A non-Hausdorff model for the complement of a complexified hyperplane arrangement

Given a hyperplane arrangement $\mathcal{A}$ in a real vector space $V$, we introduce a real algebraic prevariety $\mathcal{Z}(\mathcal{A})$, and exhibit the complement of $\mathcal{A}$ in the complexification of $V$ as the total space of an affine bundle over $\mathcal{Z}(\mathcal{A})$ with fibers modeled on the dual vector space $V^{\vee}$.