IV: Curves – AG Notes

Comments from preface:

The statement of Riemann-Roch is important; less so its proof.
Representing curves:
- A branched covering of ${\mathbf{P}}^1$,
- More generally a branched covering of another curve,
- Nonsingular projective curves: admit embeddings into ${\mathbf{P}}^3$, maps to ${\mathbf{P}}^2$ 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

A curve over $k= { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } $ is a scheme over $\operatorname{Spec}k$ 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.

The arithmetic genus of a projective curve $X$ is \begin{align*} p_a(X) \coloneqq 1 - P_X(0) \end{align*} where $P_X(t)$ is the Hilbert polynomial of $X$.

The geometric genus of a curve is \begin{align*} p+g(X) \coloneqq\dim_k H^0(X; \omega_X) \end{align*} where $\omega_X$ is the canonical sheaf.

Show that if $X$ is a curve, there is a single well-defined genus \begin{align*} g \coloneqq p_A(X) = p_G(X) = \dim_k H^1(X; {\mathcal{O}}_X) .\end{align*}

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

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 $p_a = g$.

Reviewing divisors:

The divisor group: $\operatorname{Div}(X) = {\mathbf{Z}} { \left[ \scriptstyle {X_{ \operatorname{cl}} } \right] } $
Degrees: $\deg(\sum n_i D_i) \coloneqq\sum n_i$, and
Linear equivalence: $D_1\sim D_2 \iff D_1 - D_1 = \operatorname{Div}(f)$ for some $f\in k(X)$ a rational function.
$D$ is effective if $n_i \geq 0$ for all $i$.
${\left\lvert {D} \right\rvert} \coloneqq\left\{{D'\in \operatorname{Div}(X) {~\mathrel{\Big\vert}~}D'\sim D}\right\}$ is the complete linear system of $D$.
${\left\lvert {D} \right\rvert} \cong {\mathbf{P}}H^0(X; {\mathcal{L}}(D))$
Dimensions of linear systems: $\ell(D) \coloneqq\dim_k H^0(X; {\mathcal{L}}(D))$ and $\dim {\left\lvert {D} \right\rvert} \coloneqq\ell(D) - 1$.
Relative differentials: $\Omega_X \coloneqq\Omega_{X_{/ {k}} }$ is the sheaf of relative differentials on $X$.
- The technical definition: $\Omega_{X_{/ {S}} } \coloneqq\Delta_{X_{/ {Y}} }^*({\mathcal{I}}/{\mathcal{I}}^2)$ where ${\mathcal{I}}$ is the sheaf of ideals defining the locally closed subscheme $\operatorname{im}(\Delta_{X_{/ {Y}} }) \subseteq X{ \operatorname{fp} }{Y} X$.
- On affine schemes: on the ring side, $\Omega_{B_{/ {A}} }\in {\mathsf{B}{\hbox{-}}\mathsf{Mod}}$ equipped with a differential $d: B\to \Omega{B_{/ {A}} }$, defined as $\left\langle{db{~\mathrel{\Big\vert}~}b\in B}\right\rangle_B / \left\langle{d(b_1+b_2) =db_1 + db_2, d(b_1 b_2) = d(b_1)b_2 + b_1 d(b_2), da = 0\, \forall a\in A}\right\rangle_B$.
- On curves, $\Omega_{X_{/ {Y}} }$ measures the “difference” between $K_X$ and $K_Y$.
Canonical sheaf: $\dim X = 1, \Omega_{X_{/ {k}} } \cong \omega_X$.
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.

Show that $D_1\sim D_2\implies\deg(D_1) = \deg(D_2)$.

Show that \begin{align*} {\left\lvert {D} \right\rvert} \rightleftharpoons{\mathbf{P}}H^0(X; {\mathcal{L}}(D)) ,\end{align*} so ${\left\lvert {D} \right\rvert}$ has the structure of the closed points of some projective space.

Show that if $D\in \operatorname{Div}(X)$ 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 ${\mathcal{L}}(D) \cong {\mathcal{O}}_X$.

\begin{align*} \ell(D) - \ell(K-D) = \deg(D) + (1-g) .\end{align*}

Show the following:

The divisor $K-D$ corresponds to $\omega_X \otimes{\mathcal{L}}(D) {}^{ \vee }\in \operatorname{Pic}(X)$.
$H^1(X; {\mathcal{L}}(D)) {}^{ \vee }\cong H^0(X; \omega_X \otimes{\mathcal{L}}(D) {}^{ \vee })$.
If $X$ is any projective variety, \begin{align*} H^0(X; {\mathcal{O}}_X) = k .\end{align*}

Show that if $X \subseteq {\mathbf{P}}^n$ is a curve with $\deg X = d$ and $D = X \cap H$ is a hyperplane section, then ${\mathcal{L}}(D) = {\mathcal{O}}_X(1)$ and $\chi({\mathcal{L}}(D)) = d + 1 - p_a$.

Show that if $g(X) = g$ then $\deg K_X = 2g-2$.

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

More definitions:

$X$ is rational iff birational to ${\mathbf{P}}^1$.
$X$ is elliptic if $g=1$.

Show that

If $\deg D > 2g-2$ then $D$ is nonspecial.
$p_a({\mathbf{P}}^1) = 0$.
A complete nonsingular curve is rational iff $X\cong {\mathbf{P}}^1$ iff $g(X) = 0$.
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 $\deg(D) = 0 \implies D\sim 0 \simplies p\sim q$. For (4), show $\ell(K) = p_g = 1$.

If $X$ is elliptic and $p\in X$, then there is a bijection \begin{align*} m_p: X & { \, \xrightarrow{\sim}\, }\operatorname{Pic}(X) \\ x &\mapsto {\mathcal{L}}(x-p) ,\end{align*} so $\operatorname{Pic}(X) \in {\mathsf{Grp}}$.

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