Topological invariance of intersection lattices of arrangements in $\mathbf {CP}^2$

Abstract:

Let ${\mathcal {A}^{\ast }} = \{ {l_{1}},{l_{2}},...,{l_n}\}$ be a line arrangement in $\mathbb {C}{\mathbb {P}^2}$, i.e., a collection of distinct lines in $\mathbb {C}{\mathbb {P}^2}$. Let $L({\mathcal {A}^{\ast }})$ be the set of all intersections of elements of ${A^{\ast }}$ partially ordered by $X \leq Y \Leftrightarrow Y \subseteq X$. Let $M({\mathcal {A}^{\ast }})$ be $\mathbb {C}{\mathbb {P}^2} - \cup {\mathcal {A}^{\ast }}$ where $\cup {\mathcal {A}^{\ast }} = \cup \{ {l_i}:1 \leq i \leq n\}$. The central problem of the theory of arrangement of lines in $\mathbb {C}{\mathbb {P}^2}$ is the relationship between $M({\mathcal {A}^{\ast }})$ and $L({\mathcal {A}^{\ast }}).$