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For any nonnegative class ξ in H 2 (CP 2 #nCP 2 , Z), the minimal genus of smoothly embedded surfaces which represent ξ is given for n ≤ 9, and in some cases with n ≥ 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 ≤ 8 with g ≥ 1 and for n ≤ 9 with g ≥ 2.doi:10.1090/s0002-9947-99-02422-8 fatcat:oxfskpiagrhtzo5vcece7t3tmi