The integral monodromy of the cycle type singularities

Claus Hertling and Makiko Mase

Journal of Singularities
volume 25 (2022), 268-298

Received: 13 January 2021. Received in revised form: 17 October 2021.

DOI: 10.5427/jsing.2022.25l

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Abstract:

The middle homology of the Milnor fiber of a quasihomogeneous polynomial with an isolated singularity is a $\Z$-lattice and comes equipped with an automorphism of finite order, the integral monodromy. Orlik (1972) made a precise conjecture, which would determine this monodromy in terms of the weights of the polynomial. Here we prove this conjecture for the cycle type singularities. A paper of Cooper (1982) with the same aim contained two mistakes. Still it is very useful. We build on it and correct the mistakes. We give additional algebraic and combinatorial results.

2010 Mathematical Subject Classification:

55U15, 55T05, 58K10, 32S50

Key words and phrases:

Integral monodromy, cycle type singularity, Orlik's conjecture, spectral sequence

Author(s) information:

Claus Hertling
Universität Mannheim
Lehrstuhl für algebraische Geometrie, B6, 26
68159 Mannheim, Germany
hertling@math.uni-mannheim.de

Makiko Mase
Universität Mannheim
Lehrstuhl für algebraische Geometrie, B6, 26
68159 Mannheim, Germany
mmase@mail.uni-mannheim.de