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Let $X\to Y$ be a finite morphism which is Galois in the category of smooth projective connected curves over $\mathbf{C}$.

Assume $g=g(X) \geq 2$.

Is the degree of $X\to Y$ bounded by $84(g-1)$?

I think the answer is yes. In fact, the degree of $X\to Y$ is the cardinality of $\# \mathrm{Aut}(X/Y)$. This is bounded from above by $\# \mathrm{Aut}(X)$ and this is known to be bounded from above by $84(g-1)$ (if $g>1$).

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You gave the right answer... Just be aware that this bound is true only in characteristic 0. – user18119 Mar 9 '12 at 9:08
So what changes if the base field is $\overline{ \mathbf{F}_p}$? – seporhau Mar 9 '12 at 10:59
In positive characteristic, the order of Aut(X) is bounded by $16g^4$ with a few exceptions (see works of Stichtenoth and of Singh in 1974 - 1975). – user18119 Mar 9 '12 at 12:30

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