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Rational singularity

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Source updated May 16, 2026

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).

01Formulations

Alternately, one can say that $X$ has rational singularities if and only if the natural map in the derived category

${\mathcal {O}}_{X}\rightarrow Rf_{*}{\mathcal {O}}_{Y}$

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

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.

02Examples

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.

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Sources and credits

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Rational singularity

01Formulations

02Examples

Sources and credits

Related topics

Elliptic singularity