From Wikipedia, the free encyclopedia
In
mathematics , more particularly in the field of
algebraic geometry , a
scheme
X
{\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
f
:
Y
→
X
{\displaystyle f\colon Y\rightarrow X}
from a
regular scheme
Y
{\displaystyle Y}
such that the
higher direct images 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 ).
Formulations
Alternately, one can say that
X
{\displaystyle X}
has rational singularities if and only if the natural map in the
derived category
O
X
→
R
f
∗
O
Y
{\displaystyle {\mathcal {O}}_{X}\rightarrow Rf_{*}{\mathcal {O}}_{Y}}
is a
quasi-isomorphism . Notice that this includes the statement that
O
X
≃
f
∗
O
Y
{\displaystyle {\mathcal {O}}_{X}\simeq f_{*}{\mathcal {O}}_{Y}}
and hence the assumption that
X
{\displaystyle X}
is normal.
There are related notions in positive and mixed
characteristic of
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.
[1]
Examples
An example of a rational singularity is the singular point of the
quadric cone
x
2
+
y
2
+
z
2
=
0.
{\displaystyle x^{2}+y^{2}+z^{2}=0.\,}
Artin
[2] showed that
the rational
double points of
algebraic surfaces are the
Du Val singularities .
See also
References
Artin, Michael (1966), "On isolated rational singularities of surfaces",
American Journal of Mathematics , 88 (1), The Johns Hopkins University Press: 129–136,
doi :
10.2307/2373050 ,
ISSN
0002-9327 ,
JSTOR
2373050 ,
MR
0199191
Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties , Cambridge Tracts in Mathematics, vol. 134,
Cambridge University Press ,
doi :
10.1017/CBO9780511662560 ,
ISBN
978-0-521-63277-5 ,
MR
1658959
Lipman, Joseph (1969),
"Rational singularities, with applications to algebraic surfaces and unique factorization" ,
Publications Mathématiques de l'IHÉS (36): 195–279,
ISSN
1618-1913 ,
MR
0276239