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

Author

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  • Kristopher Williams

    (Department of Mathematics, Doane College, 1014 Boswell Ave, Crete, NE 68333, USA)

Abstract

Let C 1 and C 2 be algebraic plane curves in ℂ 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 π1( ℂ 2 C 1 U C 2 )) ≅ π1 ( ℂ 2 C 1 ) × π1 ( ℂ 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 ℂ 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.

Suggested Citation

  • Kristopher Williams, 2013. "A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements," Mathematics, MDPI, vol. 1(1), pages 1-15, March.
    • Handle: RePEc:gam:jmathe:v:1:y:2013:i:1:p:31-45:d:24295
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