A converse to a theorem of Oka and Sakamoto for complex line arrangements

Kristopher Williams

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arXiv · 1208.2898

A converse to a theorem of Oka and Sakamoto for complex line arrangements

Kristopher Williams

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Abstract

Let C_1 and C_2 be algebraic plane curves in the complex plane such that the curves intersect in d_1\cdot d_2 points where d_1,d_2 are the degrees of the curves respectively. Oka and Sakamoto proved that the fundamental group of the complement of C_1 \cup C_2 is isomorphic to the direct of product of the fundamental group of the complement of C_1 and the fundamental group of the complement of C_2. 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 complex plane such that the fundamental group of the complement of A_1 \cup A_2 is isomorphic to the direct product of the complements of the arrangements A_1 and A_2. Then, the intersection of A_1 and A_2 consists of |A_1| \cdot |A_2| points of multiplicity two.

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Kristopher Williams. 2012-08-14. A converse to a theorem of Oka and Sakamoto for complex line arrangements. https://arxiv.org/abs/1208.2898

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