Example
Consider three hyperplanes \(H_1,H_2,H_3\subset \mathbb P^2\) meeting at a point \(\pt\), as in this example for instance. We compute the Chow groups of the wonderful compactification \(\overline Y_{\mathcal G}\), where \(\mathcal G = \{H_1,H_2,H_3, \pt\}\) is the building set of the intersection lattice associated to the arrangement.
The exact sequence in [Intersection Theory, Fulton, Proposition 6.7 (e)] makes short work of this calculation, as we only need to blow up \(\mathbb P^2\) once at \(\pt\) to obtain \(\overline Y_{\mathcal G}\). Denoting by \(E\) the exceptional divisor, we have a fiber diagram
which gives us an exact sequence \[0 \to A_k(\pt) \xrightarrow{\alpha} A_k E \oplus A_k\mathbb P^2\xrightarrow{\beta} \overline Y_{\mathcal G} \to 0\] for each \(k\) with \[\alpha(x) = (c_{d-1}(\mathcal E) \cap g^*x, -i_*x)\] and \[\beta(x,y) = j_*x + f^*y.\] Here, \(\mathcal E = g^*(N_{\pt/\mathbb P^2})/N_{E/\overline Y_{\mathcal G}} = g^*T_{\pt}\mathbb P^2 / \mathcal O_E(-1)\) – it is the universal quotient bundle on \(E\). Since \(E \cong \mathbb P^1\) and all but the Chow ring of \(\overline Y_{\mathcal G}\) is well understood, we can compute without actually using the maps \(\alpha\) and \(\beta\).
- For \(k = 0\), we get \(A_0\overline Y_{\mathcal G} \cong \mathbb Z^{\oplus 2}/\mathbb Z \cong \mathbb Z.\)
- For \(k = 1\), the left term in the exact sequence disappears and we get \(A_1\overline Y_{\mathcal G} \cong \mathbb Z[E] \oplus \mathbb Z[\ell]\) where \(\ell\) is a line in \(\mathbb P^2 \setminus \{\pt\}\).
- For \(k = 2\), the Chow group of \(E\) disappears and we get \(A_2\overline Y_{\mathcal G} \cong A_2\mathbb P^2 \cong \mathbb Z\).