a consequence of a more general result concerning [*nice*]{} arrangements of lines in ${\mathbb{P}}^2$ (see [@CDP Theorem 1.6]).

\[rem:more\] It is easy to see that $l({{\mathcal A}})=r$, for a split solvable arrangement of type $m=(m_1, \dots, m_r)$. By Theorem \[thm:lr\], ${{\mathcal A}}$ is not supersolvable, if $r>3$. Clearly, ${{\mathcal A}}$ cannot be generic, if there is some $i$ such that $m_i>1$.

Cohomology jumping loci {#sect:VR}
=======================

Let $M$ be a connected finite type CW-complex with torsion free first homology group $H_1(M):=H_1(M, {\mathbb{Z}})$, for instance the complement of a complex hyperplane arrangement. The description of its [*characteristic varieties*]{} $$\label{eq:charvar}
{{\mathcal V}}^q_k(M, \Bbbk):= \{ \rho \in \operatorname{Hom}(\pi_1(M), \Bbbk^*) \;|\;\dim_{\Bbbk} H^q(M,\;_{\rho}\Bbbk) \geq k\},$$ for an arbitrary field $\Bbbk$, is equivalent to the computation of twisted, rank one, $\Bbbk$-cohomology of $M$.

An analogous concept was introduced by Falk ([@Fa3]), under the name of [*resonance varieties*]{}: $$\label{res}
{{\mathcal R}}_{k}^{q}(M, \Bbbk) :=\{ \omega \in H^1(M, \Bbbk) \;|\; \dim_{\Bbbk}H^q(H^*(M,\Bbbk), \mu_{\omega})\geq k\},$$ $\mu_{\omega}$ being left-multiplication by $\omega$ in the algebra $H^*(M,\Bbbk)$. (Here, one uses the assumption on $H_1(M)$ to check that the [*Aomoto complex*]{}, $(H^*(M,\Bbbk), \mu_{\omega})$, is indeed a chain complex, that is, $\omega \cdot \omega =0$.)

\[varieties\] The sets ${{\mathcal V}}^q_k(M, \Bbbk)$ are closed algebraic subvarieties of 