## IV.2: Hurwitz

### 2.1 #to-work

Use (2.5.3) to show that $\mathbf{P}^n$ is simply connected.

### 2.2 Classification of Curves of Genus 2 . #to-work

Fix an algebraically closed field $k$ of characteristic $\neq 2$.

a. If $X$ is a curve of genus 2 over $k$, the canonical linear system $|K|$ determines a finite morphism $f: X \rightarrow \mathbf{P}^1$ of degree 2 (Ex. 1.7). Show that it is ramified at exactly 6 points, with ramification index 2 at each one. Note that $f$ is uniquely determined, up to an automorphism of $\mathbf{P}^1$, so $X$ determines an (unordered) set of 6 points of $\mathbf{P}^1$, up to an automorphism of $\mathbf{P}^1$.

b. Conversely, given six distinct elements $\alpha_1, \ldots, \alpha_6 \in k$, let $K$ be the extension of $k(x)$ determined by the equation $z^2=\left(x-\alpha_1\right) \cdots\left(x-\alpha_6\right)$. Let $f: X \rightarrow \mathbf{P}^1$ be the corresponding morphism of curves. Show that $g(X)=2$, the map $f$ is the same as the one determined by the canonical linear system, and $f$ is ramified over the six points $x=\alpha_i$ of $\mathbf{P}^1$, and nowhere else. (Cf. (II, Ex. 6.4).)

c. Using (I, Ex. 6.6), show that if $P_1, P_2, P_3$ are three distinct points of $\mathbf{P}^1$, then there exists a unique $\varphi \in$ Aut $\mathbf{P}^1$ such that $\varphi\left(P_1\right)=0, \varphi\left(P_2\right)=1, \varphi\left(P_3\right)=\infty$. 
Thus in (a), if we order the six points of $\mathbf{P}^1$, and then normalize by sending the first three to $0,1, x$, respectively, we may assume that $X$ is ramified over $0,1, \infty, \beta_1, \beta_2, \beta_3$, where $\beta_1, \beta_2, \beta_3$ are three distinct elements of $k, \neq 0,1$.

d. Let $\Sigma_6$ be the symmetric group on 6 letters. Define an action of $\Sigma_6$ on sets of three distinct elements $\beta_1, \beta_2, \beta_3$ of $k, \neq 0,1$, as follows: reorder the set $0,1, \infty, \beta_1, \beta_2, \beta_3$ according to a given element $\sigma \in \Sigma_6$, then renormalise as in (c) so that the first three become $0,1, \infty$ again. Then the last three are the new $\beta_1^{\prime}, \beta_2^{\prime}, \beta_3^{\prime}$.

e. Summing up, conclude that there is a one-to-one correspondence between the set of isomorphism classes of curves of genus 2 over $k$, and triples of distinct elements $\beta_1, \beta_2, \beta_3$ of $k, \neq 0,1$, modulo the action of $\Sigma_6$ described in (d). In particular, there are many non-isomorphic curves of genus 2 . We say that curves of genus 2 depend on three parameters, since they correspond to the points of an open subset of $\mathbf{A}_k^3$ modulo a finite group.

### 2.3 Plane Curves. #to-work

Let $X$ be a curve of degree $d$ in $\mathbf{P}^2$. For each point $P \in X$, let $T_P(X)$ be the tangent line to $X$ at $P$ (I, Ex. 7.3). Considering $T_P(X)$ as a point of the dual projective plane $\left(\mathbf{P}^2\right)^*$, the map $P \rightarrow T_P(X)$ gives a morphism of $X$ to its **dual curve** $X^*$ in $\left(\mathbf{P}^2\right)^*$ (I, Ex. 7.3). 

Note that even though $X$ is nonsingular, $X^*$ in general will have singularities. We assume char $k=0$ below.

a. Fix a line $L \subseteq \mathbf{P}^2$ which is not tangent to $X$. Define a morphism $\varphi: X \rightarrow L$ by $\varphi(P)=T_P(X) \cap L$, for each point $P \in X$. Show that $\varphi$ is ramified at $P$ if and only if either 
	(1) $P \in L$, or 
	(2) $P$ is an inflection point of $X$, which means that the intersection multiplicity (I, Ex. 5.4) of $T_P(X)$ with $X$ at $P$ is $\geqslant 3$. 
Conclude that $X$ has only finitely many inflection points.

b. A line of $\mathbf{P}^2$ is a **multiple tangent** of $X$ if it is tangent to $X$ at more than one point. It is a **bitangent** if it is tangent to $X$ at exactly two points. If $L$ is a multiple tangent of $X$, tangent to $X$ at the points $P_1, \ldots, P_r$, and if none of the $P_i$ is an inflection point, show that the corresponding point of the dual curve $X^*$ is an ordinary $r$-fold point, which means a point of multiplicity $r$ with distinct tangent directions (I, Ex. 5.3). 
Conclude that $X$ has only finitely many multiple tangents.

c. Let $O \in \mathbf{P}^2$ be a point which is not on $X$, nor on any inflectional or multiple tangent of $X$. Let $L$ be a line not containing $O$. Let $\psi: X \rightarrow L$ be the morphism defined by projection from $O$. 
Show that $\psi$ is ramified at a point $\mathrm{P} \in X$ if and only if the line $O P$ is tangent to $X$ at $P$, and in that case the ramification index is 2. 
Use Hurwitz's theorem and (I, Ex. 7.2) to conclude that there are exactly $d(d-1)$ tangents of $X$ passing through $O$. Hence the degree of the dual curve (sometimes called the **class** of $X)$ is $d(d-1)$.

d. Show that for all but a finite number of points of $X$, a point $O$ of $X$ lies on exactly $(d+1)(d-2)$ tangents of $X$, not counting the tangent at $O$.

e. Show that the degree of the morphism $\varphi$ of a. is $d(d-1)$. 
Conclude that if $d \geqslant 2$, then $X$ has $3 d(d-2)$ inflection points, properly counted. (If $T_P(X)$ has intersection multiplicity $r$ with $X$ at $P$, then $P$ should be counted $r-2$ times as an inflection point. If $r=3$ we call it an ordinary inflection point.) 
Show that an ordinary inflection point of $X$ corresponds to an ordinary cusp of the dual curve $X^*$.

f. Now let $X$ be a plane curve of degree $d \geqslant 2$, and assume that the dual curve $X^*$ has only nodes and ordinary cusps as singularities (which should be true for sufficiently general $X)$. Then show that $X$ has exactly $\frac{1}{2} d(d-2)(d-3)(d+3)$ bitangents.[^hint_2_2_3_f]

g. For example, a plane cubic curve has exactly 9 inflection points, all ordinary. The line joining any two of them intersects the curve in a third one.

h. A plane quartic curve has exactly 28 bitangents. (This holds even if the curve has a tangent with four-fold contact, in which case the dual curve $X^*$ has a tacnode.)

[^hint_2_2_3_f]: 
Hint: Show that $X$ is the normalization of $X^*$. Then calculate $p_a\left(X^*\right)$ two ways: once as a plane curve of degree $d(d-1)$, and once using (Ex. 1.8).

### 2.4 A Funny Curve in Characteristic $p$. #to-work

Let $X$ be the plane quartic curve $x^3 y+y^3 z+ z^3 x = 0$ over a field of characteristic 3 . Show that $X$ is nonsingular, every point of $X$ is an inflection point, the dual curve $X^*$ is isomorphic to $X$, but the natural map $X \rightarrow X^*$ is purely inseparable.

### 2.5 Automorphisms of a Curve of Genus $\geqslant 2$. #to-work

Prove the theorem of Hurwitz that a curve $X$ of genus $g \geqslant 2$ over a field of characteristic 0 has at most $84(g-1)$ automorphisms. 

We will see later (Ex. 5.2) or (V, Ex. 1.11) that the group $G=$ Aut $X$ is finite. So let $G$ have order $n$. Then $G$ acts on the function field $K(X)$. Let $L$ be the fixed field. Then the field extension $L \subseteq K(X)$ corresponds to a finite morphism of curves $f: X \rightarrow Y$ of degree $n$.

a. If $P \in X$ is a ramification point, and $e_P=r$, show that $f^{-1} f(P)$ consists of exactly $n / r$ points, each having ramification index $r$. Let $P_1, \ldots, P_s$ be a maximal set of ramification points of $X$ lying over distinct points of $Y$, and let $e_{P_1}=r_i$. Then show that Hurwitz's theorem implies that
$$
{2 g-2 \over  n } =2 g(Y)-2+\sum_{i=1}^s\left(1- {1\over r_i}\right)
$$

b. Since $g \geqslant 2$, the left hand side of the equation is $>0$. Show that if $g(Y) \geqslant 0$, $s \geqslant 0, r_i \geqslant 2, i=1, \ldots, s$ are integers such that
$$
2g(Y)-2+\sum_{i=1}^s\left(1- {1\over r_i}\right)>0,
$$
then the minimum value of this expression is $1 / 42$. Conclude that $n \leqslant 84(g-1)$.[^max_achieved]

[^max_achieved]: 
See (Ex. 5.7) for an example where this maximum is achieved.
Note: It is known that this maximum is achieved for infinitely many values of $g$ (Macbeath [1]). Over a field of characteristic $p>0$, the same bound holds, provided $p>g+1$, with one exception, namely the hyperelliptic curve $y^2=x^p-x$, which has $p=2 g+1$ and $2 p\left(p^2-1\right)$ automorphisms (Roquette). For other bounds on the order of the group of automorphisms in characteristic $p$, see Singh and Stichtenoth.


### 2.6 $f_*$ for Divisors. #to-work

Let $f: X \rightarrow Y$ be a finite morphism of curves of degree $n$. We define a homomorphism $f_*: \operatorname{Div} X \rightarrow$ Div $Y$ by $f_*\left(\sum n_i P_i\right)=\sum n_i f\left(P_i\right)$ for any divisor $D=\sum n_i P_i$ on $X$.

a. For any locally free sheaf $\mathscr{E}$ on $Y$, of rank $r$, we define $\det \mathscr{E}=\wedge^r \mathscr{E} \in \Pic Y$ (II, Ex. 6.11). In particular, for any invertible sheaf $\mathscr{M}$ on $X, f_* \mathscr{M}$ is locally free of rank $n$ on $Y$, so we can consider $\det f_* \mathscr{M} \in \Pic Y$. Show that for any divisor $D$ on $X$,[^hint.2.2.6]
$$
\operatorname{det}\left(f_* \mathscr{L}(D)\right) \cong\left(\operatorname{det} f_*\left(\OO_X\right) \otimes \mathscr{L}\left(f_* D\right) .\right.
$$
Note in particular that $\operatorname{det}\left(f_* \mathscr{L}(D)\right) \neq \mathscr{L}\left(f_* D\right)$ in general!

b. Conclude that $f_* D$ depends only on the linear equivalence class of $D$, so there is an induced homomorphism $f_*: \Pic X \rightarrow \Pic Y$. Show that $f_* f^*: \Pic Y \rightarrow \Pic Y$ is just multiplication by $n$.

c. Use duality for a finite flat morphism (III, Ex. 6.10) and (III, Ex. 7.2) to show that 
$$\det f_* \Omega_X \cong \qty{ \det f_* \OO_X}\inv  \otimes \Omega_Y^{\otimes n}.$$

d. Now assume that $f$ is separable, so we have the ramification divisor $R$. We define the **branch divisor** $B$ to be the divisor $f_* R$ on $Y$. Show that 
$$\left(\operatorname{det} f_* \mathcal{O}_X\right)^2 \cong \mathscr{L}(-B).$$

[^hint.2.2.6]: 
Hint: First consider an effective divisor $D$, apply $f_*$ to the exact sequence $0 \rightarrow \mathscr{L}(-D) \rightarrow$ $\OO_X \rightarrow \mathcal{O}_D \rightarrow 0$, and use (II, Ex. 6.11).]

### 2.7 Etale Covers of Degree 2. #to-work

Let $Y$ be a curve over a field $k$ of characteristic $\neq 2$. We show there is a one-to-one correspondence between finite étale morphisms $f: X \rightarrow Y$ of degree 2, and 2-torsion elements of $\Pic Y$, i.e., invertible sheaves $\mathscr{L}$ on $Y$ with $\mathscr{L}^2 \cong \mathcal{O}_Y$.

a. Given an étale morphism $f: X \rightarrow Y$ of degree 2, there is a natural map $\OO_Y \rightarrow$ $f_* \OO_X$. Let $\mathscr{L}$ be the cokernel. Then $\mathscr{L}$ is an invertible sheaf on $Y, \mathscr{L} \cong \operatorname{det} f_* \OO_X$, and so $\mathscr{L}^2 \cong \mathcal{O}_Y$ by (Ex. 2.6). Thus an étale cover of degree 2 determines a 2-torsion element in $\Pic Y$.

b. Conversely, given a 2-torsion element $\mathscr{L}$ in  $\Pic Y$, define an $\OO_Y\dash$algebra structure on $\mathcal{O}_Y \oplus \mathscr{L}$ by $\langle a, b\rangle \cdot\left\langle a^{\prime}, b^{\prime}\right\rangle=\left\langle a a^{\prime}+\varphi\left(b \otimes b^{\prime}\right), a b^{\prime}+a^{\prime} b\right\rangle$, where $\varphi$ is an isomorphism of $\mathscr{L} \otimes \mathscr{L} \rightarrow \mathcal{O}_Y$. Then take $X=\operatorname{Spec}\left(\OO_Y \oplus \mathscr{L}\right)$ (II, Ex. 5.17). Show that $X$ is an étale cover of $Y$.

c. Show that these two processes are inverse to each other.[^note_4.2.7][^hint4.2.7]

[^hint4.2.7]: 
Hint: Let $\tau: X \rightarrow X$ be the involution which interchanges the points of each fibre of $f$. Use the trace map $a \mapsto a+\tau(a)$ from $f_* \mathcal{O}_X \rightarrow \mathcal{O}_Y$ to show that the sequence of $\mathcal{O}_{Y^{-}}$ modules in a. is split exact.
$$
0 \rightarrow \mathcal{O}_Y \rightarrow f_* \mathcal{O}_X \rightarrow \mathscr{L} \rightarrow 0
$$

[^note_4.2.7]: 
Note. This is a special case of the more general fact that for $(n$, char $k)=1$, the étale Galois covers of $Y$ with group $\mathbf{Z} / n \mathbf{Z}$ are classified by the étale cohomology group $H_{\mathrm{et}}^1(Y, \mathbf{Z} / n \mathbf{Z})$, which is equal to the group of $n$-torsion points of $\Pic Y$. See Serre [6].