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  "body": "\n## Summary\n\nA rational singularity is a singular point that looks, from the point of view of sheaf cohomology, like a smooth point.\nIt is defined by the vanishing of the higher direct images of the structure sheaf under a [[resolution|MB52NDA6WN]].\nQuotient singularities, toric singularities and the [[ADE singularities|HQNXQBZ6C7]] are rational.\nRational singularities play a central role in the [[minimal model program|Q8CTDCZH01]], since the log terminal singularities it uses are rational.\n\n## Explanations\n\n### L1 human\n\nWhen a singular point is resolved, the new smooth variety may carry extra cohomology above the point.\nA singularity is rational when no such extra cohomology appears in the structure sheaf.\nThe tip of the cone $x^2 + y^2 = z^2$ is rational: its resolution adds a single projective line, which has no first cohomology.\nThe tip of the cone over a smooth plane cubic is not: its resolution adds an elliptic curve, whose first cohomology is one-dimensional.\n",
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