μ-constant monodromy groups and Torelli results for the quadrangle singularities and the bimodal series

2017/10/10 by Gauss, Falko, Hertling, Claus
#14D22 #32S15 #32S40 #58K70 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1710.03507

Abstract

This paper is a sequel to [He11] and [GH17]. In [He11] a notion of marking of isolated hypersurface singularities was defined, and a moduli space M_μ^mar for marked singularities in one μ-homotopy class of isolated hypersurface singularities was established. It is an analogue of a Teichmüller space. It comes together with a μ-constant monodromy group G^mar⊂ G_ℤ. Here G_ℤ is the group of automorphisms of a Milnor lattice which respect the Seifert form. It was conjectured that M_μ^mar is connected. This is equivalent to G^mar= G_ℤ. Also Torelli type conjectures were formulated. In [He11] and [GH17] M_μ^mar, G_ℤ and G^mar were determined and all conjectures were proved for the simple, the unimodal and the exceptional bimodal singularities. In this paper the quadrangle singularities and the bimodal series are treated. The Torelli type conjectures are true. But the conjecture G^mar= G_ℤ and M_μ^mar connected does not hold for certain subseries of the bimodal series.

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