On complex supersolvable line arrangements

We show that the number of lines in an $m$--homogeneous supersolvable line arrangement is upper bounded by $3m-3$ and we classify the $m$--homogeneous supersolvable line arrangements with two modular points up-to lattice-isotopy. A lower bound for the number of double points $n_2$ in an $m$--homogeneous supersolvable line arrangement of $d$ lines is also considered. When $3 leq m leq 5$, or when $m geq frac{d}{2}$, or when there are at least two modular points, we show that $n_2 geq frac{d}{2}$, as conjectured by B. Anzis and S. O. Tohu aneanu. This conjecture is shown to hold also for supersolvable line arrangements obtained as cones over generic line arrangements, or cones over arbitrary line arrangements having a generic vertex.