSHZ3N"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/1XG9GSHZ3N/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/1XG9GSHZ3N.md",
  "msc": [
    "14C17",
    "14N05"
  ],
  "name": "Degree of a projective variety",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1XG/1XG9GSHZ3N.md",
  "status": null,
  "tag": "1XG9GSHZ3N",
  "uri": "m2://1XG9GSHZ3N",
  "url": "https://tamnd.github.io/math2/wiki/degree-of-a-projective-variety/",
  "writing": {
    "tag": "1XG9GSHZ3N",
    "kind": "concept",
    "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
      }
    ]
  }
}