I was reading the arithmetic of elliptic curves by Silverman and I found problems with the following exercise:
Let $F(x,y,z)$ an homogeneus polynomial of degree $d$. Let $C$ be a curve in $\mathbb{P}^2$ given by the equation $F=0$ which is non singular. Prove that its genus is $\frac{(d-1)(d-2)}{2}$
The idea of the exercise is to define a map $\phi:C\to\mathbb{P}^1$, for example $\phi([x:y:z])=[x:y]$ and then use Hurwit's formula. This gives us: $$2g(C)-2=-2\deg(\phi)+\sum_{P\in C}(e_\phi(P)-1)$$ And from here I have two things to prove:
- $\deg(\phi)=d$
- $\sum_{P\in C}(e_\phi(P)-1)=d(d-1)$
I've seen this post Proving that the genus of a nonsingular plane curve is $\frac{(d-1)(d-2)}{2}$, but this is very geometric, and I've seen books that proves this with Pluker's formula, but this looks very sophisiticated. Since I am studying a particular book, I would like to have a solution just involving results of the book, how can I prove those two things usind the definitions given in the book?
Many thanks.
Edit: Here is my attempt:
WLOG we can assume $[0:0:1]\notin C$ (if $[0:0:1]\in C$ we can take a change of coordinates) and define $\phi([x:y:z])=[x:y]$. Notice that $\forall~[x_0:y_0]\in\mathbb{P}^1$ we have: $$\phi^{-1}([x_0,y_0])=\lbrace [x_0:y_0:z]\mid F(x_0,y_0,z)=0\rbrace$$ So $p(z)=F(x_0,y_0,z)$ is a polynomial in $z$ of degree $d$. Since we are in an algebraic closed field $p(z)$ has $d$ roots (not necessarily different) and so $\deg(\phi)=d$.
To study the ramification points we need to study the points where $|\phi^{-1}([x_0:y_0])|<d$, that is, the points $[x:y]\in\mathbb{P}^1$ such that the system of equations $F(x,y,z)=0,\frac{\partial F}{\partial z}(x.y,z)=0$ has solutions. By Bezout's theorem since $\deg(F)=d$ and $\deg\left(\frac{\partial F}{\partial z}\right)=d-1$ if they do not have common components (i.e the multiplicity of the roots is at most $2$) they intersect in $d(d-1)$ points. Then there are $d(d-1)$ zeroes of order $2$ and thus there are $d(d-1)$ points with ramification index $2$, which implies $\sum_{P\in C}(e_\phi(P)-1)=d(d-1)$.
It left to study the cases where there are roots of multiplicity $>2$. But this is ok? My issue is that I have not used the fact that $F$ is non singular.
