- 0-geometry: Curves
- 0-geometry: Genus
- 0-geometry: Hurwitz
- ABC-theorem for Curves
The genus formula of Bernhard Riemann (left) and Adolf Hurwitz (right) asserts that if $\phi: C_1 \rightarrow C_2$ is a separable cover of curves, we have an inequality relating their genera:
$$2 g_{C_1} – 2 \geq \deg(\phi)(2 g_{C_2} -2) + \sum_{P \in C_1} (e_{\phi}(P)-1)$$
Let’s first understand all these terms. The cover $\phi: C_1 \rightarrow C_2$ is separable if the induced field-extension $\phi^{\ast}(\overline{k}(C_2)) \subset \overline{k}(C_1)$ is finite and separable. The dimension $[\overline{k}(C_1) : \phi^{\ast}(\overline{k}(C_2))]$ is called the degree of $\phi$. By using a discriminant argument as in these notes we know that for all but finitely many points $Q \in C_2$ there are exactly $\deg(\phi)$ points of $C_1$ lying over it.
In general, let $P \in C_1$ with corresponding discrete valuation ring $\mathcal{O}_P$ in $\overline{k}(C_1)$, then $\mathcal{O}_P \cap \phi^*(\overline{k}(C_2))$ is a discrete valuation ring in $\overline{k}(C_2) \simeq \phi^*(\overline{k}(C_2))$ and thus of the form $\mathcal{O}_Q$ for some $Q \in C_2$. Naturally we have $\phi(P)=Q$.
If $R$ is the integral closure of $\mathcal{O}_Q$ in $\overline{k}(C_1)$, then $R$ is a semi-local Dedekind domain and a PID. If $t_Q$ is a uniformizer of $\mathcal{O}_Q$ we have
$$(t_Q) = P_1^{e_1} \cdots P_r^{e_r}$$
where the $P_i$ are the maximal ideals of $R$ which corresponds to points $P_i \in C_1$. The integer $e_i$ is called the ramification index of $\phi$ in $P_i$ and will be denoted $e_{\phi}(P_i)$. Clearly we have that $\deg(\phi) = \sum_i e_{\phi}(P_i)$.
Further, for almost all $P \in C_1$ we will have $e_{\phi}(P)=1$. With these notations we can now begin the
Proof of the Riemann-Hurwitz inequality: Because $\phi$ is separable, we have an inclusion
$$\phi^*~:~\Omega_{C_2} \hookrightarrow \Omega_{C_1} \qquad \phi^*(f\,\mathrm{d}x)=\phi^*(f)\,\mathrm{d} \phi^*(x)$$
Take a point $Q \in C_2$ with uniformizer $t_Q \in \mathcal{O}_Q$ and write $\omega = f d t_Q \in \Omega_{C_2}$. For the finitely many $P_i \in C_1$ lying over $Q$ we have (as before) that
$\phi^*(t_Q) = u t_{P_i}^{e_i}$ with $e_i = e_{\phi}(P_i)$ and $u$ a unit in the discrete valuation ring $\mathcal{O}_{P_i}$. But then,
$$\phi^*(\omega) = \phi^*(f)\,\mathrm{d} \phi^*(t_Q) = \phi^*(f)\,\mathrm{d}(u t_{P_i}^{e_i}) = \phi^*(f)\left(e_i u t_{P_i}^{e_i-1} + \frac{\mathrm{d}u}{\mathrm{d}t_{P_i}} t_{P_i}^{e_i}\right)\,\mathrm{d}t_{P_i}$$
The valuation $\operatorname{ord}_{P_i}$ of $e_i u t_{P_i}^{e_i-1}$ is $e_i-1$ (unless $e_i=0$ in $k$, that is $char(k) | e_i$) whereas the valuation of $(\tfrac{\mathrm{d}u}{\mathrm{d}t_{P_i}}) t_{P_i}^{e_i}$ is $\geq e_i$. But then,
$\operatorname{ord}_{P_i}(\phi^* \omega) \geq \operatorname{ord}_{P_i}(\phi^*(f)) + e_i -1 = \operatorname{ord}_Q(f) e_i + e_i – 1$
$= \operatorname{ord}_Q(\omega) e_{\phi}(P_i) + e_{\phi}(P_i) -1$
Summing these inequalities over all $P \in C_1$ we get for the degree of the divisor
$\deg(\operatorname{div}(\phi^*(\omega))) \geq \sum_{P \in C_1} (e_{\phi}(P) \operatorname{ord}_{\phi(P)}(\omega)+e_{\phi}(P) -1)$
$=\sum_{Q \in C_2} (\sum_{P \in \phi^{-1}(Q)} (e_{\phi}(P) \operatorname{ord}_Q(\omega) + e_{\phi}(P) -1))$
and because we already know that $\deg(\phi) = \sum_{P \in \phi^{-1}(Q)} e_{\phi}(P)$ for all $Q \in C_2$, this is equal to
$=(\sum_{Q \in C_2} \deg(\phi) \operatorname{ord}_Q(\omega))+(\sum_{P \in C_1} (e_{\phi}(P)-1))$
$=(\deg(\phi))(\deg(\operatorname{div}(\omega))) + \sum_{P \in C_1} (e_{\phi}(P)-1)$
Plugging in the relation between the genus and the degree of the divisor of a nonzero differential form, we have here the Riemann-Hurwitz inequality!
