In mathematics, more particularly in the field of algebraic geometry, a scheme ${\displaystyle X}$ has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a proper birational map

from a regular scheme ${\displaystyle Y}$ such that the higher direct images of ${\displaystyle f_{*}}$ applied to ${\displaystyle {\mathcal {O}}_{Y}}$ are trivial. That is,

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).

Alternately, one can say that ${\displaystyle X}$ 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 ${\displaystyle {\mathcal {O}}_{X}\simeq f_{*}{\mathcal {O}}_{Y}}$ and hence the assumption that ${\displaystyle X}$ is normal.

There are related notions in positive and mixed characteristic of

and