Topological invariance of intersection lattices of arrangements in CP^2

Abstract

Let $\scr A^*=\{l_1,l_2,\cdots,l_n\}$ be a line arrangement in $\Bbb{CP}^2$, i.e., a collection of distinct lines in $\Bbb{CP}^2$. Let $L(\scr A^*)$ be the set of all intersections of elements of $A^*$ partially ordered by $X\leq Y\Leftrightarrow Y\subseteq X$. Let $M(\scr A^*)$ be $\Bbb{CP}^2-\bigcup\scr A^*$ where $\bigcup\scr A^*= \bigcup\{l_i\colon\ 1\leq i\leq n\}$. The central problem of the theory of arrangement of lines in $\Bbb{CP}^2$ is the relationship between $M(\scr A^*)$ and $L(\scr A^*)$.