IV.2: Hurwitz
Preface:
- Degree: for a finite morphism of curves \(X \xrightarrow{f} Y\), set \(\operatorname{det}(f) \coloneqq[k(X): k(Y)]\), the degree of the extension of function fields.
- Ramification indices \(e_p\): for \(p\in X\), let \(q=f(p)\) and \(t \in {\mathcal{O}}_q\) a local coordinate. Pull back to \(t\in {\mathcal{O}}_p\) via \(f^\sharp\) and define \(e_p \coloneqq v_p(t)\) using the valuation \(v_p\) for the DVR \({\mathcal{O}}_p\).
- Ramified: \(e_p > 1\), and unramified if \(e_p = 1\).
- Branch points any \(q = f(p)\) where \(f\) is ramified.
- Tame ramification: for \(\operatorname{ch}(k) = p\), tame if \(p\notdivides e_P\).
- Wild ramification: when \(p\divides e_P\).
-
Pullback maps on divisor groups:
\begin{align*}
f^*: \operatorname{Div}(Y) &\to \operatorname{Div}(X) \\
Q &\mapsto \sum_{P \xrightarrow{f} q} e_P [P]
.\end{align*}
- This commutes with taking line bundles (exercise), so induces a well-defined map \(f^*: \operatorname{Pic}(X) \to \operatorname{Pic}(Y)\).
- \(f\) is separable if \(k(X) / k(Y)\) is a separable field extension.
Misc:
- Show that if \(f\) is everywhere unramified then it is an étale morphism.
- Show that \(f^* {\mathcal{L}}(D) = {\mathcal{L}}(f^* D)\)
Show that if \(X \xrightarrow{f} Y\) is a finite separable morphism of curves, there is a SES \begin{align*} f^* \Omega_Y \hookrightarrow\Omega_X \twoheadrightarrow\Omega_{X_{/ {Y}} } .\end{align*}
Definitions:
- Derivatives: for \(f: X\to Y\), let \(t\) be a parameter at \(Q = f(P)\) and \(u\) at \(P\). Then \(\Omega_{Y, Q} = \left\langle{dt}\right\rangle_{{\mathcal{O}}_Q}\) and \({\mathcal{O}}_{X, P} = \left\langle{du}\right\rangle_{{\mathcal{O}}_P}\) and \(\exists ! g\in {\mathcal{O}}_P\) such that \(f^* dt = du\) so we write \({\frac{\partial t}{\partial u}\,} \coloneqq g\).
- Ramification divisor: \(R \coloneqq\sum_{P\in X} \mathop{\mathrm{length}}(\Omega_{X_{/ {Y}} })_P [P] \in \operatorname{Div}(X)\)
For \(X \xrightarrow{f} Y\) a finite separable morphism of curves,
- \(\Omega_{X_{/ {Y}} }\) is a torsion sheaf on \(X\) with support equal to the ramification locus of \(f\). Thus \(f\) is ramified at finitely many points.
- The stalks \((\Omega_{X_{/ {Y}} })_P\) are principal \({\mathcal{O}}_P{\hbox{-}}\)modules of finite length equal to \(v_p\qty{{\frac{\partial t}{\partial u}\,}}\)
-
[
\mathop{\mathrm{length}}(\Omega
_{X\slice
Y})_P
\begin{cases}
= e_p - 1 & f \text{ is tamely ramified at } P
\\
> e_p -1 & f \text{ is wildly ramified at } P.
\end{cases}
.]
If \(X \xrightarrow{f} Y\) is a finite separable morphism of curves, then \begin{align*} K_X \sim f^* K_Y + R ,\end{align*} where \(R\) is the ramification divisor of \(f\).
If \(X \xrightarrow{f} Y\) is a finite separable morphism of curves, then \begin{align*} 2g(X) -2 = \deg(f)(2g(Y) - 2) + \deg(R) ,\end{align*} and if \(f\) has only tame ramification then \(\deg(R) = \sum_{P\in X}(e_P - 1)\).
Take degrees of the divisor equation: \begin{align*} \deg(K_X ) &= \deg(f^* K_Y + R) \\ \implies \chi_{\mathsf{Top}}(X) &= \deg(f^* K_Y) + \deg(R) \\ \implies 2g(X) - 2 &= \deg(f) \deg(K_Y) + \deg(R) \\ \implies 2g(X) - 2 &= \deg(f) \chi_{\mathsf{Top}}(Y) + \deg(R) \\ \implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \deg(R) \\ \implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \sum_{P\in X} (e_P - 1) \\ ,\end{align*} using tame ramification in the last step which implies \(\mathop{\mathrm{length}}(\Omega_{X_{/ {Y}} })_P = (e_p - 1)\).
Consider the purely inseparable case.
- Frobenius morphism: for \(X \in{\mathsf{Sch}}\) where \({\mathcal{O}}_P \supseteq{\mathbf{Z}}/p{\mathbf{Z}}\) for all \(P\), define \(\operatorname{Frob}: X\to X\) by \(F({\left\lvert {X} \right\rvert}) = {\left\lvert {X} \right\rvert}\) on spaces and \(F^\sharp: {\mathcal{O}}_X \to {\mathcal{O}}_X\) is \(f\mapsto f^p\). This is a morphism since \(F^\sharp\) induces a morphism on all local rings, which are all characteristic \(p\).
-
The \(k{\hbox{-}}\)linear Frobenius morphism: define \(X_p\) to be \(X\) with the structure morphism \(F\circ \pi\), so \(k\curvearrowright{\mathcal{O}}_{X_p}\) by \(p\)th powers and \(F\) becomes a \(k{\hbox{-}}\)linear morphism \(F': X_p\to X\).
- Why this is necessary: \(F\) as before is not a morphism in ${\mathsf{Sch}}_{/ {k}} $, and instead forms a commuting square involving \(F: \operatorname{Spec}k\to \operatorname{Spec}k\) and the structure maps \(X \xrightarrow{\pi} \operatorname{Spec}k\).
Find examples where
- $X_p \cong X \in {\mathsf{Sch}}_{/ {k}} $, and
- $X_p \not\cong X \in {\mathsf{Sch}}_{/ {k}} $.
Hint: consider \(X = \operatorname{Spec}k[t]\) for \(k\) perfect.
Show that if \(X \xrightarrow{f} Y\) is separable then \(\deg(R)\) is always even.
Skipped some stuff around Example 2.4.2, I don’t necessarily need characteristic \(p\) things right now.
Definitions:
- Étale covers: \(X \xrightarrow{f} Y\) is an étale cover if \(f\) is a finite étale morphism,, i.e. \(f\) is flat and \(\Omega^1_{X_{/ {Y}} } = 0\).
- \(Y\) is a trivial cover if \(X \cong {\textstyle\coprod}_{i\in I} Y\) a finite disjoint union of copies of \(Y\),
- \(Y\) is simply connected if there are no nontrivial étale covers.
In parts:
- Show that a connected regular curve is irreducible.
- Show that if \(f\) is etale then \(X\) is smooth over \(k\).
- Show that if \(f\) is finite, \(X\) must be a curve.
- Show that if \(f\) is étale, then \(f\) must be separable.
- Show that \(\pi_1^\text{ét}({\mathbf{P}}^1) = 0\).
Hint: use Hurwitz and that when \(f\) is unramified, \(R = 0\).
In parts:
- Show that the genus of a curve doesn’t change under purely inseparable extensions.
- Show that if \(f:X\to Y\) is a finite morphism of curves then \(g(X) \geq g(Y)\).
Show that if \(L\) is a subfield of a purely transcendental extension \(k(t) / k\) where $k = { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } $, then \(L\) is also purely transcendental. 1
Hint: Assume \([L: k]_{\mathrm{tr}}= 1\), so \(L = k(X)\) for \(Y\) a curve and \(L \subseteq k(t)\) corresponds to a finite morphism \(f: {\mathbf{P}}^1\to Y\). Conclude \(g(Y) = 0\) so \(Y\cong {\mathbf{P}}^1\) and \(L\cong k(u)\) for some \(u\).