  Let $\Cal C,\Cal C'$ be curves over a base scheme $S$ with $g(\Cal C)\ge 2$.
Then the functor $T\mapsto\{$generically smooth $T$-morphisms $T\times_S\Cal
C'\to T\times_S\Cal C\}$ from $((S$-schemes)) to ((sets)) is represented by a
quasi-finite unramified $S$-scheme. From this one can deduce that for any two
integers $g\ge 2$ and $g'$, there is an integer $M(g,g')$ such that for any two
curves $C,C'$ over any field $k$ with $g(C)=g$, $g(C')=g'$, there are at most
$M(g,g')$ separable $k$-morphisms $C'\to C$. It is conjectured that the
arithmetic function $M(g,g')$ is bounded by a linear function of $g'$.
