Rational singularity

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

<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 .

Formulations

Alternately, one can say that <math>X</math> has rational singularities if and only if the natural map in the derived category

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

is a quasi-isomorphism. Notice that this includes the statement that <math>mathcal_X simeq f_* mathcal_Y</math> and hence the assumption that <math>X</math> is normal.

There are related notions in positive and mixed characteristic of

• pseudo-rational

and

• F-rational

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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