The integral monodromy of isolated quasihomogeneous singularities

TL;DR

This paper proves Orlik's conjecture on the integral monodromy of certain isolated quasihomogeneous singularities, using algebraic tools involving lattices, automorphisms, and cyclotomic polynomials.

Contribution

It establishes the conjecture for all iterated Thom-Sebastiani sums of chain and cycle type singularities, advancing understanding of monodromy in singularity theory.

Findings

Proves Orlik's conjecture for specific singularities

Develops algebraic tools for lattice and automorphism analysis

Utilizes properties of cyclotomic polynomials and roots

Abstract

The integral monodromy on the Milnor lattice of an isolated quasihomogeneous singularity is subject of an almost untouched conjecture of Orlik from 1972. We prove this conjecture for all iterated Thom-Sebastiani sums of chain type singularities and cycle type singularities. The main part of the paper is purely algebraic. It provides tools for dealing with sums and tensor products of ${\mathbb Z}$-lattices with automorphisms of finite order and with cyclic generators. The calculations are involved. They use fine properties of unit roots, cyclotomic polynomials, their resultants and discriminants.