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  "body": "\n## Problem\n\nLet $k$ be an [[algebraically closed field|0PF6TM8TWT]] and let $X$ be an integral [[algebraic variety|HV6YBS0YDH]] over $k$.\nSuppose that $X$ is [[unirational|YF1SE8WJK3]].\nMust $X$ be [[rational|XV4K9GREJ9]]?\n\n## Remarks\n\nIn dimension $1$ the answer is yes, by [[Lüroth's theorem|AVCZ6HVBA7]].\nIn dimension $2$ over an algebraically closed field of characteristic $0$, the answer is yes, by [[Castelnuovo's rationality criterion|QXHMMQX1D7]].\nIn dimension $2$ over an algebraically closed field of characteristic $p \u003e 0$, the answer is no: Zariski found unirational surfaces that are not rational.\nIn dimension $3$ over $\\C$, the answer is no.\nClemens and Griffiths showed that a smooth cubic threefold in $\\mathbb{P}^4$ is unirational but not rational.\nIskovskikh and Manin showed that a smooth quartic threefold in $\\mathbb{P}^4$ is not rational, and some of these are unirational.\nArtin and Mumford gave another unirational threefold that is not rational, detected by torsion in its third cohomology group.\nSo the problem is settled in the negative in general, and the open questions now concern specific families of varieties.\n\n## Explanations\n\n### L1 human\n\nA variety is rational when it can be parametrised one-to-one, up to small exceptions, by free parameters.\nIt is unirational when it can be covered by such a parametrisation, perhaps many-to-one.\nThe problem asks whether a many-to-one parametrisation can always be improved to a one-to-one parametrisation.\nFor curves and for complex surfaces it can.\nFor complex threefolds it cannot, and the cubic threefold is the standard example.\n",
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    "history": {
      "first_stated": "1876",
      "posed_by": "Jacob Lüroth"
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      "14E08",
      "14M20"
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    "name": "Lüroth problem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/L%C3%BCroth%27s_theorem"
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    "tag": "WGGZEKF224",
    "uses": [
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  "link": "https://tamnd.github.io/math2/node/WGGZEKF224/",
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  "markdown": "https://tamnd.github.io/math2/node/WGGZEKF224.md",
  "msc": [
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  "name": "Lüroth problem",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/WGG/WGGZEKF224.md",
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  "uri": "m2://WGGZEKF224",
  "url": "https://tamnd.github.io/math2/wiki/luroth-problem/",
  "writing": {
    "tag": "WGGZEKF224",
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