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Rational singularity

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

:f \colon Y \rightarrow X

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

:R^i f_* \mathcal{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 .

Formulations

Alternately, one can say that X has rational singularities if and only if the natural map in the derived category :\mathcal{O}X \rightarrow R f* \mathcal{O}_Y is a quasi-isomorphism. Notice that this includes the statement that \mathcal{O}X \simeq f* \mathcal{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.

Examples

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

:x^2 + y^2 + z^2 = 0. ,

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

References

References

  1. {{harv. Kollár. Mori. 1998
  2. {{harv. Artin. 1966
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