{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $k$ be a field and let $X$ be a [[projective|A4H4CZ2V35]] scheme over $k$ of dimension $n$.\nLet\n$$\\chi(\\mathcal{O}_X) = \\sum_{i \\ge 0} (-1)^i \\dim_k H^i(X, \\mathcal{O}_X)$$\nbe the [[Euler characteristic|RNDDWDAFJ5]] of its structure sheaf, computed with [[sheaf cohomology|PQ6Q143ARJ]].\nThe *arithmetic genus* of $X$ is\n$$p_a(X) = (-1)^n \\left( \\chi(\\mathcal{O}_X) - 1 \\right).$$\n\n## Remarks\n\nThe sum is finite because $X$ is projective, so each $H^i(X, \\mathcal{O}_X)$ is finite-dimensional and vanishes for $i \u003e n$.\nEquivalently, $p_a(X) = (-1)^n (P_X(0) - 1)$, where $P_X$ is the Hilbert polynomial of $X$ for a chosen embedding.\nFor a curve $C$ with $H^0(C, \\mathcal{O}_C) = k$, for example a connected reduced projective curve over an algebraically closed field, $p_a(C) = \\dim_k H^1(C, \\mathcal{O}_C)$.\nExample: $H^1(\\mathbb{P}^1, \\mathcal{O}) = 0$, so $p_a(\\mathbb{P}^1) = 0$.\nExample: a plane cubic curve, smooth or nodal, has $p_a = 1$; this follows from the exact sequence $0 \\to \\mathcal{O}_{\\mathbb{P}^2}(-3) \\to \\mathcal{O}_{\\mathbb{P}^2} \\to \\mathcal{O}_C \\to 0$.\nNon-example: the affine line $\\mathbb{A}^1$ is not projective, and $H^0(\\mathbb{A}^1, \\mathcal{O}) = k[t]$ is infinite-dimensional, so $\\chi$ is not defined.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "H03GKRVBJ1",
    "kind": "definition",
    "msc": [
      "14H99",
      "14C40"
    ],
    "name": "Arithmetic genus",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Arithmetic_genus"
      }
    ],
    "tag": "ENMB2J4JJ6",
    "uses": [
      "A4H4CZ2V35",
      "PQ6Q143ARJ",
      "RNDDWDAFJ5"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/ENMB2J4JJ6/",
  "links": [
    {
      "field": "defines",
      "target": "H03GKRVBJ1"
    },
    {
      "field": "uses",
      "target": "A4H4CZ2V35"
    },
    {
      "field": "uses",
      "target": "PQ6Q143ARJ"
    },
    {
      "field": "uses",
      "target": "RNDDWDAFJ5"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/ENMB2J4JJ6.md",
  "msc": [
    "14H99",
    "14C40"
  ],
  "name": "Arithmetic genus",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ENM/ENMB2J4JJ6.md",
  "status": null,
  "tag": "ENMB2J4JJ6",
  "uri": "m2://ENMB2J4JJ6",
  "url": "https://tamnd.github.io/math2/wiki/arithmetic-genus-definition/",
  "writing": {
    "tag": "ENMB2J4JJ6",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}