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
- 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).
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
An example of a rational singularity is the singular point of the quadric cone
- 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, 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