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

# Abstract

Let C_{1}and C

_{2}be algebraic plane curves in ${\u2102}^{2}$ such that the curves intersect in

*d*points where

_{1}· d_{2}*d*are the degrees of the curves respectively. Oka and Sakamoto proved that π1(${\u2102}^{2}$ \ C

_{1}, d_{2}_{1}U C

_{2})) ≅ π1 (${\u2102}^{2}$ \ C

_{1}) × π1 (${\u2102}^{2}$ \ 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 ${\u2102}^{2}$ 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.

*Keywords:*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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