Definition
Let \(\mathcal A = \{H_1,...,H_k\}\) be a hyperplane arrangement in \(\mathbb P^N\), \(\mathcal L\) its intersection lattice (which is in particular a finite lattice) ordered by reverse inclusion and \(\mathcal G\) a buildilng set for \(\mathcal L\). The wonderful compactification of \(\mathcal A\) with respect to \(\mathcal G\) is a series of blowups of \(\mathbb P^N\). The order or these blowups is important but allows for some flexibility; choose an ordering \[\mathcal G = \{G_1, ..., G_\ell\}\] for \(\mathcal G\) so that whenever \(i \leq j\) \(G_i \subseteq G_{j}\) (confusingly, this means that \(G_i \geq G_j\) as lattice elements, since \(\mathcal L\) is ordered by reverse inclusion). Then define
- \(\overline Y_{\mathcal G, 0} = \mathbb P^N\)
- \(\overline Y_{\mathcal G, i} = \operatorname{Bl}_{\overline G_i}\overline Y_{\mathcal G, i-1}\) where \(\overline G_i\) is the strict transform of \(G_i\) in \(\overline Y_{\mathcal G, i-1}\)
- \(\overline Y_{\mathcal G} = \overline Y_{\mathcal G, \ell}\).
It is a (probably not too hard to verify) fact that any two choices of orderings of \(\mathcal G\) satisfying the above requirement produce isomorphic wonderful compactifications. Probably easy to see from the universal property of blow ups.