Rational singularity

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

${\displaystyle f\colon Y\rightarrow X}$

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

${\displaystyle R^{i}f_{*}{\mathcal {O}}_{Y}=0}$ for ${\displaystyle i>0}$ .

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 (Artin 1966).

Formulations

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

${\displaystyle {\mathcal {O}}_{X}\rightarrow Rf_{*}{\mathcal {O}}_{Y}}$

is a quasi-isomorphism. Notice that this includes the statement that ${\displaystyle {\mathcal {O}}_{X}\simeq f_{*}{\mathcal {O}}_{Y}}$ and hence the assumption that ${\displaystyle X}$ 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.^[1]

Examples

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

${\displaystyle x^{2}+y^{2}+z^{2}=0.\,}$

Artin^[2] showed that the rational double points of algebraic surfaces are the Du Val singularities.

See also

• Elliptic singularity

References

1. (Kollár & Mori 1998, Theorem 5.22.)
2. (Artin 1966)

• Artin, Michael (1966), "On isolated rational singularities of surfaces", American Journal of Mathematics, 88 (1), The Johns Hopkins University Press: 129–136, doi:10.2307/2373050, ISSN 0002-9327, JSTOR 2373050, MR 0199191
• Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, doi:10.1017/CBO9780511662560, ISBN 978-0-521-63277-5, MR 1658959
• Lipman, Joseph (1969), "Rational singularities, with applications to algebraic surfaces and unique factorization", Publications Mathématiques de l'IHÉS (36): 195–279, ISSN 1618-1913, MR 0276239