The space of stability conditions for quivers with two vertices

The purpose of this article is to study the space of stability conditions $\text{stab}(P_n)$ on the bounded derived category ${\text{D}}^b(P_n)$ of finite dimensional representations of a quiver $P_n$ with two vertices and $n$ parallel arrows. There is a local homeomorphism $\text{z}:\text{stab}(P_n)\to {\text{C}}^2$. We show that, when the number of arrows is one or two, the map is a covering map if we restrict it to the complement of a line arrangement. When the number of arrows is greater than two we need to remove uncountably many lines to obtain a covering map.