Let $S$ be a complex surface of general type. Are there infinitely many smooth rational curves on $S$? And more general, what if $V$ is a variety of general type?



Rational curves on varieties of general type
I think that the best result in this direction is the following theorem due to Bogomolov:



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> **Theorem.** Let $S$ be a surface of general type with $c\_1^2(S) > c\_2(S)$. Then for any $g$ the curves of geometric genus $g$ on $S$ form a bounded family. 
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In particular, since a surface of general type cannot be covered by rational or elliptic curves, these curves cannot deform. So Bogomolov's result implies that if $c\_1^2(S) > c\_2(S)$ then $S$ contains only finitely many rational or elliptic curves.


In general, it is conjectured than rational curves are never Zariski dense on a variety $V$ of general type, and more precisely it is expected that they are contained in a proper subvariety (hyperbolicity conjecture).


If $\dim V \geq 3$, you can obviously have infinitely many of them: for instance, take $V= S\times C$, where $S$ is a surface of general type containing a smooth rational curve and $C$ is a curve of genus at least $2$. 


