Representing nonnegative homology classes of ℂℙ²#𝕟\overline{ℂℙ}² by minimal genus smooth embeddings

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