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  "body": "\n## Definition\n\nLet $X$ be a smooth projective [[algebraic surface|KFTECAATKG]] over an algebraically closed field.\nAn *elliptic fibration* on $X$ is a surjective [[morphism|CDS1VWB7A3]] $f \\colon X \\to C$ onto a smooth projective [[curve|XJM2MT4S3Y]] $C$ such that:\n\n1. every fibre of $f$ is connected, and\n2. the fibre $f^{-1}(c)$ is a smooth curve of [[genus|6W8TDV9VVN]] $1$ for all but finitely many $c \\in C$.\n\nAn *elliptic surface* is a surface $X$ together with an elliptic fibration on it.\n\n## Remarks\n\nMany authors also require that $f$ be relatively minimal, meaning that no fibre contains a smooth rational curve of self-intersection $-1$, and some require a section of $f$.\nExample: for an elliptic curve $E$ and a smooth projective curve $C$, the projection $E \\times C \\to C$ has every fibre isomorphic to $E$, so it is an elliptic fibration.\nExample: blowing up $\\mathbb{P}^2$ at the nine base points of a general pencil of plane cubics gives a rational elliptic surface, whose fibres are the cubics of the pencil.\nNon-example: the projection $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\to \\mathbb{P}^1$ has all fibres of genus $0$, so it fails clause 2.\nIn characteristics $2$ and $3$ one also meets quasi-elliptic fibrations, whose general fibre is a cuspidal rational curve.\n",
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