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  "body": "\n## Definition\n\nLet $k$ be a field of characteristic $0$ and let $X$ be a normal [[algebraic variety|HV6YBS0YDH]] over $k$ (see [[normal scheme|EK261G1XP9]]).\nLet $f \\colon Y \\to X$ be a [[resolution of singularities|MB52NDA6WN]], that is, a [[proper|9537RKBZPP]] [[birational|KA5890R9YM]] morphism from a nonsingular variety $Y$.\nThe variety $X$ has *rational singularities* if\n$$R^i f_* \\mathcal{O}_Y = 0 \\quad \\text{for all } i \u003e 0,$$\nwhere $R^i f_*$ are the higher direct images, computed by [[sheaf cohomology|PQ6Q143ARJ]] on preimages of open sets.\nA point $x \\in X$ is a *rational singularity* if some open neighbourhood of $x$ has rational singularities.\n\n## Remarks\n\nThe condition does not depend on the chosen resolution.\nSince $X$ is normal and $f$ is proper and birational, one also has $f_* \\mathcal{O}_Y = \\mathcal{O}_X$, by Zariski's main theorem.\nEvery nonsingular variety has rational singularities, with $f = \\id$.\nExample: for the cone $X \\colon xy = z^2$ in $\\mathbb{A}^3$, the blow-up of the vertex is a resolution whose exceptional curve is $E \\cong \\mathbb{P}^1$ with normal bundle $\\mathcal{O}(-2)$.\nHere $Y$ is the total space of the line bundle $\\mathcal{O}(-2)$ on $E$, and $X$ is affine, so $R^1 f_* \\mathcal{O}_Y$ is determined by $H^1(Y, \\mathcal{O}_Y) = \\bigoplus_{m \\ge 0} H^1(\\mathbb{P}^1, \\mathcal{O}(2m)) = 0$.\nSo the singularity is rational.\nNon-example: for the cone $X \\colon x^3 + y^3 + z^3 = 0$ over a smooth plane cubic $E$, the blow-up of the vertex is the total space of $\\mathcal{O}_E(-1)$, and the same computation gives $H^1(Y, \\mathcal{O}_Y) = \\bigoplus_{m \\ge 0} H^1(E, \\mathcal{O}_E(m))$.\nThe term $m = 0$ is $H^1(E, \\mathcal{O}_E) \\cong k$, so $R^1 f_* \\mathcal{O}_Y \\ne 0$ and the singularity is not rational.\n",
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