Rational singularity

In mathematics, more particularly in the field of algebraic geometry, a scheme 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 such that the higher direct images of applied to are trivial. That is,

for .

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 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 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.

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.

See also Elliptic singularity

References