Example
Take target \(X = \mathbb P^2\) thought of as a toric variety with fan \(\Sigma\) generated by rays \(u_0 = (1,0),u_1 = (0,1),u_2 = (-1,-1)\). Let’s consider genus \(g = 0\) curves mapping by degree \(2\) with simple contact order, so \((g,d,n) = (0,2,6)\) and \(u = (u_0,u_0, u_1, u_1, u_2, u_2)\). Our moduli space is \(\mathcal M(X, D)\) of basic stable log maps to \(X\) with \(D\) the full toric divisor. The base stack is \(\mathfrak M = \mathfrak M(\mathcal A_X)\), the stack of basic prestable log maps to the Artin fan \(\mathcal A_X = [X/T]\) of \(X\).
The perfect obstrution theory of \(\mathcal M(X,D)\to \mathfrak M\) is \(Rp_*f^*T_X(-\log D)\):