I'm studying from Joseph Silverman's book The Arithmetic Of Elliptic Curves and I'm trying to do as many exercises as I can. Right now I'm trying to do Exercise 2.7 from chapter II which reads as follows.
Let $f(X, Y, Z) \in K[X, Y, Z]$ be a homogeneous polynomial of degree $d \geq 1$ and assume that the curve $C$ in $\mathbb{P}^2$ given by the equation $F = 0$ is nonsingular. Prove that $ \text{genus}(C) = \frac{(d-1)(d-2)}{2} $
The Exercise has a hint which tells me to define a map $C \rightarrow \mathbb{P}^1$ and use Theorem II.5.9 from that same chapter. For completeness I include the Theorem.
Theorem 5.9 (Hurwitz) Let $\phi : C_1 \rightarrow C_2$ be a nonconstant separable map of smooth curves of genera $g_1$ and $g_2$, respectively. Then $ 2g_1 - 2 \geq (\deg{\phi})(2g_2 - 2) + \sum_{P \in C_1} (e_\phi (P) - 1) $ Further, equality holds if and only if one of the following two conditions holds:
- $\text{char}(K) = 0$
- $\text{char}(K) = p > 0$ and $p$ does not divide $e_\phi (P)$ for all $P \in C_1$.
My attempts
I consider a map $\phi : C \rightarrow \mathbb{P}^1$, which has the form $\phi = [f, g]$ for some $f, g \in K[X, Y, Z]$ homogeneous of the same degree. Since $\mathbb{P}^1$ has genus $0$ the equality becomes $ 2 \text{genus}(C) - 2 = -2\deg{\phi} + \sum_{P \in C} (e_\phi (P) - 1) $
so that $ 2 \text{genus}(C) = 2 - 2\deg{\phi} + \sum_{P \in C} (e_\phi (P) - 1) $
and we have to prove that $2 - 2\deg{\phi} + \sum_{P \in C} (e_\phi (P) - 1) = (d-1)(d-2)$. I have also tried to use the formula from Proposition 2.6 in Chapter II of the book which for this particular case tells us that
$ \sum_{P \in \phi^{-1} (Q)} e_{\phi} (P) = \deg{\phi} $
for any point $Q \in \mathbb{P}^1$.
Then using this we get
$ 2 \text{genus}(C) = 2 - 2\sum_{P \in \phi^{-1} (Q)} e_{\phi} (P) + \sum_{P \in C} (e_\phi (P) - 1) $
for some point $Q \in \mathbb{P}^1$. Now the problem for me is that I'm not really sure how to relate the degree $d$ of the polynomial that defines the curve $C$ to those sums. I'm also wondering how it may be possible to compute those ramification indexes in the sums because I suspect that's were the problem lies.
If it is of some help, I know that we can identify any map $\phi : C \rightarrow \mathbb{P}^1$ with a function in $K(C)$ or with the constant map $\infty = [1, 0]$. I was thinking that maybe there's a way for me to get the degree $d$ to come into play here, but up until now I'm stuck and I run out of ideas.
Questions
So I would very much appreciate some hints and advice as to how to proceed with this exercise. I'm not looking for a full solution but for some advice and hints that may guide me in the right direction.
Sorry for the long post, I know that it will not get many people to read it but I'm trying to get the most out of the exercise.
Thank you very much for any help.