Rational singularity
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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 .
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 . ==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 R f_* \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. ==Examples== An example of a rational singularity is the singular point of the quadric cone x^2 + y^2 + z^2 = 0. \, Artin showed that the rational double points of algebraic surfaces are the Du Val singularities. ==See also== Elliptic singularity ==References==
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