Harman Patil (Editor)

Rational singularity

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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

Contents

f : Y → X

from a regular scheme Y such that the higher direct images of f ∗ applied to O Y are trivial. That is,

R i f ∗ O Y = 0 for i > 0 .

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

Alternately, one can say that X has rational singularities if and only if the natural map in the derived category

O X → R f ∗ O Y

is a quasi-isomorphism. Notice that this includes the statement that O X ≃ f ∗ O Y and hence the assumption that X 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, (Kollár, Mori, 1998, Theorem 5.22.)

    Examples

    An example of a rational singularity is the singular point of the quadric cone

    x 2 + y 2 + z 2 = 0.

    (Artin 1966) showed that the rational double points of a algebraic surfaces are the Du Val singularities.

    References

    Rational singularity Wikipedia
    (Text) CC BY-SA