$\mu$-constant monodromy groups and Torelli results for the quadrangle singularities and the bimodal series

TL;DR

This paper investigates the structure of moduli spaces and monodromy groups associated with certain hypersurface singularities, confirming Torelli conjectures for some series while identifying exceptions in others.

Contribution

It extends previous work by analyzing quadrangle singularities and the bimodal series, proving Torelli conjectures and identifying cases where the monodromy group equality does not hold.

Findings

Torelli conjectures are confirmed for quadrangle and bimodal series.

The conjecture that the moduli space is connected does not hold for some bimodal subseries.

The monodromy group equality $G^{mar} = G_{\mathbb{Z}}$ fails in certain cases.

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 $M_\mu^{mar}$ for marked singularities in one $\mu$-homotopy class of isolated hypersurface singularities was established. It is an analogue of a Teichm\"uller space. It comes together with a $\mu$-constant monodromy group $G^{mar}\subset G_{\mathbb{Z}}$. Here $G_{\mathbb{Z}}$ is the group of automorphisms of a Milnor lattice which respect the Seifert form. It was conjectured that $M_\mu^{mar}$ is connected. This is equivalent to $G^{mar}= G_{\mathbb{Z}}$. Also Torelli type conjectures were formulated. In [He11] and [GH17] $M_\mu^{mar}, G_{\mathbb{Z}}$ 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 $G^{mar}= G_{\mathbb{Z}}$ and $M_\mu^{mar}$ connected does not hold for certain subseries of the bimodal series.