I am trying to find a (smooth) compact complex surface $X$ so that the set of irreducible curves $C$ on $X$ for which $C.C<0$ is infinite. Do any of you know of an example. Thanks. 



Compact complex surfaces having infinitely many negative curves?
Blow up $\mathbb P^2$ at 9 points. See e.g. Hartshorne exercise 5.4.15e) and the reference there.


