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A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements
Department of Mathematics, Doane College, 1014 Boswell Ave, Crete, NE 68333, USA
Received: 19 February 2013; in revised form: 7 March 2013 / Accepted: 7 March 2013 / Published: 14 March 2013
Abstract: Let C1 and C2 be algebraic plane curves in 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( \ C1 U C2)) ≅ π1 ( \ C1) × π1 ( \ C2) . 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 such that π1 (M(A1 U A2)) ≅ π1 (M(A1)) × π1 (M(A2)) Then, the intersection of A1 and A2 consists of /A1/ · /A2/ points of multiplicity two.
Keywords: line arrangement; hyperplane arrangement; Oka and Sakamoto; direct product of groups; fundamental groups; algebraic curves
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Williams, K. A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements. Mathematics 2013, 1, 31-45.
Williams K. A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements. Mathematics. 2013; 1(1):31-45.
Williams, Kristopher. 2013. "A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements." Mathematics 1, no. 1: 31-45.