Exceptional curves on smooth rational surfaces with -K not nef and of self-intersection zero

We prove that a smooth rational surface X defined over the field of complex numbers having an anti-canonical divisor not nef and of self-intersection zero has a finite number of (-1)-curves. A (-1)-curve is a smooth rational curve of self-interaction -1. By giving an example, we also show that X may have no (-2)-curves, a (-2)-curve is a smooth rational curve of self-intersection -2. (author)