Irreducibility of the $(-1)$-classes on smooth rational surfaces

We give a characterization for a $(-1)$-divisor $D$ on a smooth rational surface $X$ to be irreducible under the assumption that an anticanonical divisor $-K_X$ of $X$ is nef. Here $-K_X$ is nef means $K_X . C \leq 0$ for every effective divisor $C$ on $X$, and a $(-1)$-divisor $D$ is a divisor such that the two numerical conditions $D^2 =-1=D.K_X$ 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 $(-1)$-curves. A $(-1)$-curve is a smooth rational curve of self-intersection $-1$.