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A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements

Author(s): Kristopher Williams

Journal: Mathematics
ISSN 2227-7390

Volume: 1;
Issue: 1;
Start page: 31;
Date: 2013;
Original page

Keywords: line arrangement | hyperplane arrangement | Oka and Sakamoto | direct product of groups | fundamental groups | algebraic curves

Let C1 and C2 be algebraic plane curves in C2 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(C2 C1 U C2)) ≅ π1 (C2   C1) × π1 (C2   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 C2 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.
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