  We study a relation between roots of characteristic polynomials and
intersection points of line arrangements. Using these results, we obtain a lot
of applications for line arrangements. Namely, we give (i) a generalized
addition theorem for line arrangements, (ii) a generalization of
Faenzi-Vall\`{e}s' theorem over a field of arbitrary characteristic, (iii) a
partial result on the conjecture of Terao of line arrangements, and (iv) a new
sufficient condition for freeness over finite fields. Also, a higher
dimensional version of main results is considered.
