## IV.2: Hurwitz $\star$ :::{.remark} Summary of results: - For curves, complete = projective. - Riemann-Hurwitz: for $f:X\to Y$ finite separable, \[ K_X \sim f^* K_Y + R \implies \deg(K_X) = \deg(f^*K_Y) + \deg(R) \implies \\ \\ \chi(X) = \deg (f)\cdot \chi(Y) + \deg R, \qquad \deg R = \sum_{p\in X} (e_p - 1) .\] - $\deg f \da [K(X): K(Y)]$ for finite morphisms of curves. - $e_p \da v_p(f^\sharp_* t)$ where $t$ is uniformizer in $\OO_{f(p)}$ and $f^\sharp: \OO_{Y, f(p)}\to \OO_{X, p}$ for $f:X\to Y$. - $e_p > 1 \implies$ ramification. - Unramified everywhere implies etale (since automatically flat). - $p\divides e_{x_0}\implies$ wild ramification, otherwise tame. - $\exists f^*: \Div(Y)\to \Div(X)$ where $q\mapsto \sum_{p\mapsto q} e_p p$. - Pullback commutes with forming line bundles: \[ f^* \mcl(D) \cong \mcl(f^* D) \] where the LHS $f^*: \Pic(Y) \to \Pic(X)$. - The fundamental SES for relative differentials: if $f:X\to Y$ is finite separable, \[ f^* \Omega_{Y} \injects \Omega_{X} \surjects \Omega_{X/Y} .\] - $\dd{t}{u}$ for $t$ a uniformizer at $f(p)$ and $u$ a uniformizer at $p$ is defined by noting $\Omega{Y, f(p)} = \gens{\dt}, \Omega_{X, p} = \gens{\du}$, and there is some $g\in \OO_{X, p}$ such that $f^* \dt = g\du$; set $g \da \dd t u$. - For finite separable morphisms of curves $f:X\to Y$, - $\supp \Omega_{X/Y} = \mathrm{Ram}(f)$ is the ramification locus, and $\Omega_{X/Y}$ is torsion so $\Ram(f)$ is finite. - $\length (\Omega_{X, Y})_p = v_p\qty{\dd t u}$ for any $p\in X$ - Tamely ramified $\implies \length(\Omega_{X/Y})_p = e_p - 1$, and wild ramification increases this length. Recall that length is the largest size of chains of submodules. - The ramification divisor: \[ R \da \sum_{p\in X} \length (\Omega_{X/Y})_p p .\] - $K_X \sim f^*K_Y + R$ - $\PP^1$ can't admit an unramified cover: for $n\geq 1$, \[ \chi(X) = n\chi(\PP^1) + \deg R \implies \chi(X) = -2n + \deg R \implies \chi(X) = -2n \leq -2 ,\] which forces $g(X) = 0, n=1, X = \PP^1, f=\id$. - The Frobenius morphism on schemes is defined by taking $f^\sharp: \OO_X\to \OO_X$ to be the $p$th power map; pullback yields a definition of $X_p$, the Frobenius twist of $X$. - $F: X_p\to X$ is finite, $\deg F = p$, and corresponds to $K(X) \injects K(X)^{1\over p}$ - If $f:X\to Y$ induces a purely inseparable extension $K(X)/K(Y)$, then $X\iso Y$ as schemes, $g(X) = g(Y)$, and $f$ is a composition of Frobenii. - Everywhere ramified extensions: $f:Y_p \to Y$, where $e_{q} = p$ for every $q\in X$. Induces $\Omega_{X/Y}\cong \Omega_{X}$. - $\deg R$ is always even. - Finite implies proper: finite implies separated, of finite type, closed by "going up" and universally closed by since finiteness is preserved under base change. - $\PP^1$ no nontrivial etale covers. - If $f:X\to Y$ then $g(X) \geq g(Y)$. - Thus $\exists \PP^1\to Y$ finite $\implies g(Y) = 0$. ::: :::{.remark} Preface: - **Degree**: for a finite morphism of curves $X \mapsvia{f} Y$, set $\det(f) \da [k(X): k(Y)]$, the degree of the extension of function fields. - **Ramification indices $e_p$**: for $p\in X$, let $q=f(p)$ and $t \in \OO_q$ a local coordinate. Pull back to $t\in \OO_p$ via $f^\sharp$ and define $e_p \da v_p(t)$ using the valuation $v_p$ for the DVR $\OO_p$. - **Ramified**: $e_p > 1$, and **unramified** if $e_p = 1$. - **Branch points** any $q = f(p)$ where $f$ is ramified. - **Tame ramification**: for $\characteristic(k) = p$, tame if $p\notdivides e_P$. - **Wild ramification**: when $p\divides e_P$. - Pullback maps on divisor groups: \[ f^*: \Div(Y) &\to \Div(X) \\ Q &\mapsto \sum_{P \mapsvia{f} q} e_P [P] .\] - This commutes with taking line bundles (exercise), so induces a well-defined map $f^*: \Pic(X) \to \Pic(Y)$. - $f$ is **separable** if $k(X) / k(Y)$ is a separable field extension. ::: :::{.exercise title="?"} Misc: - Show that if $f$ is everywhere unramified then it is an étale morphism. - Show that $f^* \mcl(D) = \mcl(f^* D)$ ::: :::{.exercise title="Prop 2.1"} Show that if $X \mapsvia{f} Y$ is a finite separable morphism of curves, there is a SES \[ f^* \Omega_Y \injects \Omega_X \surjects \Omega_{X\slice Y} .\] ::: :::{.remark} Definitions: - **Derivatives**: for $f: X\to Y$, let $t$ be a parameter at $Q = f(P)$ and $u$ at $P$. Then $\Omega_{Y, Q} = \gens{dt}_{\OO_Q}$ and $\OO_{X, P} = \gens{du}_{\OO_P}$ and $\exists ! g\in \OO_P$ such that $f^* dt = du$ so we write $\dd{t}{u} \da g$. - **Ramification divisor**: $R \da \sum_{P\in X} \length(\Omega_{X\slice Y})_P [P] \in \Div(X)$ ::: :::{.exercise title="Prop 2.2"} For $X \mapsvia{f} Y$ a finite separable morphism of curves, a. $\Omega_{X\slice Y}$ is a torsion sheaf on $X$ with support equal to the ramification locus of $f$. Thus $f$ is ramified at finitely many points. b. The stalks $(\Omega_{X\slice Y})_P$ are principal $\OO_P\dash$modules of finite length equal to $v_p\qty{\dd t u}$ c. \[ \length(\Omega_{X\slice Y})_P \begin{cases} = e_p - 1 & f \text{ is tamely ramified at } P \\ > e_p -1 & f \text{ is wildly ramified at } P. \end{cases} .\] ::: :::{.exercise title="Prop 2.3"} If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then \[ K_X \sim f^* K_Y + R ,\] where $R$ is the ramification divisor of $f$. ::: :::{.theorem title="Hurwitz"} If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then \[ 2g(X) -2 = \deg(f)(2g(Y) - 2) + \deg(R) ,\] and if $f$ has only tame ramification then $\deg(R) = \sum_{P\in X}(e_P - 1)$. ::: :::{.remark title="proof of Hurwitz"} Take degrees of the divisor equation: \[ \deg(K_X ) &= \deg(f^* K_Y + R) \\ \implies \chi_\Top(X) &= \deg(f^* K_Y) + \deg(R) \\ \implies 2g(X) - 2 &= \deg(f) \deg(K_Y) + \deg(R) \\ \implies 2g(X) - 2 &= \deg(f) \chi_\Top(Y) + \deg(R) \\ \implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \deg(R) \\ \implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \sum_{P\in X} (e_P - 1) \\ ,\] using tame ramification in the last step which implies $\length(\Omega_{X\slice Y})_P = (e_p - 1)$. ::: :::{.remark} Consider the purely inseparable case. - **Frobenius morphism**: for $X \in\Sch$ where $\OO_P \contains \ZZ/p\ZZ$ for all $P$, define $\Frob: X\to X$ by $F(\abs{X}) = \abs{X}$ on spaces and $F^\sharp: \OO_X \to \OO_X$ 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 $k\dash$linear Frobenius morphism**: define $X_p$ to be $X$ with the structure morphism $F\circ \pi$, so $k\actson \OO_{X_p}$ by $p$th powers and $F$ becomes a $k\dash$linear morphism $F': X_p\to X$. - Why this is necessary: $F$ as before is not a morphism in $\Sch\slice k$, and instead forms a commuting square involving $F: \spec k\to \spec k$ and the structure maps $X \mapsvia{\pi} \spec k$. ::: :::{.exercise title="?"} Find examples where - $X_p \cong X \in \Sch\slice k$, and - $X_p \not\cong X \in \Sch\slice k$. > Hint: consider $X = \spec k[t]$ for $k$ perfect. ::: :::{.exercise title="?"} Show that if $X \mapsvia{f} Y$ 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. :::{.remark} Definitions: - **Étale covers**: $X \mapsvia{f} Y$ is an étale cover if $f$ is a finite étale morphism,, i.e. $f$ is flat and $\Omega^1_{X\slice Y} = 0$. - $Y$ is a **trivial** cover if $X \cong \disjoint_{i\in I} Y$ a finite disjoint union of copies of $Y$, - $Y$ is **simply connected** if there are no nontrivial étale covers. ::: :::{.exercise title="?"} \envlist - 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 $\pi_1^\et(\PP^1) = 0$. > Hint: use Hurwitz and that when $f$ is unramified, $R = 0$. ::: :::{.exercise title="?"} \envlist - 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 $g(X) \geq g(Y)$. ::: :::{.exercise title="Lüroth"} Show that if $L$ is a subfield of a purely transcendental extension $k(t) / k$ where $k = \kbar$, then $L$ is also purely transcendental.[^more_luroth_stuff] > Hint: Assume $[L: k]_\tr = 1$, so $L = k(X)$ for $Y$ a curve and $L \subseteq k(t)$ corresponds to a finite morphism $f: \PP^1\to Y$. Conclude $g(Y) = 0$ so $Y\cong \PP^1$ and $L\cong k(u)$ for some $u$. [^more_luroth_stuff]: This is true over any field $k$ in dimension 1, over $k=\kbar$ in dimension 2, and false in dimension 3 by the existence of nonrational unirational threefolds. :::