Gysin morphisms for non-transversal hyperplane sections with an application to line arrangements

TL;DR

This paper establishes Gysin morphisms for certain hyperplane sections lacking typical transversality conditions and applies this to prove the triviality of Alexander polynomials in specific line arrangements, supporting a conjecture.

Contribution

It introduces Gysin morphisms in non-transversal settings and applies them to line arrangements, advancing understanding of Alexander polynomials.

Findings

Gysin morphisms exist for non-transversal hyperplane sections

Alexander polynomial is trivial for certain non-symmetric line arrangements

Supports conjecture of Papadima and Suciu

Abstract

We prove the existence of Gysin morphisms for hyperplane sections that may not satisfy the usual hypotheses of the Lefschetz hyperplane theorem. As an application, we show the triviality of the Alexander polynomial of a particular class of non-symmetric line arrangements, thus providing positive evidence for a conjecture of Papadima and Suciu.