A note on distinguished bases of singularities

TL;DR

This paper investigates the structure of distinguished bases of vanishing cycles in certain hypersurface singularities, revealing infinite sets of Coxeter-Dynkin diagrams for most cases, contrasting with the finite sets in simple types.

Contribution

It demonstrates the existence of infinite distinguished bases and Coxeter-Dynkin diagrams for non-simple singularities, extending understanding beyond simple and simple elliptic cases.

Findings

Infinite Coxeter-Dynkin diagrams for most singularities.

Existence of distinguished bases with arbitrary intersection numbers.

Hyperbolic unimodal singularities share diagrams with exceptional unimodal ones.

Abstract

For an isolated hypersurface singularity which is neither simple nor simple elliptic, it is shown that there exists a distinguished basis of vanishing cycles which contains two basis elements with an arbitrary intersection number. This implies that the set of Coxeter-Dynkin diagrams of such a singularity is infinite, whereas it is finite for the simple and simple elliptic singularities. For the simple elliptic singularities, it is shown that the set of distinguished bases of vanishing cycles is also infinite. We also show that some of the hyperbolic unimodal singularities have Coxeter-Dynkin diagrams like the exceptional unimodal singularities.