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Revista Matemática Complutense

, Volume 27, Issue 2, pp 447–460 | Cite as

A geometric definition of Gabrielov numbers

  • Wolfgang EbelingEmail author
  • Atsushi Takahashi
Article
  • 98 Downloads

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 \(\mathrm{SL}(3,{\mathbb {C}})\) using the Gabrielov numbers of the cusp singularity and data of the group \(G\). Here we consider a crepant resolution \(Y \rightarrow {\mathbb {C}}^3/G\) and the preimage \(Z\) of the image of the Milnor fibre of the cusp singularity under the natural projection \({\mathbb {C}}^3 \rightarrow {\mathbb {C}}^3/G\). Using the McKay correspondence, we compute the homology of the pair \((Y,Z)\). We construct a basis of the relative homology group \(H_3(Y,Z;{\mathbb {Q}})\) with a Coxeter–Dynkin diagram where one can read off the Gabrielov numbers.

Keywords

Cusp singularity Group action Crepant resolution  McKay correspondence Coxeter–Dynkin diagram Gabrielov numbers 

Mathematics Subject Classification (2010)

32S25 32S55 14E16 14L30 

Notes

Acknowledgments

This work has been supported by the DFG-programme SPP1388 “Representation Theory” (Eb 102/6-1). The second named author is also supported by JSPS KAKENHI Grant Number 24684005. We would like to thank the referees for their useful comments.

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Copyright information

© Universidad Complutense de Madrid 2013

Authors and Affiliations

  1. 1.Institut für Algebraische GeometrieLeibniz Universität HannoverHannoverGermany
  2. 2.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonakaJapan

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