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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 . ==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, 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「rational singularity」の詳細全文を読む スポンサード リンク
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