On the homotopy type of the complement of an arrangement that is a 2-generic section of the parallel connection of an arrangement and a pencil of lines

Abstract

Let $\mathcal{A}$ be a complexified-real arrangement of lines in ${\mathbb{C}}^2.$ Let $H$ be any line in $\mathcal{A}$. Then, form a new complexified-real arrangement ${\mathcal{B}}_H=\mathcal{A}\cup \mathcal{C}$ where $\mathcal{C}\cup \{H\}$ is a pencil of lines with multiplicity $m\ge 3$, the intersection point in the pencil is not a multiple point in $\mathcal{A},$ and every line in $\mathcal{C}$ intersects every line in $\mathcal{A}\setminus \{H\}$ in points of multiplicity two. In this article, we show that for $H_1,H_2\in \mathcal{A}$ we may have that ${\mathcal{B}}_{H_1}$ and ${\mathcal{B}}_{H_2}$ do not have diffeomorphic complements, but the complements of the arrangements will always be homotopy equivalent.

13 pages, 5 figures