Rational singularity

In mathematics, more particularly in the field of algebraic geometry, a scheme formula_1 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 formula_3 such that the higher direct images of formula_4 applied to formula_5 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 formula_1 has rational singularities if and only if the natural map in the derived category
is a quasi-isomorphism. Notice that this includes the statement that formula_10 and hence the assumption that formula_1 is normal.
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 rational singularity is the singular point of the quadric cone 
 showed that 
the rational double points of algebraic surfaces are the Du Val singularities.