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  "body": "\n## Summary\n\nAn elliptic surface is an [[algebraic surface|KFTECAATKG]] fibred over a curve so that almost every fibre is a smooth curve of genus one.\nSo it is a family of [[elliptic curves|427N6S6QQW]], or of genus-one curves, parametrized by a curve, degenerating over finitely many points.\nKodaira classified the possible singular fibres.\nElliptic surfaces include all minimal surfaces of Kodaira dimension $1$ and many K3 and rational surfaces.\nThey connect geometry with the arithmetic of elliptic curves over function fields.\n\n## Explanations\n\n### L1 human\n\nPicture a curve, and above each of its points place a doughnut-shaped elliptic curve that varies with the point.\nAt a few special points the doughnut is pinched or broken into pieces.\nThe whole family forms a surface.\nA simple example comes from the equation $y^2 = x^3 + t$, which for each nonzero value of the parameter $t$ is an elliptic curve and degenerates to a cusp at $t = 0$.\n",
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