{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $k$ be an [[algebraically closed field|0PF6TM8TWT]] and let $C$ be a smooth projective connected [[curve|XJM2MT4S3Y]] over $k$.\nLet $\\Omega_C$ be its [[sheaf of differentials|6VC6HQTFSA]].\nThe *genus* of $C$ is\n$$g(C) = \\dim_k H^0(C, \\Omega_C),$$\nthe dimension of the space of regular differential $1$-forms on $C$.\nLet $C$ be an integral curve, possibly singular or not projective.\nThe *geometric genus* of $C$ is the genus of the smooth projective curve with the same function field.\nThis curve is the [[normalization|M2FAMD9HZN]] of a projective closure of $C$.\n\n## Remarks\n\nFor a smooth projective connected curve, Serre duality gives $g(C) = \\dim_k H^1(C, \\mathcal{O}_C)$, so the genus equals the [[arithmetic genus|H03GKRVBJ1]].\nExample: on $\\mathbb{P}^1$ one has $\\Omega = \\mathcal{O}(-2)$, which has no nonzero global sections, so $g(\\mathbb{P}^1) = 0$.\nExample: the smooth projective curve $E$ with affine equation $y^2 = x^3 - x$, in characteristic not $2$, has the form $\\omega = dx/y$, regular and nowhere zero on $E$, and $g(E) = 1$.\nNon-example of the equality with the arithmetic genus: the nodal cubic $y^2 = x^2(x + 1)$ is singular, its normalization is $\\mathbb{P}^1$, so its geometric genus is $0$, while its arithmetic genus is $1$.\nOver $\\C$, the genus equals the topological genus of $C(\\C)$ as a closed orientable surface.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "6W8TDV9VVN",
    "kind": "definition",
    "msc": [
      "14H99"
    ],
    "name": "Genus of a curve",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Geometric_genus"
      }
    ],
    "tag": "J7K42A9F8A",
    "uses": [
      "XJM2MT4S3Y",
      "6VC6HQTFSA",
      "M2FAMD9HZN",
      "H03GKRVBJ1",
      "0PF6TM8TWT"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/J7K42A9F8A/",
  "links": [
    {
      "field": "defines",
      "target": "6W8TDV9VVN"
    },
    {
      "field": "uses",
      "target": "XJM2MT4S3Y"
    },
    {
      "field": "uses",
      "target": "6VC6HQTFSA"
    },
    {
      "field": "uses",
      "target": "M2FAMD9HZN"
    },
    {
      "field": "uses",
      "target": "H03GKRVBJ1"
    },
    {
      "field": "uses",
      "target": "0PF6TM8TWT"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/J7K42A9F8A.md",
  "msc": [
    "14H99"
  ],
  "name": "Genus of a curve",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/J7K/J7K42A9F8A.md",
  "status": null,
  "tag": "J7K42A9F8A",
  "uri": "m2://J7K42A9F8A",
  "url": "https://tamnd.github.io/math2/wiki/genus-of-a-curve-definition/",
  "writing": {
    "tag": "J7K42A9F8A",
    "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
      }
    ]
  }
}