combinatorics and geometry; for this reason, in this introduction and in Section 3 we focus only on the aspects we care the most about.

The Alexander polynomial $\Delta_C$ of a reduced plane curve $C=V(f)$ of degree $d$ is the characteristic polynomial of the algebraic monodromy acting on $H^1(F,{\mathbb{C}})$, where $F$ is any fibre of the Milnor fibration $f:{\mathbb{C}}^3\setminus f^{-1}(0)\rightarrow{\mathbb{C}}^*$; it is known that $\Delta_C$ depends on the type and relative position of the singular points of ${\mathbb{C}}$, and that it can be written as

$$\Delta_C(t)=(t-1)^{r-1}\prod_{1<k|d}\Phi_k(t)^{e_k}=(t-1)^{r-1}q(t)$$

where $r$ is the number of irreducible components of $C$ and $\Phi_k$ is the $k$-th cyclotomic polynomial. When $f$ factors into linear forms $C$ is called a line arrangement, and is usually denoted by ${\overline{{\mathcal{A}}}}$. Such curves have been intensively studied by mathematicians interested in the Alexander polynomial, as one hopes to relate the latter to the combinatorial structure of the line arrangement encoded in its intersection semilattice $L({\overline{{\mathcal{A}}}})$; in particular, it is natural to ask the following questions:

1.  Does $\Delta_{{\overline{{\mathcal{A}}}}}$ depend only on $L({\overline{{\mathcal{A}}}})$?

2.  Are there necessary or sufficient conditions in order to have $q(t)\neq 1$ that depend only on $L({\overline{{\mathcal{A}}}})$?

Regarding question 1., in [@PS Conjecture 1.9] Papadima and Suciu formulated 