Suppose $X \subset \mathbb{CP}^n$ is a smooth projective curve. Why is $G/H$ a meromorphic function on $X$ if $G$ and $H$ are homogeneous polynomials of degree $d$ and $H$ does not vanish identically on $X$.
It's been a while since I did Riemann surfaces, so I forgot what kind of approach should be used. I was initially thinking that perhaps Bezout would be useful, but now I'm doubting that. Should one explicitly do something with the homogenous function $F(x,y,z)$ which defines $X$ (and then embeds it into $\mathbb{CP}^n$. Or perhaps I should do something with affine patches or charts and show that $G/H$ has no essential singularities. But then again, I'm not sure how to show that all singularities are either poles or removable.
I was also thinking about the fact that $G/H$ being meromorphic, means that it's holomorphic when extended to the Riemann sphere $C_{\infty}$ and not the constant $\infty$ function. The latter is clear, as $H$ is not identically zero on $X$, but then again I'd have no idea how to show that the map to $C_{\infty}$ would be holomorphic.
I used the search function here as well, but I didn't really find posts tackling the same problem, so I hope it's not a duplicate.