Rational singularity

In mathematics, more particularly in the field of algebraic geometry, a scheme <math>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>f \colon Y \rightarrow X</math>

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

:<math>R^i f_* \mathcal{O}_Y = 0</math> for <math>i > 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 {{harv|Artin|1966}}.

==Formulations== Alternately, one can say that <math>X</math> has rational singularities if and only if the natural map in the derived category :<math>\mathcal{O}_X \rightarrow R f_* \mathcal{O}_Y</math> is a quasi-isomorphism. Notice that this includes the statement that <math>\mathcal{O}_X \simeq f_* \mathcal{O}_Y</math> and hence the assumption that <math>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.<ref>{{harv|Kollár|Mori|1998|loc=Theorem 5.22.}}</ref>

==Examples==

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

:<math>x^2 + y^2 + z^2 = 0. \,</math>

Artin<ref>{{harv|Artin|1966}}</ref> showed that the rational double points of algebraic surfaces are the Du Val singularities.

==See also== *Elliptic singularity

==References== {{reflist}} *{{Citation | doi=10.2307/2373050 | last1=Artin | first1=Michael | author1-link=Michael Artin | title=On isolated rational singularities of surfaces |mr=0199191 | year=1966 | journal=American Journal of Mathematics | issn=0002-9327 | volume=88 | pages=129–136 | issue=1 | publisher=The Johns Hopkins University Press | jstor=2373050}} *{{Citation | last1=Kollár | first1=János | last2=Mori | first2=Shigefumi | title=Birational geometry of algebraic varieties | publisher=Cambridge University Press | series=Cambridge Tracts in Mathematics | isbn=978-0-521-63277-5 |mr=1658959 | year=1998 | volume=134 | doi=10.1017/CBO9780511662560}} *{{Citation | last1=Lipman | first1=Joseph |author-link=Joseph Lipman | title=Rational singularities, with applications to algebraic surfaces and unique factorization | url=http://www.numdam.org/item?id=PMIHES_1969__36__195_0 |mr=0276239 | year=1969 | journal=Publications Mathématiques de l'IHÉS | issn=1618-1913 | issue=36 | pages=195–279}}

Category:Algebraic surfaces Category:Singularity theory