In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, more particularly in the field of
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, a
scheme has rational singularities, if it is
normal, of finite type over a field of
characteristic zero, and there exists a
proper birational map
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational fu ...
:
from a
regular scheme
In algebraic geometry, a regular scheme is a locally Noetherian scheme whose local rings are regular everywhere. Every smooth scheme is regular, and every regular scheme of finite type over a perfect field is smooth..
For an example of a regul ...
such that the
higher direct image
In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topo ...
s of
applied to
are trivial. That is,
:
for
.
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 .
Formulations
Alternately, one can say that
has rational singularities if and only if the natural map in the
derived category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pr ...
:
is a
quasi-isomorphism
In homological algebra, a branch of mathematics, a quasi-isomorphism or quism is a morphism ''A'' → ''B'' of chain complexes (respectively, cochain complexes) such that the induced morphisms
:H_n(A_\bullet) \to H_n(B_\bullet)\ (\text H^n(A^\bull ...
. Notice that this includes the statement that
and hence the assumption that
is normal.
There are related notions in positive and mixed
characteristic of
*
pseudo-rational
and
*
F-rational
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.
Examples
An example of a rational singularity is the singular point of the
quadric cone
:
Artin
showed that
the rational
double points of
algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
s are the
Du Val singularities.
See also
*
Elliptic singularity
References
*
*
*{{Citation , last1=Lipman , first1=Joseph , title=Rational singularities, with applications to algebraic surfaces and unique factorization , url=http://www.numdam.org/item?id=PMIHES_1969__36__195_0 , mr=0276239 , year=1969 , journal=
Publications Mathématiques de l'IHÉS
''Publications Mathématiques de l'IHÉS'' is a peer-reviewed mathematical journal. It is published by Springer Science+Business Media on behalf of the Institut des Hautes Études Scientifiques, with the help of the Centre National de la Recher ...
, issn=1618-1913 , issue=36 , pages=195–279
Algebraic surfaces
Singularity theory