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