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

f : Y \rightarrow X

from a non-singular scheme Y such that the higher right derived functors of f * applied to OY are trivial.

That is,

Rif * OY = 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 another.

[edit] Formulations

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

O_X \rightarrow R f_* O_Y

is a quasi-isomorphism.

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.

[edit] Examples

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

x2 + y2 + z2 = 0.

A Du Val singularity is a rational double point of an algebraic surface; they are also called a Klein-Du Val singularity.