{{\mathcal A}}$ increases the rank with at most $1$, that is $r({{\mathcal A}}) \leq r({{\mathcal B}})+1$. Accordingly, singular ($r({{\mathcal A}}) = r({{\mathcal B}})$) or nonsingular ($r({{\mathcal A}}) =r({{\mathcal B}})+1$) extensions appear. Singular extensions may be deformed, by constructing an arrangement $\tilde{{{\mathcal A}}} \subset V \times {\mathbb{C}},\; r(\tilde{{{\mathcal A}}})= r({{\mathcal B}})+1$, with the same fundamental group of the complement as ${{\mathcal A}}$ and the same collinearity relations. In the end this process produces for an arbitrary hypersolvable ${{\mathcal A}}$ a fiber-type deformation $\tilde{{{\mathcal A}}} \subset V \times {\mathbb{C}}^s$, where $s$ is the number of singular extensions in a composition series of ${{\mathcal A}}$ ([@JP2]).

Another large family of examples of hypersolvability is described next. [*Split solvable*]{} arrangements are line arrangements ${{\mathcal A}}=\{H_0, \dots,H_n\} \subset {\mathbb{C}}{\mathbb{P}}^2$ with a simple intuitive combinatorics. Take $H_0$ to be the line at infinity. Let $p_1, \dots, p_r$ be the intersection points on $H_0$, with multiplicities $m_1+1, \dots, m_r+1$. By definition, ${{\mathcal A}}$ is split solvable if it has only double intersection points outside $H_0$. Equivalently, the affine arrangement in ${\mathbb{C}}{\mathbb{P}}^2\setminus H_0 \cong {\mathbb{C}}^2$ can be pictured as $n$ lines with $r$ parallel directions, each direction containing $m_i$ lines, $1 \leq i \leq r$, and $d{{\mathcal A}}$ 