The supersolvable Sylvester–Gallai conjecture over the complex numbers

Let ${\mathbb K}={\mathbb C}$, and let ${\mathcal A}$ be a supersolvable projective line arrangement. Write $n_2({\mathcal A})$ for the number of its double points.

Supersolvable Sylvester–Gallai conjecture. One has

$$n_2({\mathcal A})>0.$$

This is the supersolvable version of the Sylvester–Gallai theorem and is weaker than the preceding double-point lower-bound conjecture. Since the stronger lower bound is established in the paper, this conjecture is also solved.