Admissible local systems for a class of line arrangements

A rank one local system $\text{LL}$ on a smooth complex algebraic variety $M$ is admissible roughly speaking if the dimension of the cohomology groups $H^m(M,\text{LL})$ can be computed directly from the cohomology algebra $H^{\ast }(M,\text{C})$. We say that a line arrangement $\text{A}$ is of type ${\text{CC}}_k$ if $k\ge 0$ is the minimal number of lines in $\text{A}$ containing all the points of multiplicity at least 3. We show that if $\text{A}$ is a line arrangement in the classes ${\text{CC}}_k$ for $k\le 2$, then any rank one local system $\text{LL}$ on the line arrangement complement $M$ is admissible. Partial results are obtained for the class ${\text{CC}}_3$.