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On the arc filtration for the singularities of Arnold's lists

Published online by Cambridge University Press:  24 February 2005

WOLFGANG EBELING
Affiliation:
Universität Hannover, Institut für Mathematik, Postfach 6009, D-30060 Hannover, Germany. e-mail: ebeling@math.uni-hannover.de
SABIR M. GUSEIN-ZADE
Affiliation:
Moscow State University, Faculty of Mechanics and Mathematics, Moscow, 119992, Russia. e-mail: sabir@mccme.ru
Partially supported by the DFG-programme “Global methods in complex geometry” (Eb 102/4-2), grants RFBR-01-01-00739, INTAS-00-259, NSh-1972.2003.1.

Abstract

In a previous paper, the authors introduced a filtration on the ring ${\cal O}_{V,0}$ of germs of functions on a germ $(V,0)$ of a complex analytic variety defined by arcs on the singularity and called the arc filtration. The Poincaré series of this filtration were computed for simple surface singularities in the 3-space. Here they are computed for surface singularities from Arnold's lists including uni- and bimodular ones. The classification of the unimodular singularities by these Poincaré series turns out to be in accordance with their hierarchy defined by E. Brieskorn using the adjacency relations. We also give a general formula for the Poincaré series of the arc filtration for isolated surface singularities which are stabilizations of plane curve ones.

Type
Research Article
Copyright
2005 Cambridge Philosophical Society

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