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 ${\mathcal {O}}_{Y}$ are trivial. That is,

R^{i}f_{*}{\mathcal {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 a third.

For surfaces, rational singularities were defined by (Artin 1966).

Formulations

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

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

is a quasi-isomorphism. Notice that this includes the statement that ${\mathcal {O}}_{X}\simeq f_{*}{\mathcal {O}}_{Y}$ 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, (Kollár, Mori, 1998, Theorem 5.22.)

Examples

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

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

(Artin 1966) showed that the rational double points of algebraic surfaces are the Du Val singularities.

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, JSTOR 2373050, MR 0199191
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, 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

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