algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, a Du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is an isolated singularity of a complex surface which is modeled on a double branched cover of the plane, with minimal resolution obtained by replacing the singular point with a tree of smooth rational curves, with intersection pattern dual to a Dynkin diagram of A-D-E singularity type. They are the

canonical singularities In mathematics, canonical singularities appear as singularities of the canonical model of a projective variety, and terminal singularities are special cases that appear as singularities of minimal models. They were introduced by . Terminal singula ...

(or, equivalently, rational Gorenstein singularities) in dimension 2. They were studied by

Patrick du Val Patrick du Val (March 26, 1903 – January 22, 1987) was a British mathematician, known for his work on algebraic geometry, differential geometry, and general relativity. The concept of Du Val singularity of an algebraic surface is named aft ...

and

Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...

. The Du Val singularities also appear as quotients of

\Complex^2

by a finite subgroup of SL₂

(\Complex)

; equivalently, a finite subgroup of

SU(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...

, which are known as

binary polyhedral group In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries ...

s. The rings of

invariant polynomial In mathematics, an invariant polynomial is a polynomial P that is invariant under a group \Gamma acting on a vector space V. Therefore, P is a \Gamma-invariant polynomial if :P(\gamma x) = P(x) for all \gamma \in \Gamma and x \in V. Cases of p ...

s of these finite group actions were computed by Klein, and are essentially the coordinate rings of the singularities; this is a classic result in

invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...

Classification

The possible Du Val singularities are (up to analytical isomorphism): *

A_n: \quad w^2+x^2+y^=0

D_n: \quad w^2+y(x^2+y^) = 0 \qquad (n\ge 4)

E_6: \quad w^2+x^3+y^4=0

E_7: \quad w^2+x(x^2+y^3)=0

E_8: \quad w^2+x^3+y^5=0.

References

External links

* * {{lowercase Algebraic surfaces Singularity theory

Classification

See also

References

External links