Let $X$ be a compact connected Riemann surface of genus at least two, or let $X$ be a smooth projective connected curve over an algebraically closed field of characateristic zero.
Then Hurwitz proved that the number of automorphisms of $X$ is at most $84(g-1)$.
I would like to know why Aut$(X)$ is finite (without appealing to Hurwitz' result) by an "elementary" argument.
That is, why should the automorphism group of $X$ be finite?
Once I have such an elementary argument, I believe it should be applicable to certain higher-dimensional varieties such as varieties with ample canonical sheaf. Why should they have only finitely many automorphisms?