I have this curve $C =$ { $[X,Y,Z] \in \mathbb{P^2C} | Y^3Z = X^4 - X^2Z^2$}.
I've found that it's singular in $[0,0,0]$. So, I have to blow up this point. I don't know if $[0,0,1]$ is a base point (How to find one ?), but I've considered the equation for the affine curve related which is $Y^3 = X^4 - X^2$. Then for one chart of the blow-up I put $Y = ut$ and $X = t$ and I obtain $u^2(ut^3 - u^2 +1)$. I take $ut^3 - u^2 + 1 = 0$ which is $ut^3 -u^2w^2 + w^4$ in the projective space (Is this the curve without singularities I obtain blowing up the singular point ?)
If it is correct I've found a Riemann surface related to the curve $C$, on this surface I consider the meromorphic function $f = \dfrac{X}{Y}$. Is it possible that f has a zero in $[0,0,1]$ and a pole in $[0,1,0]$ both of order one ?
I can consider the holomorphic function $G : C{\setminus}[0,0,0] \longrightarrow \mathbb{P^1C}$ which sends the poles in $\infty$. I have then in $mult_{[1,0,0]}G =1 = mult_{[0,1,0]}G$.
This Riemann surface has genus 3 by Plücker's formula. So I have to verify Hurwitz's formula $$2g(C{\setminus}[0,0,0]) -2 = degG(2g(\mathbb{P^1} - 2) + \sum_p (mult_pG - 1).$$ So, I don't know how to compute the degree of G and if G has any branch points.
Sorry for the long question but I'm a little bit confused. Thank you.