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In problem numerated by 2.7. in Silverman's book "The Arithmetic of Elliptic Curves" is required to calculate genus of the nonsingluar curve $C$ given by equation $F=0$, where $F=F(x,y,z)$ is homogeneous polynomial of degree $d$. There is the suggestion to use Hurwitz formula and I am interested in solution which actually uses this formula.

I agree that we can suppose that point $[0,1,0]$ is not on the curve $C$, therefore we can define map $C\longrightarrow \mathbb{P}^1$ by $[x,y,z]\mapsto[x,z]$, but I cannot determine indices of ramification of points on the curve. I do not know the uniformizers at points on the curve. How can I determine these uniformizers since $F$ can be almost any homogeneous polynomial of degree $d$?

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One approach to working out the ramification is on pages 9-11 of these notes, where it is related to the degree of the dual curve.

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