Rational singularity

In mathematics, more particularly in the field of algebraic geometry, a scheme [math]\displaystyle{ X }[/math] has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a proper birational map

[math]\displaystyle{ f \colon Y \rightarrow X }[/math]

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

[math]\displaystyle{ R^i f_* \mathcal{O}_Y = 0 }[/math] for [math]\displaystyle{ i \gt 0 }[/math].

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 (Artin 1966).

Formulations

Alternately, one can say that [math]\displaystyle{ X }[/math] has rational singularities if and only if the natural map in the derived category

[math]\displaystyle{ \mathcal{O}_X \rightarrow R f_* \mathcal{O}_Y }[/math]

is a quasi-isomorphism. Notice that this includes the statement that [math]\displaystyle{ \mathcal{O}_X \simeq f_* \mathcal{O}_Y }[/math] and hence the assumption that [math]\displaystyle{ X }[/math] 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

[math]\displaystyle{ x^2 + y^2 + z^2 = 0. \, }[/math]

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

See also

• Elliptic singularity

References

• Artin, Michael (1966), "On isolated rational singularities of surfaces", American Journal of Mathematics (The Johns Hopkins University Press) 88 (1): 129–136, doi:10.2307/2373050, ISSN 0002-9327 
• Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, doi:10.1017/CBO9780511662560, ISBN 978-0-521-63277-5 
• Lipman, Joseph (1969), "Rational singularities, with applications to algebraic surfaces and unique factorization", Publications Mathématiques de l'IHÉS (36): 195–279, ISSN 1618-1913, http://www.numdam.org/item?id=PMIHES_1969__36__195_0