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  "body": "\n## Definition\n\nLet $k$ be an algebraically closed field, and let $\\mathbb{P}^1$ be the [[projective line|ZT9A6JK8QF]] over $k$.\nA *ruled surface* over $k$ is a smooth projective [[algebraic surface|KFTECAATKG]] $X$ together with a surjective [[morphism|CDS1VWB7A3]] $\\pi \\colon X \\to C$, with the following properties.\nThe target $C$ is a smooth projective [[curve|XJM2MT4S3Y]], and every fibre $\\pi^{-1}(c)$, $c \\in C$, is isomorphic to $\\mathbb{P}^1$.\n\n## Remarks\n\nSome books also ask for a section of $\\pi$; over an algebraically closed field a section always exists, by Tsen's theorem.\nEvery ruled surface is isomorphic over $C$ to the projectivization $\\mathbb{P}(\\mathcal{E})$ of a vector bundle $\\mathcal{E}$ of rank $2$ on $C$.\nAnother usage calls a surface *ruled* when it is only birational to $C \\times \\mathbb{P}^1$; the surfaces of the definition are then called geometrically ruled.\nExample: for any smooth projective curve $C$, the projection $C \\times \\mathbb{P}^1 \\to C$ has every fibre $\\set{c} \\times \\mathbb{P}^1 \\cong \\mathbb{P}^1$, so $C \\times \\mathbb{P}^1$ is ruled.\nNon-example: the projective plane $\\mathbb{P}^2$ is not ruled, because every morphism from $\\mathbb{P}^2$ to a curve is constant.\nIndeed, two fibres of a nonconstant one would be disjoint curves in $\\mathbb{P}^2$, but any two curves in $\\mathbb{P}^2$ meet.\n",
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