Definition
Fix a primitive \(r\)th root of unity \(\zeta\). The \(r\)-braid arrangement in \(\mathbb A^n\) (or \(\mathbb P^{n-1}\)) is the collection of hyperplanes in \(\mathbb A^n\) defined in coordinates by \[H_{ij}^k = \{(x_1,...,x_n) ~ \mid ~ x_i = \zeta^kx_j\}\subset \mathbb A^n\] for \(1\leq i \leq j \leq n\) and \(0 \leq k \leq r - 1\). When \(i = j\), we require \(k\neq 0\) (since then \(H^0_{ii}\) as defined above is the entire ambient space and not a hyperplane) in which case \[H^k_{ii} = \{(x_1,...,x_n) ~ \mid ~ x_i = 0\}\] is the \(i\)th coordinate hyperplane. The braid arrangement is the arrangement for \(r = 1\).
We typically project this arrangement down to \(\mathbb P^{n-1}\).