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  "body": "\n## Definition\n\nLet $k$ be an [[algebraically closed field|0PF6TM8TWT]], and let $\\mathbb{P}^2$ be the [[projective plane|ZT9A6JK8QF]] over $k$.\nAn [[algebraic surface|KFTECAATKG]] $X$ over $k$ is *rational* if there is a [[birational map|KA5890R9YM]] $X \\dashrightarrow \\mathbb{P}^2$.\n\n## Remarks\n\nEquivalently, the function field of $X$ is isomorphic to the field $k(s, t)$ of rational functions in two variables.\nExample: $\\mathbb{P}^1 \\times \\mathbb{P}^1$ is rational: on the open subsets $\\mathbb{A}^1 \\times \\mathbb{A}^1$ and $\\mathbb{A}^2$ the identity $(s, t) \\mapsto (s, t)$ is an isomorphism, and it extends to a birational map $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\dashrightarrow \\mathbb{P}^2$.\nNon-example: for an elliptic curve $E$, the surface $E \\times \\mathbb{P}^1$ is not rational, because the irregularity $q = h^1(\\mathcal{O})$ is a birational invariant, equal to $0$ for $\\mathbb{P}^2$ and to $1$ for $E \\times \\mathbb{P}^1$.\nEvery smooth rational surface is obtained from $\\mathbb{P}^2$ or from a Hirzebruch surface by a finite sequence of blow-ups of points.\n",
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