Before revision:  The <i>k-level<i> of {\cal A } consists of all intersection points v of lines in A which have exactly k lines of {\cal A } passing below v. The complexity  of the k-level in a line arrangement has been widely studied.
After revision:  The <i>k-level<i> of {\cal A } consists of all intersection points v of lines in A which have exactly k lines of {\cal A } passing below v. The complexity (the maximum size) of the k-level in a line arrangement has been widely studied.