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 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 X 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 and hence the assumption that X 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 Artin showed that the rational double points of algebraic surfaces are the Du Val singularities.