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  "body": "\n## Summary\n\nA normal scheme is a [[scheme|5A35TYZD02]] whose local rings are [[integrally closed domains|H20P600HQS]].\nNormality is a mild regularity condition.\nOver an algebraically closed field, a normal curve is smooth, and a normal variety is smooth outside a closed set of codimension at least $2$.\nEvery integral scheme has a normal model, its normalization.\n\n## Explanations\n\n### L1 human\n\nNormal schemes have no singularities of the simplest kinds, such as a curve crossing itself or a cusp.\nThe test is algebraic: a fraction that satisfies a monic polynomial equation over the local ring must already lie in the ring.\nOn the cusp $y^2 = x^3$, the fraction $t = y/x$ satisfies $t^2 = x$ but is not a polynomial function on the cusp, so the cusp is not normal.\nThe plane and $\\operatorname{Spec} \\Z$ are normal.\n",
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