A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements

Let C 1 and C 2 be algebraic plane curves in C 2 such that the curves intersect in d1 · d2 points where d1, d2 are the degrees of the curves respectively. Oka and Sakamoto proved that π1( C 2 \ C 1 U C 2)) ？？π1？( C 2 \？ C 1) × π1？( C 2 \？ C 2) [1]. In this paper we prove the converse of Oka and Sakamoto’s result for line arrangements. Let A 1 and A 2 be non-empty arrangements of lines in C 2 such that π1？(M( A 1 U？ A 2)) ？？ π1？(M( A 1)) ×？ π1？(M( A 2)) Then, the intersection of A 1 and A 2 consists of / A 1/ ·？ / A 2/ points of multiplicity two.