I have been trying to do an exercise on Rick Miranda's book on Riemann surfaces, and the way i am solving the exercise will not make sense in the end but i dont know whats wrong.
So let $X$ be a smooth projective plane curve defined by a homogeneous polynomial $F(x,y,z)=0$. Define $f(u,v)=F(u,v,1)$ be the associated smooth affine plane curve. Show that $du$ and $dv$ define meromorphic $1$-forms on all of $X$.
So we know that we have 6 coordinates charts, and i will write some here wich are the ones that are giving me something that doenst make sense , $\phi_1^{-1}(x)=[x:h(x):1]$, $\phi_3^{-1}(y)=[1:y:g(y)]$. So we know that in local coordinates when we do the pull back we have that $w_1=dx$ and $w_3=0$. But since we should have that $\phi_1\circ \phi_3^{-1}$ should be a transition map we should have $w_3 = -\frac{g'(y)}{g(y)^2}dy$, this makes more sense also because he says that the form will be meromorphic and in my case i think i would have a holomorphic form, so something might be wrong because i think otherwise he would ask specifically to show its holomorphic. So why does my pullback of $w_3=0$, maybe i am interpreting what $du$ does the wrong way but i think that it basically just wants the $x$-coordinte ?
I guess i am confused what he means by the differential form $du$, is it already defined on one chart and im a supposed to see what it is on all the other charts, or is it just a general thing for all of $X$ and now i am supposed to see the local representation on all charts?
Any help is appreciated , Thanks in advance.