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

Kristopher Williams
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Kristopher Williams: Department of Mathematics, Doane College, 1014 Boswell Ave, Crete, NE 68333, USA

Mathematics, 2013, vol. 1, issue 1, 1-15

Abstract: Let C 1 and C 2 be algebraic plane curves in ? 2 such that the curves intersect in d 1 · d 2 points where d 1 , d 2 are the degrees of the curves respectively. Oka and Sakamoto proved that ?1( ? 2 \ C 1 U C 2 )) ? ?1 ( ? 2 \ C 1 ) × ?1 ( ? 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 ? 2 such that ?1 (M( A 1 U A 2 )) ? ?1 (M( A 1 )) × ?1 (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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2013
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