We construct many pairs of smoothly embedded complex curves with the same genus and self-intersection number in the rational complex surfaces C P 2 # n C P ¯ 2 \mathbb {C} P^{2}\# n\overline {\mathbb {C} P}^{2} with the property that no self-diffeomorphism of C P 2 # n C P ¯ 2 \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 C P 2 # 13 C P ¯ 2 \mathbb {C} P^{2} \# 13\overline {\mathbb {C} P}^{2} .