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 (Artin 1966).

Formulations[edit]

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

and

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.[citation needed]

Examples[edit]

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

x^2 + y^2 + z^2 = 0. \,

(Artin 1966) showed that the rational double points of a algebraic surfaces are the Du Val singularities.

References[edit]