%0 Journal Article
%T A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements
%A Kristopher Williams
%J Mathematics
%D 2013
%I MDPI AG
%R 10.3390/math1010031
%X Let C 1 and C 2 be algebraic plane curves in C 2 such that the curves intersect in d1 ¡¤ d2 points where d1, d2 are the degrees of the curves respectively. Oka and Sakamoto proved that ¦Ð1( C 2 \ C 1 U C 2)) £¿£¿¦Ð1£¿( C 2 \£¿ C 1) ¡Á ¦Ð1£¿( 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 ¦Ð1£¿(M( A 1 U£¿ A 2)) £¿£¿ ¦Ð1£¿(M( A 1)) ¡Á£¿ ¦Ð1£¿(M( A 2)) Then, the intersection of A 1 and A 2 consists of / A 1/ ¡¤£¿ / A 2/ points of multiplicity two.
%K line arrangement
%K hyperplane arrangement
%K Oka and Sakamoto
%K direct product of groups
%K fundamental groups
%K algebraic curves
%U http://www.mdpi.com/2227-7390/1/1/31