{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "35KB1QCD3Z"
    },
    {
      "field": "uses",
      "target": "QQ717VSFNN"
    },
    {
      "field": "uses",
      "target": "RSW29AVKGM"
    },
    {
      "field": "uses",
      "target": "TP1RQPCETV"
    },
    {
      "field": "uses",
      "target": "WMVAT9MBZ1"
    },
    {
      "field": "uses",
      "target": "XR0M3DFFMG"
    },
    {
      "field": "defines",
      "target": "8WAC71N0C2"
    },
    {
      "field": "about",
      "target": "RSW29AVKGM"
    },
    {
      "field": "about",
      "target": "WMVAT9MBZ1"
    }
  ],
  "body": "\n## Summary\n\nA Weil divisor on a suitable scheme $X$ is a formal sum with integer coefficients of finitely many closed subvarieties of codimension one.\nThe zeros and poles of a rational function, counted with multiplicity, form a principal Weil divisor.\nThe quotient of all Weil divisors by the principal ones is the divisor class group, which equals the Picard group on smooth varieties.\n\n## Explanations\n\n### L1 human\n\nOn a curve, a divisor is a finite list of points with integer weights, such as $2P - 3Q$.\nOn a surface, the points are replaced by curves, and in general by subvarieties of codimension one.\nA rational function gives a divisor: its zeros with positive weight and its poles with negative weight.\nFor example, on the projective line, the function $x^2/(x - 1)$ has divisor $2[0] - [1] - [\\infty]$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "14C20"
    ],
    "name": "Weil divisor",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Divisor_(algebraic_geometry)"
      }
    ],
    "tag": "Z9DN23DAE2"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/Z9DN23DAE2/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/Z9DN23DAE2.md",
  "msc": [
    "14C20"
  ],
  "name": "Weil divisor",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Z9D/Z9DN23DAE2.md",
  "status": null,
  "tag": "Z9DN23DAE2",
  "uri": "m2://Z9DN23DAE2",
  "url": "https://tamnd.github.io/math2/wiki/weil-divisor/",
  "writing": {
    "tag": "Z9DN23DAE2",
    "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
      }
    ]
  }
}
