Rational singularity
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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
from a regular scheme Y such that the higher direct images of f * applied to OY are trivial. That is,
- Rif * OY = 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).
[edit] Formulations
Alternately, one can remove the normality hypothesis on X and say that X has rational singularities if and only if the natural map in the derived category
is a quasi-isomorphism.
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.
[edit] Examples
An example of a rational singularity is the singular point of the quadric cone
(Artin 1966) showed that the rational double points of a algebraic surfaces are the Du Val singularities.
