# IV: Curves

## IV.1: Riemann-Roch

### 1.1. #completed

Let $X$ be a curve, and let $P \in X$ be a point. Then there exists a nonconstant rational function $f \in K(X)$, which is regular everywhere except at $P$.

> [!solution] Solution
> Such a function is a section $f\in \globsec{X; \mcl(nD) }$ for some large enough $n$ measuring the order of the pole at $P$, where $D = [P]$ is the divisor of $P$. So it suffices to show that $h^0(\mcl(nD)) > 0$ for some $n$. RR says 
> $$\chi(\mcl(nD)) = \deg D + 1 - g \implies h^0(\mcl(nD)) - h^1(\mcl(nD)) = n + 1-g.$$
> Claim: if $n$ is large, $h^1(\mcl(nD)) = 0$ and thus $h^0(\mcl(nD)) = n+1 -g$, and since $g$ is fixed and $n$ can vary, $h^0(\mcl(nD)) > 0$ for some $n$.
> Why this claim is true: by Serre duality, 
> $$
> H^1(X; \mcl(nD)) \cong H^0(X; K_X \tensor \mcl(nD)\dual )\dual =
> H^0(X; K_X \tensor \mcl(-nD))\dual = H^0(X; \mcl(K_X - nD))\dual
> $$
> so $h^1(\mcl(nD)) = h^0(\mcl(K_X - nD)) =0$ as soon as $\deg(K_X - nD) < 0$, which happens for large $n$ since $\deg(K_X - nD) = 2g-2-n$.

### 1.2. #to-work

Again let $X$ be a curve, and let $P_1, \ldots P_r \in X$ be points. Then there is a rational function $f \in K(X)$ having poles (of some order) at each of the $P_1$, and regular elsewhere.

> [!solution] Solution
> Let $D= P_1 + \cdots + P_r$, we then want $f\in \globsec{X; \mcl(nD)}$ for $n\gg 0$. By RR,
> $$
> \chi(\mcl(nD)) = \deg D + 1-g \implies h^0(\mcl(nD)) = rn + 1 - g
> ,$$
> which is non-negative for $n\gg 0$.

### 1.3. #to-work

Let $X$ be an integral, separated, regular, one-dimensional scheme of finite type over $k$, which is **not** proper over $k$. Then $X$ is affine.[^hint_4.1.3]

[^hint_4.1.3]: 
Hint: Embed $X$ in a (proper) curve $\bar{X}$ over $k$, and use (Ex. 1.2) to construct a morphism $f: \bar{X} \rightarrow \mathbf{P}^1$ such that $f^{-1}\left(\mathbf{A}^1\right)=X$

### 1.4. #to-work

Show that a separated, one-dimensional scheme of finite type over $k$, none of whose irreducible components is proper over $k$, is affine.[^hint4.1.4]

[^hint4.1.4]: 
Hint: Combine (Ex. 1.3) with (III, Ex. 3.1, Ex. 3.2, Ex. 4.2).

### 1.5. #to-work

For an effective divisor $D$ on a curve $X$ of genus $g$, show that $\operatorname{dim}|D| \leqslant \operatorname{deg} D$. Furthermore, equality holds if and only if $D=0$ or $g=0$.

### 1.6. #to-work

Let $X$ be a curve of genus $g$. Show that there is a finite morphism $f: X \rightarrow \mathbf{P}^1$ of degree[^recall_degreefinitemorphism] $\leqslant g+1$. 

[^recall_degreefinitemorphism]: 
Recall that the degree of a finite morphism of curves $f: X \rightarrow Y$ is defined as the degree of the field extension $[K(X): K(Y)]$ (II.6).

### 1.7. #to-work

A curve $X$ is called **hyperelliptic** if $g \geqslant 2$ and there exists a finite morphism $f: X \rightarrow \mathbf{P}^1$ of degree 2.

a. If $X$ is a curve of genus $g=2$, show that the canonical divisor defines a complete linear system $|K|$ of degree 2 and dimension 1, without base points. Use (II, 7.8.1) to conclude that $X$ is hyperelliptic.

b. Show that the curves constructed in (1.1.1) all admit a morphism of degree 2 to $\mathbf{P}^1$. Thus there exist hyperelliptic curves of any genus $g \geqslant 2$.[^non_hyperell]

[^non_hyperell]: 
Note: we will see later (Ex. 3.2) that there exist non-hyperelliptic curves. See also (V, Ex. 2.10).

### 1.8. $p_a$ of a Singular Curve. #to-work

Let $X$ be an integral projective scheme of dimension 1 over $k$, and let $\tilde{X}$ be its normalization (II, Ex. 3.8). Then there is an exact sequence of sheaves on $X$,
$$
0 \rightarrow \OO_X \rightarrow f_* \OO_X \rightarrow \sum_{P \in X} \tilde\OO_P/\OO_P \rightarrow 0
$$
where $\tilde\OO_P$ is the integral closure of $\OO_P$. For each $P \in X$, let $\delta_P=\operatorname{length}(\tilde\OO_P/\OO_P)$.

a. Show that $p_a(X)=p_a(\tilde{X})+\sum_{p \in X} \delta_p$.[^hint_4.1.8.a]

b. If $p_a(X)=0$, show that $X$ is already nonsingular and in fact isomorphic to $\mathbf{P}^1$.[^4.1.8.b.rmk]


c. \* If $P$ is a node or an ordinary cusp (I, Ex. 5.6, Ex. 5.14), show that $\delta_P=1$.[^hint_4.1.8.c]

[^4.1.8.b.rmk]: 
This strengthens (1.3.5).

[^hint_4.1.8.c]: 
Hint: Show first that $\delta_P$ depends only on the analytic isomorphism class of the singularity at $P$. Then compute $\delta_P$ for the node and cusp of suitable plane cubic curves. See $(\mathrm{V}, 3.9 .3)$ for another method.

[^hint_4.1.8.a]: 
Hint: Use (III, Ex. 4.1) and (III, Ex. 5.3).

### 1.9. \* Riemann-Roch for Singular Curves. #to-work

Let $X$ be an integral projective scheme of dimension 1 over $k$. Let $X_{\reg}$ be the set of regular points of $X$.

a. Let $D=\sum n_i P_i$ be a divisor with support in $X_{\reg}$, i.e., all $P_i \in X_{\reg}$. Then define deg $D=\sum n_i$. Let $\mathscr{L}(D)$ be the associated invertible sheaf on $X$, and show that
$$
\chi(\mathscr{L}(D))=\operatorname{deg} D+1-p_a .
$$
b. Show that any Cartier divisor on $X$ is the difference of two very ample Cartier divisors. [^4.1.9_rmk]


c. Conclude that every invertible sheaf $\mathscr{L}$ on $X$ is isomorphic to $\mathscr{L}(D)$ for some divisor $D$ with support in $X_{\reg}$.
d. Assume furthermore that $X$ is a locally complete intersection in some projective space. Then by (III, 7.11) the dualizing sheaf $\omega_X$ is an invertible sheaf on $X$, so we can define the canonical divisor $K$ to be a divisor with support in $X_{\reg}$ corresponding to $\omega_X$. Then the formula of a. becomes
$$
l(D)-l(K-D)=\operatorname{deg} D+1-p_a
$$

[^4.1.9_rmk]: 
Use (II, Ex. 7.5).

### 1.10. #to-work

Let $X$ be an integral projective scheme of dimension 1 over $k$, which is locally complete intersection, and has $p_a=1$. Fix a point $P_0 \in X_{\text {reg. }}$. Imitate (1.3.7) to show that the map $P \rightarrow \mathscr{L}\left(P-P_0\right)$ gives a one-to-one correspondence between the points of $X_{\reg}$ and the elements of the group $\Pic X$.[^rmk_4.1.10]

[^rmk_4.1.10]: 
This generalizes (II, 6.11.4) and (II, Ex. 6.7).