Root Systems and Quotients of Deformations of Simple Singularities

TL;DR

This paper investigates quotients of deformations of simple singularities, linking them to root systems, and proves a conjecture about singular configurations for specific types.

Contribution

It introduces a conjecture relating singular configurations to sub-root systems and proves it for certain types of singularities.

Findings

Quotients of deformations relate to specific simple singularities.

Not all subdiagrams appear as singular configurations.

Conjecture proven for types B2, B3, C3, F4, G2.

Abstract

In this article we study quotients of deformations of simple singularities, and attempt to characterize them in terms of subsystems of simple root systems. The quotient of a semiuniversal deformation of a simple singularity of inhomogeneous type $B_r$ ($r \geq 2$), $C_r$ ($r \geq 3$), $F_4$ or $G_2$ by the natural symmetry of the associated Dynkin diagram is a deformation of a simple singularity of homogeneous type $X = D_s$, $E_6$ or $E_7$, but not semiuniversal anymore. Therefore not all subdiagrams of $X$ appear as singular configurations of the fibers of the deformation. We propose a conjecture for the types of singular configurations in terms of sub-root systems of a root system of type $X$. The conjecture is then proved for the types $B_2$, $B_3$, $C_3$, $F_4$ and $G_2$.