to a central arrangement ${{\mathcal A}}\subset {\mathbb{C}}^{l+1}$ to obtain an affine arrangement $d{{\mathcal A}}\subset {\mathbb{C}}^l$. Choose coordinates such that ${{\mathcal A}}$ contains a hyperplane $H$ of equation $x_0=0$. Then $Q(d{{\mathcal A}})$ is the evaluation of $Q({{\mathcal A}})$ at $(1, x_1, \dots, x_l)$.

There is a topological reason behind this construction, emerging from projectivisation. The Hopf fibration $p: {\mathbb{C}}^{l+1} \setminus \{0\} \longrightarrow  {\mathbb{C}}{\mathbb{P}}^{l}$, with fiber ${\mathbb{C}}^*$, sends the complement $M_{c{{\mathcal A}}}$ to $M_{{{\mathcal A}}}$ and produces a decomposition $M_{c{{\mathcal A}}} \cong M_{{{\mathcal A}}} \times {\mathbb{C}}^*$, or, equivalently, for central ${{\mathcal A}}$, $M_{{{\mathcal A}}} \cong M_{d{{\mathcal A}}} \times {\mathbb{C}}^*$(see [@OT Proposition 5.1]). Here, we see the complement of an arbitrary arrangement ${{\mathcal A}}\subset {\mathbb{C}}^l$ as the complement of the projectivised arrangement ${\mathbb{P}}(c{{\mathcal A}}) \subset {\mathbb{C}}{\mathbb{P}}^l$: first remove the hyperplane at infinity, $H_0= \{ x_0=0\}$, then the rest of the hyperplanes from ${\mathbb{C}}{\mathbb{P}}^l \setminus H_0 \cong {\mathbb{C}}^l$.

\[modular\] Let ${{\mathcal A}}$ be central. An element $X \in \mathcal L({{\mathcal A}})$ is called [*modular*]{} if $X+Y \in \mathcal L({{\mathcal A}})$, for any $Y\in \mathcal L({{\mathcal A}})$. For other equivalent definitions see [@OT §2.1].

\[modular1\] All hyperplanes are modular elements. The center of the lattice is also modular. All elements in the lattice of the 