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  "body": "\n## Summary\n\nA rational variety is a variety that is birationally equivalent to projective space.\nIts points can be parametrised, on a dense open set, by rational functions of independent parameters.\nDeciding which varieties are rational is one of the oldest problems of [[birational geometry|51YVV85GWB]].\nEvery rational variety is [[unirational|YF1SE8WJK3]], and the converse fails in dimension $3$ over $\\C$.\n\n## Explanations\n\n### L1 human\n\nA variety is rational when you can describe almost all of its points by free coordinates, with no hidden constraints.\nThe circle $x^2 + y^2 = 1$ is rational: each point other than $(-1, 0)$ is\n$$\\left( \\frac{1 - t^2}{1 + t^2}, \\frac{2t}{1 + t^2} \\right)$$\nfor exactly one $t$, which is the slope of the line through $(-1, 0)$ and the point.\nA smooth plane cubic, such as $y^2 = x^3 - x$, is not rational.\nIts genus is $1$, while a rational curve has genus $0$.\n",
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