Representing nonnegative homology classes of ${\mathbb C}P^2\#n{\overline{{\mathbb C}P}}{}^2$ by minimal genus smooth embeddings

For any nonnegative class $\xi$ in $H_2({\mathbb C}P^2\#n{\overline{{\mathbb C}P}}{}^2, {\mathbf Z})$, the minimal genus of smoothly embedded surfaces which represent $\xi$ is given for $n\leq 9$, and in some cases with $n\geq 10$, the minimal genus is also given. For the finiteness of orbits under diffeomorphisms with minimal genus $g$, we prove that it is true for $n\leq 8$ with $g\geq 1$ and for $n\leq 9$ with $g\geq 2$.