A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements
نویسنده
چکیده
Let C1 and C2 be algebraic plane curves in C 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 \ C1 ∪ C2)) ∼= π1(C \ C1)× π1(C \ C2) [1]. In this paper we prove the converse of Oka and Sakamoto’s result for line arrangements. Let A1 and A2 be non-empty arrangements of lines in C such that π1(M(A1 ∪ A2)) ∼= π1(M(A1))× π1(M(A2)) Then, the intersection of A1 and A2 consists of |A1| · |A2| points of multiplicity two.
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