Take the family of cubics through those 9 points. On the blow up they form a basepoint-free linear system that defines an elliptic fibration. The exceptional curves give sections, and you can use the group structure on the elliptic curves to translate these curves. Any translate of any exceptional curve is another $(-1)$-curve.

First consider the case when the $9$ points is the base locus of a pencil of cubics.

Take the family of cubics through these $9$ points. On the blow up they form a basepoint-free linear system that defines an elliptic fibration. The exceptional curves give sections, and you can use the group structure on the elliptic curves to translate these curves. Any translate of any exceptional curve is another $(-1)$-curve and clearly there are infinitely many of them.

Now consider a family of $\mathbb P^2$'s and move the above special $9$ points into general position and consider the blown-up family of surfaces. According to Kodaira exceptional subvarieties are stable, so all the nearby members of the (blown-up) family has to have infinitely many (−1)-curves.

This means that the statement is true in an open neighbourhood of the $9$ points in $\mathbb P^2$.

Take the family of cubics through those 9 points. On the blow up they form a basepoint-free linear system that defines an elliptic fibration. The exceptional curves give sections, so you have a relative group action. The exceptional curves generate a free group of rank 8. Any translate of any exceptional curve is another $(-1)$-curve.

Take the family of cubics through those 9 points. On the blow up they form a basepoint-free linear system that defines an elliptic fibration. The exceptional curves give sections, and you can use the group structure on the elliptic curves to translate these curves. Any translate of any exceptional curve is another $(-1)$-curve.