Arrangements in unitary and orthogonal geometry over finite fields

Let V be an n-dimensional vector space over F_q. Let Φ be a Hermitian form with respect to an automorphism σ with σ² = 1. If σ = 1 assume that q is odd. Let $\text{A}$ be the arrangement of hyperplanes of V which are non-isotropic with respect to Φ, and let L be the intersection lattice of $\text{A}$. We prove that the characteristic polynomial of L has n ? v roots 1, q,…, q^{n ? v? 1} where v is the Witt index of Φ.