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Mathematics 2013, 1(1), 31-45; doi:10.3390/math1010031
Article

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 / Revised: 7 March 2013 / Accepted: 7 March 2013 / Published: 14 March 2013
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Abstract

Let C1 and C2 be algebraic plane curves in 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( 2 \ C1 U C2)) ≅ π1 ( 2 \  C1) × π1 ( 2 \  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 2 such that π1 (M(A1A2)) ≅  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 line arrangement; hyperplane arrangement; Oka and Sakamoto; direct product of groups; fundamental groups; algebraic curves
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).
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Williams, K. A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements. Mathematics 2013, 1, 31-45.

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