{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1PFE27B2MG"
    },
    {
      "field": "uses",
      "target": "1X3EV2TZNN"
    },
    {
      "field": "uses",
      "target": "28DVNYNMZR"
    },
    {
      "field": "uses",
      "target": "3T9TT5SSGD"
    },
    {
      "field": "uses",
      "target": "3THE3Q96KK"
    },
    {
      "field": "uses",
      "target": "4M1SYSPGA2"
    },
    {
      "field": "uses",
      "target": "58WSWYBCWA"
    },
    {
      "field": "uses",
      "target": "5FKKCM9ESE"
    },
    {
      "field": "uses",
      "target": "5WGZPE6249"
    },
    {
      "field": "uses",
      "target": "5Z19Y02MME"
    },
    {
      "field": "uses",
      "target": "6WV71T5CK1"
    },
    {
      "field": "uses",
      "target": "7G7CSM2NZ9"
    },
    {
      "field": "uses",
      "target": "7GJ4STCNNM"
    },
    {
      "field": "uses",
      "target": "94AKWRHND0"
    },
    {
      "field": "uses",
      "target": "9CD4SGERVP"
    },
    {
      "field": "uses",
      "target": "ABE41ZYD78"
    },
    {
      "field": "uses",
      "target": "AHCXQCYW3C"
    },
    {
      "field": "uses",
      "target": "B6C3J2GJWP"
    },
    {
      "field": "uses",
      "target": "BZ2MM3CTNA"
    },
    {
      "field": "uses",
      "target": "CYPA882RB9"
    },
    {
      "field": "uses",
      "target": "DQA60Z3FD7"
    },
    {
      "field": "uses",
      "target": "E4F316AR19"
    },
    {
      "field": "uses",
      "target": "F9ZCCKATES"
    },
    {
      "field": "uses",
      "target": "FSASFB5BMN"
    },
    {
      "field": "uses",
      "target": "G0W7S1Y7XC"
    },
    {
      "field": "uses",
      "target": "GE75S5XZ9T"
    },
    {
      "field": "uses",
      "target": "GKV4RVXBRE"
    },
    {
      "field": "uses",
      "target": "GZ3XA1V7CJ"
    },
    {
      "field": "uses",
      "target": "H8J52N0P9A"
    },
    {
      "field": "uses",
      "target": "HQ0JN9W2PR"
    },
    {
      "field": "uses",
      "target": "J7NBAJMQND"
    },
    {
      "field": "uses",
      "target": "KDEVJ69P10"
    },
    {
      "field": "uses",
      "target": "KJQM4EJHV1"
    },
    {
      "field": "uses",
      "target": "KWNVF194D8"
    },
    {
      "field": "uses",
      "target": "MQCBFACHXH"
    },
    {
      "field": "uses",
      "target": "MV3PHCJ6CE"
    },
    {
      "field": "uses",
      "target": "NH7S9CHHBN"
    },
    {
      "field": "uses",
      "target": "NPNNXWMVCC"
    },
    {
      "field": "uses",
      "target": "NW7W4R0T85"
    },
    {
      "field": "uses",
      "target": "Q2V8JVPB5E"
    },
    {
      "field": "uses",
      "target": "RKJBNRKEDT"
    },
    {
      "field": "uses",
      "target": "RVX312A2D2"
    },
    {
      "field": "uses",
      "target": "T7B1WJVT9Z"
    },
    {
      "field": "uses",
      "target": "TA0NREQMPX"
    },
    {
      "field": "uses",
      "target": "V50N3JEBDP"
    },
    {
      "field": "uses",
      "target": "VQK5HQFYEM"
    },
    {
      "field": "uses",
      "target": "W0J2G7RKBY"
    },
    {
      "field": "uses",
      "target": "W94AEJV9NK"
    },
    {
      "field": "uses",
      "target": "WC8M1YAYCD"
    },
    {
      "field": "uses",
      "target": "WJ5WRGG5DE"
    },
    {
      "field": "uses",
      "target": "WKYG28Z8ER"
    },
    {
      "field": "uses",
      "target": "XK866CAZHE"
    },
    {
      "field": "uses",
      "target": "Y7RP7KSH2C"
    },
    {
      "field": "uses",
      "target": "YJMGEKHYEC"
    },
    {
      "field": "uses",
      "target": "Z3AY2NEVW3"
    },
    {
      "field": "uses",
      "target": "ZAZFMK7C67"
    },
    {
      "field": "instance_of",
      "target": "CSMB49BDPT"
    },
    {
      "field": "defines",
      "target": "H30E38799N"
    },
    {
      "field": "about",
      "target": "NPNNXWMVCC"
    }
  ],
  "body": "\n## Summary\n\nThe projective space $\\mathbb{P}^n$ over a field $k$ is the set of lines through the origin in $k^{n+1}$.\nA point is written in homogeneous coordinates $[x_0 : \\dots : x_n]$, defined up to a common nonzero factor.\nProjective space adds points at infinity to affine space.\nThen, for example, any two distinct lines in the projective plane meet in exactly one point.\nIt is the ambient space of projective varieties, the compact objects of algebraic geometry.\n\n## Explanations\n\n### L1 human\n\nStand at the origin of $k^{n+1}$ and look out: each direction you can look in is one point of $\\mathbb{P}^n$.\nThe points $(1, 2, 3)$ and $(2, 4, 6)$ lie on the same line through the origin, so they give the same point $[1 : 2 : 3]$ of the projective plane.\nThe points with $x_0 \\ne 0$ can be scaled to $[1 : a_1 : \\dots : a_n]$ and form a copy of $k^n$.\nThe remaining points, with $x_0 = 0$, are the points at infinity, where parallel lines of $k^n$ meet.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "14N05",
      "51N15"
    ],
    "name": "Projective space",
    "same_as": {
      "wikidata": "Q877775",
      "wikipedia": "Projective space"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Projective_space"
      }
    ],
    "tag": "ZT9A6JK8QF"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/ZT9A6JK8QF/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/ZT9A6JK8QF.md",
  "msc": [
    "14N05",
    "51N15"
  ],
  "name": "Projective space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZT9/ZT9A6JK8QF.md",
  "status": null,
  "tag": "ZT9A6JK8QF",
  "uri": "m2://ZT9A6JK8QF",
  "url": "https://tamnd.github.io/math2/wiki/projective-space/",
  "writing": {
    "tag": "ZT9A6JK8QF",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}