Next Article in Journal
Stability of Solutions to Evolution Problems
Previous Article in Journal
ρ — Adic Analogues of Ramanujan Type Formulas for 1/π

Article

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

Department of Mathematics, Doane College, 1014 Boswell Ave, Crete, NE 68333, USA
Mathematics 2013, 1(1), 31-45; https://doi.org/10.3390/math1010031
Received: 19 February 2013 / Revised: 7 March 2013 / Accepted: 7 March 2013 / Published: 14 March 2013
Let C1 and C2 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 π1(${ℂ}^{2}$ \ C1 U C2)) ≅ π1 (${ℂ}^{2}$ \ C1) × π1 (${ℂ}^{2}$ \ 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 ${ℂ}^{2}$ 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. View Full-Text
Show Figures

Figure 1

MDPI and ACS Style

Williams, K. A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements. Mathematics 2013, 1, 31-45. https://doi.org/10.3390/math1010031

AMA Style

Williams K. A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements. Mathematics. 2013; 1(1):31-45. https://doi.org/10.3390/math1010031

Chicago/Turabian Style

Williams, Kristopher. 2013. "A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements" Mathematics 1, no. 1: 31-45. https://doi.org/10.3390/math1010031

Find Other Styles

### Article Access Map by Country/Region

1
Only visits after 24 November 2015 are recorded.