Let $X$ be a complex projective surface of general type, $K\_X$ be the very ample canonical divisor (which is automatically minimal). Is there an example for such a $X$ that there exist a sequence of negative curves $\{C\_i\}\_{i=1}^{\infty}$ with $C\_i^2$ being fixed, but the geometric genus $g(C\_i)$ going to infinity?



Negative curves on surface of general type
Yes. Start with a rational surface $S$ containing infinitely many $(-1)$-curves $E\_n$ (for instance $\mathbb{P}^2$ blown up along 9 general points). Choose a very ample divisor $H$ on $S$, a smooth curve $B$ in the linear system $|2H|$, and consider the double covering $\pi : X\rightarrow S$ branched along $B$. Then $K\_X\cdot \pi ^\*E\_n=2(K\_S+H)\cdot E\_n=-2+2H\cdot E\_n\ $ goes to infinity with $n$, and so does $g(\pi ^\*E\_n)$. 
Note that if $\pi ^\*E\_n$ is not irreducible, it is the union of two smooth rational curves, and $X$ contains only finitely many such curves (Lu-Miyaoka); so $\pi ^\*E\_n$ is irreducible for all but finitely many $n$.


