Rational singularity

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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 (Artin 1966).

Formulations[edit]

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

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, (Kollár, Mori, 1998, Theorem 5.22.)

Examples[edit]

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.

References[edit]