Let $L/K$ be a finite (I need only Galois, but I doubt this extra condition is really needed) extension where $K$ is of transcedence degree $1$ over $F$, where $F$ a number field.
Let $O_K$ be a Dedekind subring of $K$ and $O_L$ the integral closure of $O_K$ in $L$.
Is it true that only finitely many primes of $O_K$ are ramified in $O_L$ ?
Is it true if $K$ is ring of functions of some projective variety over $F$ ?