Project IV 2022-23

The Du Val singularities

Description

The aim of this 4th year project will be to learn some basic algebraic geometry and singularity theory by studying the Du Val singularities. These are the simplest class of surface singularities and can be classified into two infinite families and three sporadic cases as follows:

• A_n : x² + y² + zⁿ⁺¹ = 0 for n ≥ 1

• D_n : x² + y²z + z^n-1 = 0 for n ≥ 4

• E₆ : x² + y³ + z⁴ = 0

• E₇ : x² + y³ + yz³ = 0

• E₈ : x² + y³ + z⁵ = 0

where these ADE labels are the same labels used to classify the simply laced Dynkin diagrams:

Curiously, these ADE labels and Dynkin diagrams also appear in the classification of many other mathematical objects, such as root systems and Lie algebras. Another curiosity is that these Du Val singularities have a large number of alternative characterisations. To list but just a few, they are also

1. the surface singularities of multiplicity 2 whose minimal resolution is a tree of rational curves (the configuration in which these curves meet is determined by the Dynkin diagram),

2. the simple 2-dimensional hypersurface singularities (meaning they deform to only a finite number of other singularities, up to isomorphism)

3. the quotient singularities of the form C²/G, for G a finite subgroup of SL(2,C).

In this project we will start by learning how to resolve surface singularities by blowing up. Then we will learn what a root system is and understand the explicit connection between these surface singularities and the root system which corresponds to the same Dynkin diagram.

After this you are free to explore some of the other characterisations of Du Val singularities that you find most interesting. Some of the more advanced material includes the McKay correspondence, which is a deep connection between the geometry of the quotient singularity X=C²/G (from point 3. above) and the representation theory of the group G.

Pre/corequisities

Algebra II and Geometry IV. Some familiarity with complex analysis and/or algebraic topology would be very useful, depending on the direction of the project.

References

For the background material in Algebraic Geometry:

• Miles Reid, Undergraduate Algebraic Geometry

For the Du Val singularities:

• Miles Reid, The Du Val singularities An, Dn, E6, E7, E8

• Alan Durfee, Fifteen characterizations of rational double points and simple critical points