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Here's an example with $g=2$ and $n=5$. Consider the hyperelliptic curve defined by the equation $$y^2=x^5-1$$ or to be more precise the corresponding desingularized projective curve. Now this is a Riemann surface of genus two and has an automorphism $(x,y)\mapsto(\zeta x,y)$ where $\zeta$ is a primitive fifth root of unity.

Because of Hurwitz's theorem this kind of construction via Riemann surface automorphisms cannot work when $n$ is large compared to $g$.

Added

Here's another example. Consider the Riemann surface corresponding to the curve $$y^2=x^5-x.$$ It has an order $8$ autmorphism $(x,y)\mapsto(\eta^2 x,\eta y)$ where $\eta$ is a primitive eighth root of unity (and so this genus two surface also has an automorphism of order $4$).
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Here's an example with $g=2$ and $n=5$. Consider the hyperelliptic curve defined by the equation $$y^2=x^5-1$$ or to be more precise the corresponding desingularized projective curve. Now this is a Riemann surface of genus zero two and has an automorphism $(x,y)\mapsto(\zeta x,y)$ where $\zeta$ is a primitive fifth root of unity.

Because of Hurwitz's theorem this kind of construction via Riemann surface automorphisms cannot work when $n$ is large compared to $s$.g$.
show/hide this revision's text 1

Here's an example with $g=2$ and $n=5$. Consider the hyperelliptic curve defined by the equation $$y^2=x^5-1$$ or to be more precise the corresponding desingularized projective curve. Now this is a Riemann surface of genus zero and has an automorphism $(x,y)\mapsto(\zeta x,y)$ where $\zeta$ is a primitive fifth root of unity.

Because of Hurwitz's theorem this kind of construction via Riemann surface automorphisms cannot work when $n$ is large compared to $s$.