Lecture notes from Karl Christ’s Algebraic Curves course, UT Austin, Spring 2024. Live-TeXed during lecture; any errors are mine.
Let \(k\) be an algebraically closed field (unless otherwise stated) of characteristic 0.
Remark
Whenever you have these two hypotheses (algebraically closed and characteristic 0), you don’t lose anything by simply taking \(\mathbb C\) to be the complex numbers. This is the Lefschetz principle. This isn’t strictly true, but is a good slogan.
Definition
A curve is a scheme of pure dimension 1 over \(k\). This is often called an abstract curve. Unless otherwise stated we will always assume that a curve is smooth and connected.
In particular, our curves will always be irreducible (this follows from the connected hypothesis + Zariski topology weirdness).
Two alternative ways to view curves
- The category of curves is equivalent to that of Riemann-surfaces, i.e. complete 1-dimensional \(\mathbb C\)-manifolds. (Included here is a picture of a genus 2 surface.) I was unfamiliar with the term “complete manifold”, so here’s a convenient link to the Wikipedia page.
- The category of curves is equivalent to the category of finitely generated field extensions of \(K\) of transcendence degree 1. This is given simply by associating to a curve its field of rational functions.
The second statement above is equivalent to saying that the field of rational functions of a curve entirely determine the curve’s isomorphism class. This in particular fails for higher dimensional varieties; any two birational varieties have isomorphic fields of rational functions. Thus
Corollary
Any birational morphism \(f:X\to Y\) between (smooth) curves is an isomorphism.
Again, this clearly fails for non-smooth curves. Take the cuspoidal cubic \(C = V(x^3 = y^2)\) for instance; it is birationally equivalent to \(K = \mathbb A^1\) via \(t \mapsto (t^2, t^3)\) (one should projectivize this argument but you get the idea). Alternatively, take a nodal cubic and blow up at the node to resolve it. The blowup map is then an isomorphism away from the singularity.
Some facts about projective varieties
Theorem
Let \(f:X\to Y\) be a morphism of projective varieties \(X\) and \(Y\). Then \(f\) is closed.
Corollary
Any non-constant morphism \(f:X\to Y\) between projective curves \(X\) and \(Y\) if finite and surjective.
Recall the difference betwen finite and finite type: it’s about generation as an algebra vs as a module. A morphism \(f:X\to Y\) is finite type if it is affine and if over each affine \(U\subseteq Y\) \(f_*\mathcal O_X(U)\) is a finitely generated algebra of \(\mathcal O_Y(U)\). The morphism is finite if it is affine and if over each affine \(U\subseteq Y\) \(f_*\mathcal O_X(U)\) is finitely generated module of \(\mathcal O_Y(U)\).
Definition
The genus of a curve \(X\) is
\begin{align*} g = 1 - \chi(\mathcal O_X) = 1 - h^0(\mathcal O_X) + h^1(\mathcal O_X) = h^1(\mathcal O_X). \end{align*}The three equalities above represent the equivalence of arithmetic and geometric genus for smooth curves.
Divisors
A divisor on a curve is a formal linear combination of points on \(X\):
\begin{align*} D = \sum_{P\in X} D_P\cdot P \end{align*}where \(D_P \in \mathbb Z\) and only finitely many \(D_P \neq 0\).
- The degree of a divisor is \(\sum_{P\in X} D_P\).
- A divisor is called effective if \(D_P\geq 0\) for all \(P\in X\).
Remark
Effective divisors correspond uniquely to 0-dimensional subschemes of \(X\).
If \(f:X\to Y\) is a non-constant morphism of curves \(X\) and \(Y\), then
\begin{align*} f^*Y = \sum_{x\in X} \text{val}_{\mathcal O_{X,x}}(\pi)\cdot x \in \operatorname{Div}(X) \end{align*}where \(\pi\) is a local parameter of \(\mathcal O_Y,y\) (also known as the uniformizer. Extending this linearly gives us a pullback of divisors on \(Y\) to divisors on \(X\):
\begin{align*} f^*:\operatorname{Div}(Y)\to \operatorname{Div}(X) \end{align*}Remark
Analytically, \(f\) is given around \(x\) as \(z\mapsto z^n\). Then \(n = \operatorname{val}_{\mathcal O_{X,x}}(\pi).\)
Proposition
If \(f:X\to Y\) is a non-constant morphism of curves, then
\begin{align*} \deg(f^*D) = \deg(\pi)\cdot \deg(D). \end{align*}Principal divisors
Let \(f\in K(X)\) [i.e. \(f:X\to \mathbb P^1\)] then define
\begin{align*} \div(f) = f^*(0 - \infty). \end{align*}Corollary
For \(f\) above, \(\deg(\div(f)) = 0\).
Any divisor \(D = \div(f)\) is called a principal divisor. Two divisors \(D\) and \(E\) are linearly equivalent if \(D - E\) is principal. In this case, we write \(D \sim E\).
Vague Note: We call this “linearly equivalence” because the two divisors vary by a linear family, since \(\mathbb P^1\) is our prototype of a “line”.
To any divisor \(D\) we can associate an invertible sheaf, denoted \(\mathcal O_X(D)\), defined to be the sheaf of rational functions \(f\) such that \(D + \div(f)\) is effective. This is invertible