Irreducibility of the -classes on smooth rational surfaces

We give a characterization for a -divisor on a smooth rational surface to be irreducible under the assumption that an anticanonical divisor of is nef. Here is nef means for every effective divisor on , and a -divisor is a divisor such that the two numerical conditions hold.

As an application we give explicit examples of blowing up the projective plane at nine points infinitely near such that the obtained surface has an infinite number of -curves. A -curve is a smooth rational curve of self-intersection .