---
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.'
author:
- Federico Venturelli
title: 'Gysin morphisms for non-transversal hyperplane sections with an application to line arrangements'
---

Introduction {#introduction .unnumbered}
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We work over the field of complex numbers ${\mathbb{C}}$. Let $X$ be a closed subvariety of ${\mathbb{P}}^n$ and $Y:=X\cap H$ be a hyperplane section; in order to compare the cohomology groups of $X$ and $Y$ one needs some form of the Lefschetz hyperplane theorem. The typical situations in which this theorem can be used are:

(a) (Classical version) Both $X$ and $Y$ are smooth.

(b) (Modern formulation) There exists a Whitney stratification ${\mathcal{A}}$ of ${\mathbb{P}}^n$ such that $X=\cup_{i=1}^k A_i$ with $A_i\in{\mathcal{A}}$ and $H$ is transversal to $A_i$ for $i=1,\ldots,k$ (see for example [@D1 Theorem 1.6.5] and references therein).

In both cases above, the comparison is provided by Gysin morphisms $H^{k}(Y)\rightarrow H^{k+2}(X)$ which are isomorphisms for $k>\dim(Y)$ and a surjection for $k=\dim(Y)$.

In this paper 