A converse to a theorem of Oka and Sakamoto for complex line arrangements

TL;DR

This paper proves a converse to a classical theorem by Oka and Sakamoto, showing that for line arrangements in the complex plane, a direct product of fundamental groups implies a specific intersection pattern.

Contribution

It establishes a new converse result for line arrangements, linking the fundamental group structure to the intersection multiplicity pattern.

Findings

Fundamental group of union is a direct product only when arrangements intersect in |A_1|*|A_2| double points.

Characterizes intersection patterns based on fundamental group decompositions.

Extends classical results to the setting of line arrangements in complex geometry.

Abstract

Let C_1 and C_2 be algebraic plane curves in the complex plane such that the curves intersect in d_1\cdot d_2 points where d_1,d_2 are the degrees of the curves respectively. Oka and Sakamoto proved that the fundamental group of the complement of C_1 \cup C_2 is isomorphic to the direct of product of the fundamental group of the complement of C_1 and the fundamental group of the complement of C_2. 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 complex plane such that the fundamental group of the complement of A_1 \cup A_2 is isomorphic to the direct product of the complements of the arrangements A_1 and A_2. Then, the intersection of A_1 and A_2 consists of |A_1| \cdot |A_2| points of multiplicity two.