A geometric definition of Gabrielov numbers

TL;DR

This paper introduces a geometric approach to defining Gabrielov numbers for cusp singularities with group actions, utilizing crepant resolutions and the McKay correspondence to relate homology and Coxeter-Dynkin diagrams.

Contribution

It provides a new geometric framework for Gabrielov numbers using crepant resolutions and homological methods, connecting singularity theory with algebraic geometry.

Findings

Computed the homology of the pair (Y,Z) using McKay correspondence.

Constructed a basis of H_3(Y,Z;Q) with a Coxeter-Dynkin diagram.

Read off Gabrielov numbers from the diagram.

Abstract

Gabrielov numbers describe certain Coxeter-Dynkin diagrams of the 14 exceptional unimodal singularities and play a role in Arnold's strange duality. In a previous paper, the authors defined Gabrielov numbers of a cusp singularity with an action of a finite abelian subgroup $G$ of ${\rm SL}(3,\CC)$ using the Gabrielov numbers of the cusp singularity and data of the group $G$. Here we consider a crepant resolution $Y \to \CC^3/G$ and the preimage $Z$ of the image of the Milnor fibre of the cusp singularity under the natural projection $\CC^3 \to \CC^3/G$. Using the McKay correspondence, we compute the homology of the pair $(Y,Z)$. We construct a basis of the relative homology group $H_3(Y,Z;\QQ)$ with a Coxeter-Dynkin diagram where one can read off the Gabrielov numbers.