Lecture notes from Karl Christ’s Algebraic Curves course, UT Austin, Spring 2024. Live-TeXed during lecture; any errors are mine.
Correction from last time
- The dimension of a variety is the degree of a Hilbert polynomial.
- The leading coefficient of the Hilbert polynomial is \(\frac{d}{n!}\).
Final two corollaries
We finished with the following observation, a corollary following from Riemann-Roch:
Corollary
- Any line bundle of degree \(\geq 2g\) is base point free
- Any line bundle of degree \(\geq 2g+1\) is very ample
- Any curve can be embedded in \(\mathbb P^{g+1}\).
- A line bundle is ample if and only if its degree is positive.
You get the dimension of the embedding above in (3) directly from Riemann Roch: RR says \(h^0(L) = d - g + 1\), so if \(d = 2g + 1\) then \(h^0(L) = g + 2\).
The last corollary we’ll mention, at least for now, is the following.
Corollary
For any set of points \(\{p_1, ..., p_k\}\) in \(X\), \(X\setminus \{p_1, ..., p_k\}\) is affine.
Proof
Set \(D = \sum_i P_i\in \textrm{Div}(X)\). Let \(L = \mathcal O_X(D)^{\otimes \ell}\) be very ample.
Embedd \(X\) in \(\mathbb P^r\) via \(\varphi_L\). Then there is a hyperplane \(H\subseteq \mathbb P^r\) such that \(H|_{\varphi_L(X)} = \ell \cdot D\). Thus \(X\setminus D = X|_{\mathbb P^r \setminus H}\) is affine.
Note that this is definitely not true in general for varieties, even if you replace “points” with “divisors”. For example, \(\mathbb P^2\) minus a point is not affine. If you take \(\mathbb P^2\) blown up at a point and then remove the exceptional divisor, you get the same thing as \(\mathbb P^2\) with a point removed; so it’s still not affine.
The proof above works because, if you take any effective divisor \(D\), then you have a global section of \(\mathcal O_X(D)\). This is because you can find a regular function which vanishes on \(D\).
Example: (genus 1 curves). Let \(X\) be a curve of genus 1, \(p\in X\), and consider \(L_n = \mathcal O_X(n\cdot p)\) for \(n\mathbb N\). Riemann Roch says that \(h^0(L_n) = d - g + 1 = d\) as soon as \(d\geq 2g - 1 = 1\). A corollary from last week says
\begin{align*} \deg(K_X) = 2g - 2 = 0 \end{align*}and
\begin{align*} h^0(K_X) = g = 1, \end{align*}hence \(K_X \cong \mathcal O_X\).
If \(n=1\), then \(h^0(L_n) = 1\).
If \(n=2\) then \(h^0(L_2) = 2\), and hence we get 2:1 a map \(X\xrightarrow{2:1} \mathbb P^1\). Thus \(H^0(L_2) = textrm{span}\{1, x\}\).
If \(n=3\), then \(h^0(L_3) = 3\). We get a new global section \(y\) with a triple pole at \(p\). \(L_3\) defines a map \(\varphi_{L_3}:X\to \mathbb P^2\) which can be written as an affine chart \([z^3 : zx : y]\), where \(x = x/z^2\) and \(y = y/z^3\). Here \(z\) is a local coordinate which vanishes to order \(1\) at \(p\). The intersection of \(z = 0\) with the image of the map \(\varphi_{L_3}\) will be \([0:0:1]\). In particular, we have a “globally defined” marked point on \(X\).
If \(n=4\), \(h^0(L_4) = 4\), the new section is \(x^2\).
If \(n=5\), \(h^0(L_5) = 5\), the new section is \(x\cdot y\).
If \(n=6\), \(h^0(L_4) = 4\), but two new sections \(x^3\) and \(y^2\). This tells you that you must have a linear relation between all the sections now; i.e. we have an equation of the form
\begin{align*} y^2 + a_1xy + a_3y = x^3 + a_2 x^2 + a_4x + a_6 \end{align*}where \(x\) and \(y\) have been rescaled so the coefficients of \(x^3\) and \(y^2\) are \(1\). This is some standard form of an elliptic curve.
Remark
A genus \(1\) curve + a marked point \(p\) is called an elliptic curve. It has a group structure compatible with the algebraic structure, making it an Abelian variety.
Not all distinct choices of \(a_1,...,a_5\) yield distinct isomorphism classes of curves, and some choices yield singular curves. Nonetheless, we get a map \(\mathbb A^5\dashrightarrow \mathcal M_{1,1}\).
Remark
This shows that \(\mathcal M_{1,1}\) (the moduli space of genus one curves with one marked point, i.e. the moduli space of elliptic curves) is unirational. It’s not particularly interesting here because \(\mathcal M_{1,1}\) is just \(\mathbb A^1\), but as the genus and number of marked points increases this becomes more interesting.
Adjunction
Suppose we have \(Y\subseteq X\) a subvariety of a variety \(X\). On any smooth variety \(X\) we can define the canonical sheaf \(K_X = \bigwedge^n_{i=1}\Omega_{X}\).
Example: \(K_{\mathbb P^n} = \mathcal O_{\mathbb P^n}(-n-1).\)
In an open standard chart \(u_1,...,u_n\) of \(\mathbb P^n\), \(K_{\mathbb P^n}\) is generated by \(du_1 \wedge ... \wedge du_n\). On a different chart with standard coordinates \(w_1,...,w_n\), the transition functions between the charts are something like \(u_1 = \frac{1}{w_i}\) and \(u_j = \frac{w_j}{w_i}\). Over this \(w_1,...,w_n\) chart \(K_X\) is generated by \(dw_1\wedge ... \wedge dw_n\), and
\begin{align*} du_1 = d\frac{1}{w_i} = -\frac{1}{w_i^2} dw_i \end{align*}and
\begin{align*} du_i = d\frac{w_i}{w_j} = \frac{1}{w_i}dw_j - \frac{w_j}{w_i^2}dw_j. \end{align*}Checking \(du_1\wedge ...\wedge du_n\), applying bilinearity and canceling accordingly, you should end up with something like
\begin{align*} du_1 \wedge ...\wedge du_n = -\frac{1}{w_1^{n+1}}\cdot \left(dw_1\wedge ...\wedge dw_n\right). \end{align*}The point of this calculation is that I end up with a pole of order \(n+1\).