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I think that the best result in this direction is the following result of Bogomolov:

Theorem Let $X$ S$be a surface of general type with$c_1^2(X) c_1^2(S) > c_2(X)$c_2(S)$. Thern for any $g$ the curves of geometric genus $g$ on $X$ S$form a bounded family. In particular, since a surface of general type cannot be covered by rational or elliptic curves, these curves cannot deform. So Bogomolov's result implies that if$c_1^2(X) c_1^2(S) > c_2(X)$c_2(S)$ then $X$ 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 X V \geq 3$, you can obviously have infinitely many of them: for instance, take$X= Y \times V= S\times C$, where$Y$S$ is a surface of general type containing a smooth rational curve and $C$ is a curve of genus at least $2$.

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I think that the best result in this direction is the following result of Bogomolov:

Theorem Let $X$ be a surface of general type with $c_1^2(X) > c_2(X)$. Thern for any $g$ the curves of geometric genus $g$ on $X$ form a bounded family.

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 $c_1^2(X) > c_2(X)$ then $X$ contains only finitely many rational or elliptic curves.

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

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