Algebraic Curves Lecture 20

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{align*} \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{align*}

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

\begin{align*} 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{align*}

and \(S_\ell = \dim(\ker \varphi)\). 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{align*} \beta_\ell - \beta_{\ell - 1} \geq S_\ell \geq \min\{d, \ell(r - 1) + 1\}. \end{align*}

This is where we stopped.

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{align*} \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{align*}

Here’s a trick inequality:

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

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{align*} 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{align*}

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.

Example:

1. \(r = 2\implies \epsilon = 0\implies g \leq \binom{m}{2} = \binom{d- 1}{2}\).
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 \(\pi(r,d) \sim \frac{d^2}{2(r-1)}\).

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

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

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

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

is surjective.

Example:

1. \(C\) is \(\ell\)-normal if and only if \(C\) is embedded by complete linear series.
2. 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{align*} \varphi_2:H^0(\mathbb P^3, \mathcal O_{\mathbb P^3}(2))\to H^0(X, L^{\oplus 2}), \end{align*}

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 \(\varphi_2(X)\) is \(1\)-normal but not \(2\)-normal.