Rational singularity

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 Rf_{*}{\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.