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

:$f\; :\; Y\; ightarrow\; X$

from a non-singular scheme $Y$ such that the higher right derived functors 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 another.

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:$O\_X\; ightarrow\; R\; f\_*\; O\_Y$is a quasi-isomorphism.

There are related notions in positive and mixed characteristic of
* pseudo-rationaland
* 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

:$x^2\; +\; y^2\; +\; z^2\; =\; 0$.

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