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| In [[mathematics]], more particularly in the field of [[algebraic geometry]], a [[scheme (mathematics)|scheme]] <math>X</math> has '''rational singularities''', if it is [[normal scheme|normal]], of finite type over a field of [[characteristic of a ring|characteristic]] zero, and there exists a [[proper morphism|proper]] [[birational map]]
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| :<math>f \colon Y \rightarrow X</math>
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| from a [[Glossary of scheme theory#Properties of schemes|regular scheme]] <math>Y</math> such that the [[higher direct image]]s of <math>f_*</math> applied to <math>\mathcal{O}_Y</math> are trivial. That is,
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| :<math>R^i f_* \mathcal{O}_Y = 0</math> for <math>i > 0</math>.
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| 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.
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| For surfaces, rational singularities were defined by {{harv|Artin|1966}}.
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| ==Formulations==
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| Alternately, one can say that <math>X</math> has rational singularities if and only if the natural map in the [[derived category]]
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| :<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.
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| There are related notions in positive and mixed [[characteristic of a ring|characteristic]] of
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| * [[Pseudo-rational (mathematics)|pseudo-rational]]
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| and
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| * [[F-rational (mathematics)|F-rational]]
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| Rational singularities are in particular [[Cohen-Macaulay ring|Cohen-Macaulay]], [[normal scheme|normal]] and [[Du Bois singularities|Du Bois]]. They need not be [[Gorenstein ring|Gorenstein]] or even [[Q-Gorenstein]].
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| [[Log terminal]] singularities are rational.{{citation needed|date=September 2012}}
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| ==Examples==
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| An example of a rational singularity is the singular point of the [[quadric cone]]
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| :<math>x^2 + y^2 + z^2 = 0. \,</math> | |
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| {{harv|Artin|1966}} showed that
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| the rational [[double point]]s of a [[algebraic surface]]s are the [[Du Val singularities]].
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| ==References==
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| *{{Citation | doi=10.2307/2373050 | last1=Artin | first1=Michael | author1-link=Michael Artin | title=On isolated rational singularities of surfaces | id={{MathSciNet | id = 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}}
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| *{{Citation | last1=Lipman | first1=Joseph | title=Rational singularities, with applications to algebraic surfaces and unique factorization | url=http://www.numdam.org/item?id=PMIHES_1969__36__195_0 | id={{MathSciNet | id = 0276239}} | year=1969 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | issue=36 | pages=195–279}}
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| [[Category:Algebraic surfaces]]
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| [[Category:Singularity theory]]
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