, more particularly in the field of algebraic geometry
, a scheme
<math>X</math> has rational singularities
, if it is normal
, of finite type over a field of characteristic
zero, and there exists a proper birational map
- <math>f colon Y rightarrow X</math>
from a regular scheme
<math>Y</math> such that the higher direct images
of <math>f_*</math> applied to <math>mathcal_Y</math> are trivial. That is,
- <math>R^i f_* mathcal_Y = 0</math> for <math>i > 0</math>.
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 .
Alternately, one can say that <math>X</math> has rational singularities if and only if the natural map in the derived category
- <math>mathcal_X rightarrow R f_* mathcal_Y</math>
is a quasi-isomorphism
. Notice that this includes the statement that <math>mathcal_X simeq f_* mathcal_Y</math> and hence the assumption that <math>X</math> is normal.
There are related notions in positive and mixed characteristic
Rational singularities are in particular Cohen-Macaulay
and Du Bois
. They need not be Gorenstein
or even Q-Gorenstein
singularities are rational.
An example of a... Read More