Rational singularity

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, more particularly in the field of <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, a <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 has rational singularities, if it is <a href="/facts/Normal_scheme/TQ6cB9By">normal</a>, of finite type over a field of <a href="/facts/Characteristic_of_a_ring/D2VzcQaG">characteristic</a> zero, and there exists a <a href="/facts/Proper_morphism/EJHIgjnr">proper</a> <a href="/facts/Birational_map/hsWqRhJ1">birational map</a>

f
        :
        Y
        →
        X
      
    
    {\displaystyle f\colon Y\rightarrow X}

from a <a href="/facts/Glossary_of_scheme_theory/eh7NLhmS">regular scheme</a> 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 such that the <a href="/facts/Higher_direct_image/z3mckxvp">higher direct images</a> of 
 
 
 
 
 f
 
 ∗
 
 
 
 
 {\displaystyle f_{*}}
 
 applied to 
 
 
 
 
 
 
 O
 
 
 
 Y
 
 
 
 
 {\displaystyle {\mathcal {O}}_{Y}}
 
 are trivial. That is,

R
 
 i
 
 
 
 f
 
 ∗
 
 
 
 
 
 O
 
 
 
 Y
 
 
 =
 0
 
 
 {\displaystyle R^{i}f_{*}{\mathcal {O}}_{Y}=0}
 
 for 
 
 
 
 i
 >
 0
 
 
 {\displaystyle 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).

Rational singularity open-in-new

Rational singularity