First homology groups of the Milnor fiber boundary for generic hyperplane arrangements in $\mathbb{C}^{3}$

Abstract

We study the Milnor fiber boundary for hyperplane arrangements in $\mathbb{C}^3$. This is one of the examples of non-isolated surface singularities, which are studied by N\'emethi--Szil\'ard. In this paper, we compute the first homology group of the Milnor fiber boundary for a generic arrangement, which gives an affirmative answer to the conjecture of Suciu. Also, we give an example of an arrangement with $n$ hyperplanes, whose torsion part in the Milnor fiber boundary homology contains a direct summand other than $\mathbb{Z}_{n}$, for certain value of $n$.