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  "body": "\n## Summary\n\nA unirational variety is a variety that is dominated by projective space through a rational map.\nIts points can be parametrised by rational functions, but the parametrisation may cover each point several times.\nEvery [[rational variety|XV4K9GREJ9]] is unirational.\nWhether the converse holds is the Lüroth problem.\nIt holds for curves, and for surfaces over an algebraically closed field of characteristic $0$, but it fails for some threefolds over $\\C$.\n\n## Explanations\n\n### L1 human\n\nA rational variety has a parametrisation that is one-to-one on a dense open set.\nA unirational variety only needs a parametrisation that covers a dense open set, possibly many times over.\nFor curves the two notions agree, by [[Lüroth's theorem|AVCZ6HVBA7]].\nOver $\\C$, a smooth cubic threefold in $\\mathbb{P}^4$ is unirational but not rational, by a theorem of Clemens and Griffiths.\n",
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