High Quality Content by WIKIPEDIA articles! In mathematics, more particularly in the field of algebraic geometry, a scheme X has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a proper birational map f colon Y rightarrow X from a regular scheme Y such that the higher direct images of f * applied to mathcal{O}_Y are trivial. That is, R^i f_* mathcal{O}_Y = 0 for i > 0.If there is one such resolution, then it follows that all resolutions share this property, since any two resolutions of singularities can be dominated by a third.For surfaces, rational singularities were defined by (Artin 1966).

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