Algebraic Curves Lecture 20

The 20th (probably?) lecture of algebraic curves by Karl Christ

Reminder from before spring break
Castlenuova's Bound
Extremal Curves

Reminder from before spring break

We had a curve $C\subseteq \mathbb P^r$ and a divisor $D = H\cap C$ given by intersecting $C$ with a hyperplane $H$ . Set $\alpha_\ell = \operatorname{rank}(\ell D)$ , $E_\ell \subseteq H^0(C,\ell \cdot D)$ given as the image of $H^0(\mathbb P^r, \mathcal O_{\mathbb P^r}(\ell))\to H^0(C, \ell\cdot D)$ and $\beta_\ell =\operatorname{rank}(E_\ell)$ . Note that in particular $\alpha_\ell \geq \beta_\ell$ .

With these definitions we have that

\begin{aligned} \beta_\ell - \beta_{\ell - 1} &= h^0(\mathbb P^r,\mathcal O_{\mathbb P^r}(\ell)) - h6)(\mathbb P^r, I_D(\ell)) \\ &=: S_\ell, \end{aligned}

which is the "number of conditions imposed by $D$ on hyperplanes of degree $\ell$ . The long exact sequence on cohomology gives

\begin{aligned} 0\to H^0(\mathbb P^r, I_D(\ell)) \to H^0(\mathbb P^r, \mathcal O_{\mathbb P^r}(\ell)) \xrightarrow{\varphi} H^0(D,\mathcal O_{D}(\ell)), \end{aligned}

and $S_\ell = \operatorname{dim}(\operatorname{ker} \varphi)$ . We wanted to estimate $S_\ell$ , and we had this theorem that said as long as the points comprising $D$ are in general linear position (which we can assume as long as $H$ is generic) then $S_\ell \geq \min\{\ell(r - 1), 1\}$ implies

\begin{aligned} \beta_\ell - \beta_{\ell - 1} \geq S_\ell \geq \min\{d, \ell(r - 1) + 1\}. \end{aligned}

This is where we stopped.

Castlenuova's Bound

Now let's set $m = \left[\frac{d-1}{r-1}\right]$ , i.e. set $m$ to be the largest integer such that $m(r - 1) \leq d-1$ . We get

\begin{aligned} \alpha_1 &\geq \beta_1 \geq r \\ \alpha_2 &\geq \beta_2 \geq r + 2(r - 1) + 1 = 3r - 1 \\ &\hspace{5pt}\vdots \\ \operatorname{rank}(m\cdot D) = \alpha_m &\geq \beta_m \geq \sum^m_{i = 1}(i \cdot (r - 1) + 1) = \binom{m+1}{2}(r - 1) + m. \end{aligned}

Here's a trick inequality:

\begin{aligned} \binom{m+1}{2}(r-1) + m = \frac{m\left((m+1)(r - 1) + 2\right)}{2} > \frac{md}{2}. \end{aligned}

This means that $m\cdot D$ is non-special, and hence $\alpha_m = d\cdot m - g + 1$ . Our original motivation for this whole thing was to find a bound on the genus of $C$ , so rearranging, we get

\begin{aligned} g = dm + 1 - \alpha_m \leq d m + 1 - \binom{m+1}{2}(r-1) - m = \binom{m}{2}(r-1) + m\cdot \epsilon \end{aligned}

where $\epsilon$ is the integer required so that $d - 1 = (r - 1)m + \epsilon$ with $0\leq \epsilon < r-1$ . This is precisely Castlenuova's bound.

Theorem: (Castelnuovo's bound): Let

C\subseteq \mathbb P^r

be a non-degenerate curve of degree

d

. Then

g(C)\leq \binom{m}{2}(r - 1) + m\cdot \epsilon =: \pi(r, d).

Example:

$r = 2\implies \epsilon = 0\implies g \leq \binom{m}{2} = \binom{d- 1}{2}$ .
$r = 3$ so then $\lfloor \frac{d-1}{2} = m\rfloor = m$ .
- Case 1: $d = 2k +1$ , $m = k$ and $\epsilon = 0$ so $2\cdot \binom{k}{2} = k(k - 1)$ .
- Case 2: $d = 2k, m = k - 1, \epsilon = 1$ so $2\cdot \binom{k-1}{2} + (k-1) = (k-1)^2$ .

Observation: Fix $r$ . for large $d$ , we get asymptotically $\pi(r,d) \sim \frac{d^2}{2(r-1)}$ .

Extremal Curves

A curve is called (Castelnuovo) extremal if it satisfies $g(C) = \pi(r, d)$ . The only way this is possible is if

\begin{aligned} \alpha_\ell = \beta_\ell = \sum^\ell_{i=1} i (r-1) + 1 \end{aligned}

for all $\ell \leq m$ . Increasing $\ell$ by one increases $\beta_\ell$ by $\min\{d, \ell(r - 1) + 1\}.$ This implies

\begin{aligned} \varphi_\ell:H^0(\mathbb P^r, \mathcal O_{\mathbb P^r}(\ell)) \to H^0(C, \mathcal O_C(\ell)) \end{aligned}

is surjective.

Definition: In this case,

C

is called projectively normal.

C

is called $\ell$ -normal if

\varphi_\ell

is surjective.

Cor: Any extremal curve is projectively normal (by the above).

Example:

$C$ is $\ell$ -normal if and only if $C$ is embedded by complete linear series.
If $X$ is a genus $4$ curve, $L$ a curve of degree $7$ , then $h^0(X,L) = 7 - 4 + 1 = 4.$

Black box: A general such $L$ in (2) above is very ample hence gives an embedding $X\to \mathbb P^3$ with image of degree $7$ . Examining the map

\begin{aligned} \varphi_2:H^0(\mathbb P^3, \mathcal O_{\mathbb P^3}(2))\to H^0(X, L^{\oplus 2}), \end{aligned}

we see that the domain has dimension

\binom{5}{3} = 10

and the codomain has dimension

14 - 4 + 1 = 11,

hence

\varphi_2

cannot be surjective. This implies

\varphi_2(X)

1

-normal but not

2

-normal.