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?
I think that the best result in this direction is the following result of Bogomolov:
Theorem Let $S$ be a surface of general type with $c_1^2(S) > c_2(S)$. Thern for any $g$ the curves of geometric genus $g$ on $S$ 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 $c_1^2(S) > c_2(S)$ 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 $2$.
|show 6 more comments|
The conjecture Francesco is referring to as the "hyperbolicity conjecture" is actually the Green-Griffiths-Lang conjecture. It states that on any given smooth projective manifold of general type $X$ there should exist a proper subvariety $Y\subsetneq X$ such that for all non-constant holomorphic map $f\colon\mathbb C\to X$ one has $f(\mathbb C)\subset Y$.
This would imply in particular that the same proper subvariety should contain every rational or elliptic curve (or, more generally, every image of a complex torus).
Beside the result of Bogomolov (which treats "only" the algebraic part of this conjecture) one should cite also Mc Quillan's theorem, which prove this conjecture for surfaces under the same assumption on the second Segre number $c_1^2-c_2>0$.
This condition is a technical hypothesis which guarantees the existence of an algebraic (multi)foliation on the surface. The core of Mc Quillan's proof is then in showing that an algebraic (multi)foliation on a surface of general type does not admit any dense parabolic leaf.
In higher dimensions, very little is known even for the algebraic part of the conjecture, apart from the case of generic complete intersections of high (multi)degree (Clemens, Ein, Voisin, Pacienza...).
To finish with, thanks to very recent results of Cano about resolution of singularities of holomorphic foliations by curves on threefolds, probably Mc Quillan will be soon able to improve his previous result with the weaker assumption $13c_1^2-9c_2>0$.