This note studies the minimal genus problem for classes which are equivalent, via the geometric diffeomorphism group, to a simplified class in CP²#kCP². It is shown that the orbit structure for primitive classes is basically determined by the self-intersection number. Making use of this result, an upper bound for the minimal genus for each orbit is determined and it is shown that for k large enough, then genus stabilizes at either 0 or 1.

The author would like to thank Tian-Jun Li for useful suggestions to improve this paper.