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  "body": "\n## Definition\n\nLet $k$ be a field and let $X$ be an integral [[algebraic variety|HV6YBS0YDH]] over $k$.\nThe variety $X$ is *unirational* over $k$ if there are an integer $N \\ge 1$ and a dominant [[rational map|4D073CCYGV]] $\\mathbb{P}^N_k \\dashrightarrow X$ from [[projective space|ZT9A6JK8QF]].\n\n## Remarks\n\nEquivalently, the [[function field|2D5SS6WM14]] $k(X)$ embeds over $k$ into a purely transcendental extension $k(t_1, \\ldots, t_N)$.\nExample: every rational variety is unirational, since a birational map is dominant.\nExample: the curve $y^2 = x^3$ in $\\mathbb{A}^2$ is unirational.\nThe map $t \\mapsto (t^2, t^3)$ is a dominant morphism from $\\mathbb{A}^1$, an open subset of $\\mathbb{P}^1$, since $(t^3)^2 = (t^2)^3$ and its image is the whole curve.\nNon-example: a smooth projective curve $E$ of genus $1$ over $\\C$ is not unirational.\nA dominant rational map $\\mathbb{P}^N \\dashrightarrow E$ restricts to a nonconstant rational map on a general line, and this extends to a nonconstant morphism $f \\colon \\mathbb{P}^1 \\to E$.\nThe Riemann–Hurwitz formula would then give $-2 = 2g(\\mathbb{P}^1) - 2 \\ge \\deg f \\cdot (2g(E) - 2) = 0$, which is false.\n",
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    "name": "Unirational variety",
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