Rational singularity explained

f \colon Y → X Y such that the higher direct images of f _{*} applied to l{O} _{Y} are trivial. That is, R ^{i} f _{*} l{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 categoryl{O} _{X} → R f _{*} l{O} _{Y} is a quasi-isomorphism. Notice that this includes the statement that l{O} _{X} \simeq f _{*} l{O} _{Y} and hence the assumption that X is normal.

There are related notions in positive and mixed characteristic of

• pseudo-rational

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.

Examples

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