TY - JOUR

T1 - A note on distinguished bases of singularities

AU - Ebeling, Wolfgang

N1 - Funding information: Partially supported by the DFG-programme SPP1388 “Representation Theory”.

PY - 2018/2/1

Y1 - 2018/2/1

N2 - 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.

AB - 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.

KW - Braid group

KW - Coxeter–Dynkin diagram

KW - Distinguished basis of vanishing cycles

KW - Intersection number

KW - Singularity

UR - http://www.scopus.com/inward/record.url?scp=85035338395&partnerID=8YFLogxK

UR - https://arxiv.org/abs/1611.06074

U2 - 10.48550/arXiv.1611.06074

DO - 10.48550/arXiv.1611.06074

M3 - Article

AN - SCOPUS:85035338395

VL - 234

SP - 259

EP - 268

JO - Topology and its applications

JF - Topology and its applications

SN - 0166-8641

ER -