Rational Singularity

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

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

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 <math>X</math> has rational singularities if and only if the natural map in the derived category

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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- <math>f colon Y rightarrow X</math>

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

- <math>R^i f_* mathcal_Y = 0</math> for <math>i > 0</math>.

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 .

- <math>mathcal_X rightarrow R f_* mathcal_Y</math>

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.

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