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# IV: Curves $\star$

:::{.remark}
Summary of major results:

- $p_a(X) \da 1 - P_X(0) = (-1)^r (1-\chi(\OO_X))$.
  - Note: $P_X(\ell)$ is defined as the Hilbert polynomial of the homogeneous coordinate ring $S(Y)$, and then defined for graded $S\dash$modules $M$ by setting $\phi_M(\ell) = \dim_k M_\ell$ and showing $\exists ! P_M(z) \in \QQ[z]$ with $\phi_M(\ell) = P_M(\ell)$ for $\ell \gg 0$.
- $p_g(X) \da h^0(\omega_X) = h^0(\mcl(K_X))$.
- Remembering these:

  ![](Reading%20Notes/4_Hartshorne/figures/2022-12-04_20-08-00.png)

> [Link to Diagram](https://q.uiver.app/?q=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)

- For curves, $p_a(X) = p_g(X) = h^1(\OO_X)$ by setting $D\da K_C$ in RR.
  - $\deg K_C = 2g-2$.
- $D_1\sim D_2 \iff D_1-D_2 = (f)$ for $f\in K(X)$ rational, $\abs{D} = \ts{D'\sim D}$, and this bijects with points of $H^0(\mcl(D))\smz\over \GG_m$.
  - Thus $\dim \abs{D} = h^0(\mcl(D)) - 1 \da \ell(D) - 1$.
- $X$ smooth $\implies \Cl(X)\iso \Pic(X)$ via $D\mapsto \mcl(D)$.
- $h^0(\mcl(D)) >0 \implies \deg(D) \geq 0$, and if $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
- RR:
\[
\chi(\mcl(D) &= h^0(\mcl(D)) - h^1(\mcl(D)) \\ 
  &= h^0(\mcl(D)) - h^0(\mcl(K-D)) \\
  &= \deg(D) + (1-g)
.\]
  - How to remember: note $g= h^1(\OO_X) = h^1(\mcl(0))$, and $H^0(\OO_X) = k$ so $h^0(\OO_X) = 1$, thus
  \[
  \chi(\OO_X) = h^0(\OO_X) - h^1(\OO_X) = 1-g = \deg \mcl(0) + 1-g
  .\]
  - For $C \subseteq \PP^n, \deg(C) = d$ and $D = C \intersect H$ a hyperplane section defining $\mcl(D) = \OO_X(1)$,
  \[
 \chi(\mcl(D)) = \deg(D) + (1-g) = d + (1-p_a(C))
 \]
- A curve is rational iff isomorphic to $\PP^1$ iff $g=0$.
- $K\sim 0$ on an elliptic curve since $\deg K = 2g-2 = 0$ and $\deg D = 0\implies D\sim 0$.
- For $X$ elliptic, $\Pic^0(X) \da \ts{D\in \Div(X) \st \deg D = 0}$ and $\abs{X} \iso \abs{\Pic^0(X)}$ via $p\mapsto \mcl(p-p_0)$ for any fixed $p_0\in X$, inducing its group structure. (This is proved with RR.)
:::

:::{.remark}
Comments from preface:

- The statement of Riemann-Roch is important; less so its proof.
- Representing curves: 
  - A branched covering of $\PP^1$, 
  - More generally a branched covering of another curve,
  - Nonsingular projective curves: admit embeddings into $\PP^3$, maps to $\PP^2$ birationally such that the image is at worst a nodal curve.
- The central result regarding representing curves: Hurwitz's theorem which compares $K_X, K_Y$ for a cover $Y\to X$ of curves.
- Curves of genus 1: elliptic curves.
- Later sections: the canonical embedding of a curve.
:::

## IV.1: Riemann-Roch

:::{.definition title="Curves"}
A **curve** over $k=\kbar$ is a scheme over $\spec k$ which is 

- Integral
- Dimension 1
- Proper over $k$
- With regular local rings

In particular, a curve is smooth, complete, and necessary projective.
A **point** on a curve is a closed point.
:::

:::{.definition title="Arithmetic genus"}
The **arithmetic genus** of a projective curve $X$ is
\[
p_a(X) \da 1 - P_X(0)
\]
where $P_X(t)$ is the **Hilbert polynomial** of $X$.
:::

:::{.definition title="Geometric genus"}
The **geometric genus** of a curve is 
\[
p+g(X) \da \dim_k H^0(X; \omega_X)
\]
where $\omega_X$ is the canonical sheaf.
:::

:::{.exercise title="?"}
Show that if $X$ is a curve, there is a single well-defined **genus** 
\[
g \da p_A(X) = p_G(X) = \dim_k H^1(X; \OO_X)
.\]

> Hint: see Ch. III Ex. 5.3, and use Serre duality for $p_g$.

:::

:::{.exercise title="?"}
Show that for any $g\geq 0$ there exists a curve of genus $g$.

> Hint: take a divisor of type $(g+1, 2)$ on a smooth quadric which is irreducible and smooth with $p_a = g$.

:::

:::{.definition title="Divisors on a curve"}
Reviewing divisors:

- The **divisor group**: $\Div(X) = \zadjoin{X_\cl}$
- **Degrees**: $\deg(\sum n_i D_i) \da \sum n_i$, and 
- **Linear equivalence**: $D_1\sim D_2 \iff D_1 - D_1 = \div(f)$ for some $f\in k(X)$ a rational function.
- $D$ is **effective** if $n_i \geq 0$ for all $i$.
- $\abs{D} \da \ts{D'\in \Div(X) \st D'\sim D}$ is the **complete linear system** of $D$.
- $\abs{D} \cong \PP H^0(X; \mcl(D))$
- **Dimensions of linear systems**: $\ell(D) \da \dim_k H^0(X; \mcl(D))$ and $\dim \abs{D} \da \ell(D) - 1$.
- **Relative differentials**: $\Omega_X \da \Omega_{X\slice k}$ is the sheaf of relative differentials on $X$.
  - The technical definition: $\Omega_{X\slice S} \da \Delta_{X\slice Y}^*(\mci/\mci^2)$ where $\mci$ is the sheaf of ideals defining the locally closed subscheme $\im(\Delta_{X\slice Y}) \subseteq X\fp{Y} X$.
  - On affine schemes: on the ring side, $\Omega_{B/{A}} \in \mods{B}$ equipped with a differential $d: B\to \Omega{B/ A}$, defined as 
    $$\gens{db\st b\in B}_B \over \gens{d(b_1+b_2) =db_1 + db_2, d(b_1 b_2) = d(b_1)b_2 + b_1 d(b_2), da = 0\, \forall a\in A}_B$$
  - On curves, $\Omega_{X\slice Y}$ measures the "difference" between $K_X$ and $K_Y$.

- **Canonical sheaf**: $\dim X = 1, \Omega_{X\slice k} \cong \omega_X$.
- **Canonical divisor**: $K_X$ 2is any divisor in the linear equivalence class corresponding to $\omega_X$ 
- $D$ is **special** iff its **index of speciality** $\ell(K-D) > 0$, otherwise $D$ is **nonspecial**.

:::

:::{.exercise title="?"}
Show that $D_1\sim D_2\implies\deg(D_1) = \deg(D_2)$.
:::

:::{.exercise title="?"}
Show that 
\[
\abs{D} \mapstofrom \PP H^0(X; \mcl(D))
,\]
so $\abs{D}$ has the structure of the closed points of some projective space.
:::

:::{.exercise title="Lemma 1.2"}
Show that if $D\in \Div(X)$ for $X$ a curve and $\ell(D) \neq 0$, then $\deg(D) \geq 0$.

Show that is $\ell(D) \neq 0$ and $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
:::

:::{.theorem title="Riemann-Roch"}
\[
\ell(D) - \ell(K-D) = \deg(D) + (1-g)
.\]

:::

:::{.exercise title="Ingredients for proof of RR"}
Show the following:

- The divisor $K-D$ corresponds to $\omega_X \tensor\mcl(D)\dual\in \Pic(X)$.
- $H^1(X; \mcl(D))\dual \cong H^0(X; \omega_X \tensor \mcl(D)\dual)$.

- If $X$ is any projective variety,
\[
H^0(X; \OO_X) = k
.\]
:::

:::{.exercise title="?"}
Show that if $X \subseteq \PP^n$ is a curve with $\deg X = d$ and $D = X \intersect H$ is a hyperplane section, then $\mcl(D) = \OO_X(1)$ and $\chi(\mcl(D)) = d + 1 - p_a$.
:::

:::{.exercise title="?"}
Show that if $g(X) = g$ then $\deg K_X = 2g-2$.

> Hint: set $D = K$ and use $\ell(K) = p_g = g$ and $\ell(0) = 1$.

:::

:::{.remark}
More definitions:

- $X$ is **rational** iff birational to $\PP^1$.
- $X$ is **elliptic** if $g=1$.
:::

:::{.exercise title="?"}
Show that

1. If $\deg D > 2g-2$ then $D$ is nonspecial.
2. $p_a(\PP^1) = 0$.
3. A complete nonsingular curve is rational iff $X\cong \PP^1$ iff $g(X) = 0$.
4. If $X$ is elliptic then $K\sim 0$

> Hint: for (3) apply RR to $D = p-q$ for points $p\neq q$, and use $\deg(K-D) = -2$ and $\deg(D) = 0 \implies D\sim 0 \implies p\sim q$.
> For (4), show $\ell(K) = p_g = 1$.

:::

:::{.exercise title="?"}
If $X$ is elliptic and $p\in X$, then there is a bijection
\[
m_p: X &\iso \Pic(X) \\
x &\mapsto \mcl(x-p)
,\]
so $\Pic(X) \in \Grp$.

> Hint: show that if $\deg(D) = 0$ then there is some $x\in X$ such that $D\sim x-p$ and apply RR to $D+p$.

:::