Rational singularity

A normal singular point $ P $ of an algebraic variety or complex-analytic space $ X $ admitting a resolution $ \pi : \ Y \rightarrow X $( cf. Resolution of singularities), under which the direct images $ R ^{i} \pi _{*} {\mathcal O} _{Y} $ of the structure sheaf $ {\mathcal O} _{Y} $ are trivial for $ i \geq 1 $. Then any resolution of the given singularity will have this property. If the ground field has characteristic 0, then a singularity is rational if and only if $ X $ is a Cohen–Macaulay variety and the imbedding $ \pi _{*} : \ \omega _{Y} \rightarrow \omega _{X} $ of dual sheaves is an isomorphism [5].

Some examples of rational singularities are the singular points of the quotient space $ \mathbf C ^{n} / G $, where $ G $ is a finite group of linear transformations, the singular point 0 of the hypersurface $ x _{0} ^{ {k} _{0}} + \dots + x _{n} ^{ {k} _{n}} = 0 $ when $ \sum _{i=1} ^{n} k _{i} ^{-1} > 1 $( see [8]), and toric singularities.

If $ P \in X $ is a Gorenstein isolated singularity (i.e. the sheaf $ \omega _{X} $ is locally free) over the field $ \mathbf C $ and $ \omega $ is a generating section of $ \omega _{X} $, then $ P $ is a rational singularity if and only if $$ \int\limits _{U} \omega \land \overline \omega < \infty $$ in a sufficiently small neighbourhood $ U $ of the point (see [7]).

In the case when $ \mathop{\rm dim}\nolimits X = 2 $, a singularity $ P $ is rational if and only if $ h ^{1} ( {\mathcal O} _{D} ) = 0 $ for every cycle $ D $ on the exceptional curve $ E = \pi ^{-1} (P) $ of the resolution $ \pi $. In this case, all the components $ E _{i} $ of $ E $ are isomorphic to the projective line $ \mathbf P ^{1} $, $ E $ is a divisor with normal intersections and the graph $ \Gamma $ of the resolution is a tree.

The fundamental cycle of a singularity is defined as the minimal cycle $ Z > 0 $ on $ E $ for which $ Z \cdot E _{i} \leq 0 $ for all $ i $. There is a criterion of rationality in terms of $ Z $: $ h ^{1} ( {\mathcal O} _{Z} ) = 0 $,

and one can calculate the multiplicity of the singularity and the dimension of the tangent space [1].

<tbody> </tbody> Notation Equation Graph $ G $ $ A _{n} $, $ n \geq 1 $ $ x ^{n+1} + y ^{2} + z ^{2} $ $ C _{n+1} $ $ D _{n} $, $ n \geq 4 $ $ xy ^{2} + x ^{n+1} + z ^{2} $ $ D _{n-2} $ $ E _{6} $ $ x ^{3} + y ^{4} + z ^{2} $ $ T $ $ E _{7} $ $ x ^{3} + xy ^{3} + z ^{2} $ $ O $ $ E _{8} $ $ x ^{3} + y ^{5} + z ^{2} $ $ I $

Rational singularities of hypersurfaces $ X $ in a three-dimensional affine space $ A ^{3} $, equivalently, two-dimensional rational singularities of multiplicity 2, are called rational double points. Rational double points admit various equivalent characterizations and have various names, such as Klein singularities, Du Val singularities and simple singularities. The equations of rational double points arose as equations relating invariants of symmetry groups of regular polyhedra (see [6]). To this corresponds a characterization of rational double points as singularities of the quotient space $ X = \mathbf C ^{2} / G \subset \mathbf C ^{3} $, where $ G $ is a finite subgroup of $ \mathop{\rm SL}\nolimits ( 2 ,\ \mathbf C ) $; that is, up to conjugacy, $ G $ is either the cyclic group $ C _{n} $ or order $ n $, the binary dihedral group $ D _{n} $, the tetrahedral group $ T $, the octahedral group $ O $, or the icosahedral group $ I $. If $ \pi $ is a minimal resolution of a rational double point, then $ E _{i} ^{2} = - 2 $ for all $ i $, and the weighted (resolution) graph $ \Gamma $ coincides with the diagram of simple roots of one of the semi-simple Lie algebras $ A _{n} $, $ D _{n} $, $ E _{6} $, $ E _{7} $, or $ E _{8} $, whose symbol is also used to denote the singularity (cf. Lie algebra, semi-simple). Such a singularity is determined up to an isomorphism by its weighted graph $ \Gamma $([3], [11]), as depicted in the table above. Rational double points can be characterized as two-dimensional Gorenstein rational singularities. They are also called canonical singularities, since they are just those singularities which appear in canonical models of algebraic surfaces of general type.

If $ P \in X $ is a Gorenstein rational singularity of arbitrary dimension, then its general hypersurface section is either a rational or an elliptic Gorenstein singularity, and this leads, in particular, to a characterization of three-dimensional rational singularities (see [8]).

The following assertions are true in all dimensions (see [4]).

1) A deformation of a rational singularity is again a rational singularity.

2) If $ f : \ X \rightarrow S $ is a flat morphism and $ x \in X $ is such that $ s = f (x) $ is a rational singularity in $ S $ and $ x $ is a rational singularity of the fibre $ X _{s} = f ^{ {\ } -1} (s) $, then $ x $ is a rational singularity in $ X $.

3) If a deformation $ f : \ X \rightarrow S $ has a smooth base $ S $ and admits a simultaneous resolution of singularities, then a point $ x \in X $ is a rational singularity if and only if $ x $ is a rational singularity in its own fibre $ f ^{ {\ } -1} ( f (x) ) $.

In the case when $ \mathop{\rm dim}\nolimits X = 2 $, every deformation of a variety $ Y $ resolving a rational singularity $ P \in X $ defines a deformation of $ P $, obtained by contracting the exceptional curves of the fibres of the given deformation. As a result, one obtains a morphism $ \phi : \ \mathop{\rm Def}\nolimits \ Y \rightarrow \mathop{\rm Def}\nolimits \ X $ of the bases of versal deformations of the variety $ Y $ and the singularity $ P $. The image $ A = \phi ( \mathop{\rm Def}\nolimits \ Y ) $ is a non-singular irreducible component of $ \mathop{\rm Def}\nolimits \ X $, called the Artin component, and $ \phi : \ \mathop{\rm Def}\nolimits \ Y \rightarrow A $ is a Galois covering whose group $ W $ can be found using the graph $ \Gamma $ of the singularity $ P $( see [2], [10]). In particular, for a double rational singularity $ \phi $ is surjective and $ W $ coincides with the Weyl group of the corresponding Lie algebra, that is, a versal deformation of a rational singularity is simultaneously resolved after the Galois covering of the base of the deformation with Weyl group $ W $( see [9]).

References

[1]	M. Artin, "On isolated rational singularities of surfaces" Amer. J. Math. , 88 (1966) pp. 129–136 MR0199191 Zbl 0142.18602
[2]	M. Artin, "Algebraic construction of Brieskorn's resolutions" J. Algebra , 29 : 2 (1974) pp. 330–348 MR354665
[3]	E. Brieskorn, "Rationale singularitäten komplexer Flächen" Invent. Math. , 4 (1968) pp. 336–358 MR0222084 Zbl 0219.14003
[4]	R. Elkik, "Singularités rationelles et déformations" Invent. Math. , 47 (1978) pp. 139–147
[5]	G. Kempf, "Cohomology and convexity" G. Kempf (ed.) et al. (ed.) , Toroidal embeddings , Lect. notes in math. , 339 , Springer (1973) pp. 41–52
[6]	F. Klein, "Lectures on the icosahedron and the solution of equations of the fifth degree" , Dover, reprint (1956) (Translated from German) MR0080930 Zbl 0072.25901
[7]	H.B. Laufer, "On rational singularities" Amer. J. Math. , 94 (1972) pp. 597–608 MR0330500 Zbl 0251.32002
[8]	M. Reid, "Canonical 3-folds" J. Geom. Alg. Angers (1980) pp. 273–310 MR0605348 Zbl 0451.14014
[9]	P.J. Slodowy, "Simple singularities and simple algebraic groups" , Lect. notes in math. , 815 , Springer (1980) MR0584445 Zbl 0441.14002
[10]	J.M. Wahl, "Simultaneous resolution of rational singularities" Compos. Math. , 38 (1979) pp. 43–54 MR0523262 Zbl 0412.14008
[11]	G.N. Tyurina, "On the tautness of rationally contractible curves on a surface" Math. USSR Izv. , 2 (1968) pp. 907–934 Izv. Akad. Nauk SSSR. Ser. Mat. , 32 (1968) pp. 943–970

Comments

See also Dynkin diagram.

References

[a1]	M. Demazure (ed.) H. Pinkham (ed.) B. Teissier (ed.) , Sem. singularités des surfaces , Lect. notes in math. , 777 , Springer (1980) MR579026
[a2]	A. Durfee, "Fifteen characterizations of rational double points and simple critical points" Ens. Math. , 25 (1979) pp. 131–163 MR0543555 Zbl 0418.14020