Abstract:
 A conjecture of Lang predicts that a variety of general type $X$
contains a proper closed
algebraic subset $Z$ such that all the rational curves and elliptic
curves on $X$ are
necessarily contained in $Z$.
Though Kodaira proved a weaker result to the effect that
the union of all the rational/elliptic curves on $X$ has Lebesgue
measure zero,
the conjecture is still wide open, even in the case where $X$ is a surface.
We discuss this problem and prove that, if $X$ satisfies a certain
topological condition, we can effectively bound the number of
rational/elliptic curves
on $X$.
