is isomorphic to a hyperplane section $Y$ of $X$. Then we choose a generic fibre $Y$ of $\psi$, we explicitly compute the monodromy action $\phi$ on it and we show that $X$ and $Y$ satisfy the hypotheses of Theorem \[thm:risultato\].

2.  We prove that the surjective Gysin morphism $H^2(Y)\twoheadrightarrow H^4(X)$ remains surjective if we restrict the domain to the fixed part of $H^2(Y)$ under the action of the algebraic monodromy $T_{\phi}$ (Proposition \[prop:monodromy\]) and that it gives a surjective morphism between the primitive parts (Lemma \[lem:primitive\]).

3.  We bound the dimension of $H^2(Y)^{T_{\phi}}$ using the inclusion relation between the Hodge and polar filtration on $H^{{\bullet}}({\mathbb{P}}^3\setminus Y)$ and the explicit description of the graded pieces of the latter in terms of differential forms; we then use a Thom-type result (Lemma \[lem:ThomIso\]) to deduce from this bound our result on $\Delta_{{\overline{{\mathcal{A}}}}}$.

The paper is organised as follows. In Section 1 we give a brief account of the theory of cubical hyperresolutions, following [@GNPP] and [@PeSt Section 5], showing in particular a sketch of their standard construction. Section 2 is devoted to the proof of Theorem \[thm:risultato\]. Section 3 deals with the Alexander polynomial of curves and line arrangements, presenting the definition of 