- Rational singularity
In

mathematics , more particularly in the field ofalgebraic geometry , a scheme $X$ has**rational singularities**, if it is normal, of finite type over a field of characteristic zero, and there exists a properbirational map :$f\; :\; Y\; ightarrow\; X$

from a

non-singular scheme $Y$ such that the higherright derived functor s 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 aquasi-isomorphism .There are related notions in positive and mixed characteristic of

* pseudo-rationaland

* F-rationalRational singularities are in particular Cohen-Macaulay, normal and Du Bois. They need not be

Gorenstein or evenQ-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 rationaldouble point of analgebraic surface ; they are also called a**Klein-Du Val singularity**.

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