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 $\mathcal_Y$ are trivial. That is,

:$R^i f_* \mathcal_Y = 0$ 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
:$\mathcal_X \rightarrow R f_* \mathcal_Y$
is a quasi-isomorphism. Notice that this includes the statement that $\mathcal_X \simeq f_* \mathcal_Y$ 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.

See also

* Elliptic singularity

References

*
*
*{{Citation , last1=Lipman , first1=Joseph , title=Rational singularities, with applications to algebraic surfaces and unique factorization , url=http://www.numdam.org/item?id=PMIHES_1969__36__195_0 , mr=0276239 , year=1969 , journal=Publications Mathématiques de l'IHÉS , issn=1618-1913 , issue=36 , pages=195–279

Algebraic surfaces
Singularity theory