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

from a regular scheme such that the higher direct images of applied to are trivial. That is,

for .

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 has rational singularities if and only if the natural map in the derived category

is a quasi-isomorphism. Notice that this includes the statement that and hence the assumption that 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

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

See also

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, 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
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