Distinguishing embedded curves in rational complex surfaces

We construct many pairs of smoothly embedded complex curves with the same genus and self-intersection number in the rational complex surfaces $\mathbb{C} P^{2}\# n\overline{\mathbb{C} P}^{2}$ with the property that no self-diffeomorphism of $\mathbb{C} P^{2} \# n \overline{\mathbb{C} P}^{2}$ sends one to the other. In particular, as a special case we answer a question originally posed by R. Gompf (1995) concerning genus two curves of self-intersection number 0 in $\mathbb{C} P^{2} \# 13\overline{\mathbb{C} P}^{2}$.