# Rational singularity

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In mathematics, more particularly in the field of algebraic geometry, a scheme [itex]X[/itex] has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a proper birational map

[itex]f colon Y rightarrow X[/itex]

from a regular scheme [itex]Y[/itex] such that the higher direct images of [itex]f_*[/itex] applied to [itex]mathcal_Y[/itex] are trivial. That is,

[itex]R^i f_* mathcal_Y = 0[/itex] for [itex]i > 0[/itex].

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 .

## Formulations

Alternately, one can say that [itex]X[/itex] has rational singularities if and only if the natural map in the derived category
[itex]mathcal_X rightarrow R f_* mathcal_Y[/itex]
is a quasi-isomorphism. Notice that this includes the statement that [itex]mathcal_X simeq f_* mathcal_Y[/itex] and hence the assumption that [itex]X[/itex] is normal.

There are related notions in positive and mixed characteristic of and

Rational singularities are in particular Cohen-Macaulay, normal and Du Bois. They need not be Gorenstein or even Q-Gorenstein.

Log terminal singularities are rational.

## Examples

An example of a...

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