  In this article, we study algebraic properties of binomial edge ideals of
Levi graphs associated with certain plane curve arrangements. Using
combinatorial properties of Levi graphs, we discuss the Cohen-Macaulayness of
binomial edge ideals of Levi graphs associated to some curve arrangements in
the complex projective plane, like the $d$-arrangement of curves and the
conic-line arrangements. We also discuss the existence of certain induced
cycles in the Levi graphs of these arrangements and obtain lower bounds for the
regularity of powers of the corresponding binomial edge ideals.
