{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1GVF19KFAK"
    }
  ],
  "body": "\n## Statement\n\nLet $X$ be a smooth projective [[algebraic surface|KFTECAATKG]] over an algebraically closed field, with [[canonical divisor|W05VYY4HBR]] $K_X$.\nLet $q = \\dim H^1(X, \\mathcal{O}_X)$ be its irregularity, in the sense of [[sheaf cohomology|PQ6Q143ARJ]], and let $P_2 = \\dim H^0(X, \\mathcal{O}_X(2K_X))$ be its second plurigenus.\nIf $q = 0$ and $P_2 = 0$, then $X$ is a [[rational surface|55A4A53TRF]].\n\n## Remarks\n\nThe converse holds as well, since $q$ and $P_2$ are birational invariants and both vanish for $\\mathbb{P}^2$.\nCastelnuovo proved it over $\\C$, and Zariski extended it to positive characteristic.\nThe proof passes to a minimal model and uses Riemann–Roch on the surface to show that $K_X$ is not nef.\nA minimal surface with this property is $\\mathbb{P}^2$ or a ruled surface, and $q = 0$ makes the base curve of the ruling rational.\nThe plurigenus $P_2$ cannot be replaced by the geometric genus $p_g = h^0(K_X)$: an [[Enriques surface|Y30JE2DC6Y]] has $q = p_g = 0$ but $P_2 = 1$, and it is not rational.\nExample: over $\\C$ a unirational surface $X$ is dominated by $\\mathbb{P}^2$.\nPulling back forms shows that $q$ and $P_2$ of $X$ are at most those of $\\mathbb{P}^2$, which vanish.\nHence every complex unirational surface is rational.\n\n## Explanations\n\n### L1 human\n\nDeciding whether a surface can be parametrized by the plane seems hard, since one would have to find the parametrization.\nCastelnuovo's criterion replaces this by two numbers that can be computed: the number of independent holomorphic $1$-forms and the number of independent sections of twice the canonical bundle.\nIf both are zero, the surface is rational.\n\n### L2 human\n\nThe criterion is the key step in the classification of surfaces of Kodaira dimension $-\\infty$.\nIt shows that, for surfaces over $\\C$, the Lüroth problem has a positive answer: unirational implies rational.\nThis fails in dimension three, where there are unirational threefolds that are not rational, and it fails for surfaces in positive characteristic.\nThe Enriques surface shows that the weaker condition $q = p_g = 0$ is not enough.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "55A4A53TRF"
    ],
    "kind": "statement",
    "msc": [
      "14J26",
      "14E08"
    ],
    "name": "Castelnuovo's rationality criterion",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Rational_surface"
      }
    ],
    "tag": "QXHMMQX1D7",
    "uses": [
      "55A4A53TRF",
      "KFTECAATKG",
      "W05VYY4HBR",
      "PQ6Q143ARJ",
      "Y30JE2DC6Y"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/QXHMMQX1D7/",
  "links": [
    {
      "field": "about",
      "target": "55A4A53TRF"
    },
    {
      "field": "uses",
      "target": "55A4A53TRF"
    },
    {
      "field": "uses",
      "target": "KFTECAATKG"
    },
    {
      "field": "uses",
      "target": "W05VYY4HBR"
    },
    {
      "field": "uses",
      "target": "PQ6Q143ARJ"
    },
    {
      "field": "uses",
      "target": "Y30JE2DC6Y"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/QXHMMQX1D7.md",
  "msc": [
    "14J26",
    "14E08"
  ],
  "name": "Castelnuovo's rationality criterion",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QXH/QXHMMQX1D7.md",
  "status": null,
  "tag": "QXHMMQX1D7",
  "uri": "m2://QXHMMQX1D7",
  "url": "https://tamnd.github.io/math2/wiki/castelnuovos-rationality-criterion/",
  "writing": {
    "tag": "QXHMMQX1D7",
    "kind": "statement",
    "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
      }
    ]
  }
}