Admissible local systems for a class of line arrangements

A rank one local system $\mathcal{L}$ on a smooth complex algebraic variety $M$ is admissible if roughly speaking the dimension of the cohomology groups $H^m(M,\mathcal{L})$ can be computed directly from the cohomology algebra $H^{\*}(M,\mathbb{C})$.

We say that a line arrangement $\mathcal{A}$ is of type $\mathcal{C}_k$ for some $k\ge 0$ if $k$ is the minimal number of lines in $\mathcal{A}$ containing all the points of multiplicity at least 3. We show that if $\mathcal{A}$ is a line arrangement in the classes $\mathcal{C}_k$ for $k\leq 2$, then any rank one local system $\mathcal{L}$ on the line arrangement complement $M$ is admissible. Partial results are obtained for the class $\mathcal{C}_3$.