Rational singularity

A normal singular point of an algebraic variety or complex-analytic space admitting a resolution (cf. Resolution of singularities), under which the direct images of the structure sheaf are trivial for . 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 is a Cohen–Macaulay variety and the imbedding of dual sheaves is an isomorphism [5].

Some examples of rational singularities are the singular points of the quotient space , where is a finite group of linear transformations, the singular point 0 of the hypersurface when (see [8]), and toric singularities.

If is a Gorenstein isolated singularity (i.e. the sheaf is locally free) over the field and is a generating section of , then is a rational singularity if and only if

in a sufficiently small neighbourhood of the point (see [7]).

In the case when , a singularity is rational if and only if for every cycle on the exceptional curve of the resolution . In this case, all the components of are isomorphic to the projective line , is a divisor with normal intersections and the graph of the resolution is a tree.

The fundamental cycle of a singularity is defined as the minimal cycle on for which for all . There is a criterion of rationality in terms of : , and one can calculate the multiplicity of the singularity and the dimension of the tangent space [1].'

<tbody> </tbody> Notation Equation Graph , ,

Rational singularities of hypersurfaces in a three-dimensional affine space , 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 , where is a finite subgroup of ; that is, up to conjugacy, is either the cyclic group or order , the binary dihedral group , the tetrahedral group , the octahedral group , or the icosahedral group . If is a minimal resolution of a rational double point, then for all , and the weighted (resolution) graph coincides with the diagram of simple roots of one of the semi-simple Lie algebras , , , , or , 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 ([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 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 is a flat morphism and is such that is a rational singularity in and is a rational singularity of the fibre , then is a rational singularity in .

3) If a deformation has a smooth base and admits a simultaneous resolution of singularities, then a point is a rational singularity if and only if is a rational singularity in its own fibre .

In the case when , every deformation of a variety resolving a rational singularity defines a deformation of , obtained by contracting the exceptional curves of the fibres of the given deformation. As a result, one obtains a morphism of the bases of versal deformations of the variety and the singularity . The image is a non-singular irreducible component of , called the Artin component, and is a Galois covering whose group can be found using the graph of the singularity (see [2], [10]). In particular, for a double rational singularity is surjective and 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 (see [9]).

References

[1]	M. Artin, "On isolated rational singularities of surfaces" Amer. J. Math. , 88 (1966) pp. 129–136
[2]	M. Artin, "Algebraic construction of Brieskorn's resolutions" J. Algebra , 29 : 2 (1974) pp. 330–348
[3]	E. Brieskorn, "Rationale singularitäten komplexer Flächen" Invent. Math. , 4 (1968) pp. 336–358
[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)
[7]	H.B. Laufer, "On rational singularities" Amer. J. Math. , 94 (1972) pp. 597–608
[8]	M. Reid, "Canonical 3-folds" J. Geom. Alg. Angers (1980) pp. 273–310
[9]	P.J. Slodowy, "Simple singularities and simple algebraic groups" , Lect. notes in math. , 815 , Springer (1980)
[10]	J.M. Wahl, "Simultaneous resolution of rational singularities" Compos. Math. , 38 (1979) pp. 43–54
[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)
[a2]	A. Durfee, "Fifteen characterizations of rational double points and simple critical points" Ens. Math. , 25 (1979) pp. 131–163