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rationalisin
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There are 12 letters in RATIONALISIN ( A1I1L1N1O1R1S1T1 )
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In mathematics, more particularly in the field of algebraic geometry, a scheme * * * * X * * * {\displaystyle 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 * : * Y * → * X * * * {\displaystyle f\colon Y\rightarrow X} * from a regular scheme * * * * Y * * * {\displaystyle Y} * such that the higher direct images of * * * * * f * * ∗ * * * * * {\displaystyle f_{*}} * applied to * * * * * * * O * * * * Y * * * * * {\displaystyle {\mathcal {O}}_{Y}} * are trivial. That is, * * * * * R * * i * * * * f * * ∗ * * * * * * O * * * * Y * * * = * 0 * * * {\displaystyle R^{i}f_{*}{\mathcal {O}}_{Y}=0} * for * * * * i * > * 0 * * * {\displaystyle 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 (Artin 1966). |