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Rational singularity

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In mathematics, more particularly in the field of algebraic geometry, a scheme X {\displaystyle X} has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a proper birational map

f : Y → X {\displaystyle f\colon Y\rightarrow X}

from a regular scheme Y {\displaystyle Y} such that the higher direct images of f ∗ {\displaystyle f_{*}} applied to O Y {\displaystyle {\mathcal {O}}_{Y}} are trivial. That is,

R i f ∗ O Y = 0 {\displaystyle R^{i}f_{*}{\mathcal {O}}_{Y}=0} for i > 0 {\displaystyle 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 {\displaystyle X} has rational singularities if and only if the natural map in the derived category

O X → R f ∗ O Y {\displaystyle {\mathcal {O}}_{X}\rightarrow Rf_{*}{\mathcal {O}}_{Y}}

is a quasi-isomorphism. Notice that this includes the statement that O X ≃ f ∗ O Y {\displaystyle {\mathcal {O}}_{X}\simeq f_{*}{\mathcal {O}}_{Y}} and hence the assumption that X {\displaystyle 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

x 2 + y 2 + z 2 = 0. {\displaystyle x^{2}+y^{2}+z^{2}=0.\,}

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

See also

References

  1. (Kollár & Mori 1998, Theorem 5.22.)
  2. (Artin 1966)
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