A nonhausdorff model for the complement of a complexified hyperplane arrangement

Given a hyperplane arrangement A in a real vector space V , we introduce a real algebraic prevariety Z(A), and exhibit the complement ofA in the complexification of V as the total space of an affine bundle over Z(A) with fibers modeled on the dual vector space V . 1 Statement Let V be a real vector space of dimension d, and let f1, . . . , fn be a collection of nonconstant affine linear functions on V such that the associated linear forms span the dual vector space V . Let A denote the collection of affine hyperplanes H1, . . . , Hn ⊆ V , where Hj = f −1 j (0) for all j ∈ {1, . . . , n}. Let M(A) := V C \ n