Rational singularity

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.

Formulations
Alternately, one can say that $$X$$ has rational singularities if and only if the natural map in the derived category
 * $$\mathcal{O}_X \rightarrow R f_* \mathcal{O}_Y$$

is a quasi-isomorphism. Notice that this includes the statement that $$\mathcal{O}_X \simeq f_* \mathcal{O}_Y$$ and hence the assumption that $$X$$ is normal.

There are related notions in positive and mixed characteristic of and
 * pseudo-rational
 * 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 rational singularity is the singular point of the quadric cone


 * $$x^2 + y^2 + z^2 = 0. \,$$

Artin showed that the rational double points of algebraic surfaces are the Du Val singularities.