Singularities and clusters

TL;DR

This paper explores a geometric connection between simple plane curve singularities and cluster varieties, showing they coincide when classified by the same simply laced Cartan matrices, through configurations of flags derived from Dynkin diagrams.

Contribution

It establishes a new geometric relation linking singularities and cluster varieties using configurations of flags based on Dynkin diagrams.

Findings

Singularities and cluster varieties are shown to coincide for simply laced types.

Construction of varieties of configurations of flags from Dynkin diagrams.

The geometric relation is explicitly described for singularities classified by Cartan matrices.

Abstract

The aim of this note is to describe a geometric relation between simple plane curve singularities classified by simply laced Cartan matrices and cluster varieties of finite type also classified by the simply laced Cartan matrices. We construct certain varieties of configurations of flags out of Dynkin diagrams and out of singularities and show that they coincide if the Dynkin diagram corresponds to the singularity.