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.
Best Answer
Let me suggest a more geometric approach than the one you are using:
Let $C$ be your degree $d$ plane curve. Choose a point $P$ not on $C$, and not on the $x$-axis. Let $\mathbb P^1$ be the $x$-axis.
Now let $\phi:C \to \mathbb P^1$ be projection from $P$ to the $x$-axis, i.e. given any point $Q$ in $C$, draw a line from $P$ to $Q$, and see where it intersects the $x$-axis; that point is the value $\phi(Q)$.
This will be a map of degree $d$ (if you take a point $R$ in the $x$-axis, and count how many points lie in its preimage, you draw the line through $P$ and $R$, and count how many times this intersects $C$; the answer will be $d$ times, because $C$ is of degree $d$).
Now you can apply the Riemann--Hurwitz formula; the only problem is to work out the ramification points.
Ramification will occur when the line through $P$ and $Q$ has a multiple intersection with $C$ at $Q$, i.e. is tangent to $C$ at $Q$. If you choose everything generically (e.g. choose $P$ generically, and/or change coordinates so that $C$ is in a general position with regard to the $x$-axis), then these intersections multiplicities will never be more than $2$ (i.e. you will get tangencies, but never higher order tangencies), and so $e_{\phi}(Q)$ will be either $1$ (if the line through $P$ and $Q$ is not tangent to $C$ at $Q$) or $2$ (if the line through $P$ and $Q$ is tangent to $C$ at $Q$).
Now you have to figure out how many times a tangency occurs.
Of course, you can work this out by assuming the answer (i.e. that $g(C) = (d-1)(d-2)/2$, and working backwards). I suggest that you also write down an explicit conic, like $x^2 + y ^2 = 1$, and then perhaps a higher degree curve, and concretely apply the above procedure and see directly how many points of tangency occur. After doing all this, you will hopefully figure out in general how many $Q$s there are for which the line through $P$ is tangent to $C$, and also see how to prove that what you have worked out is correct.
To fill in all the details you will either need to make the "general position" argument above rigorous, or deal with the possibility of higher order tangency (i.e. the case when $e_{\phi}(Q) > 2$). But I would worry about this later, after you understand the basic geometry of the situation.