In this case, $M_{{{\mathcal A}}}= ({\mathbb{C}}^*)^l$.

A basic combinatorial invariant of an arrangement is the [*rank*]{}, denoted $r({{\mathcal A}})$, defined as the codimension of a maximal element of $\mathcal L({{\mathcal A}})$. It is well defined, as we can see from [@OT Lemma 2.4]. The levels $\mathcal L_p({{\mathcal A}}), \;0 \leq p \leq r({{\mathcal A}})$ consist of the elements of rank $p$ of the lattice $\mathcal L({{\mathcal A}})$. ${{\mathcal A}}$ is called [*essential*]{} if $r({{\mathcal A}})=l$. We can always assume that a central arrangement is essential, without changing the homotopy type of $M_{{{\mathcal A}}}$, eventually seeing it as an arrangement in the quotient of the ambient vector space $V$ by the [*center*]{} of the arrangement, $\cap_{H\in {{\mathcal A}}} H$.

\[essential\]

1.  \[boolean\] The [*boolean*]{} arrangement is an essential arrangement.

2.  \[converse\] The [*braid*]{} arrangement ${{\mathcal A}}\subset {\mathbb{C}}^l,\; Q({{\mathcal A}})=\Pi_{1 \leq i<j \leq l}(x_i-x_j)$ has rank $l-1$, hence is not an essential arrangement.

\[cone\] The [*cone*]{} over an arbitrary arrangement ${{\mathcal A}}\subset {\mathbb{C}}^l$ is the central arrangement $c{{\mathcal A}}\subset {\mathbb{C}}^{l+1}$ with the defining polynomial $Q(c{{\mathcal A}})=x_0 \overline{Q({{\mathcal A}})}$, where $\overline{Q({{\mathcal A}})} \in {\mathbb{C}}[x_0 \dots x_l]$ is the homogeneization of $Q({{\mathcal A}}) \in {\mathbb{C}}[x_1 \dots x_l]$. Conversely, we can apply a deconing procedure 