Algebraic Curves Lecture 2: Basic Notions

A discussion of the fundamental notions of algebraic curves: definition of a curve, facts about maps between smooth curves, divisors.

Two alternative ways to view curves
Some facts about projective varieties
Divisors
Principal divisors

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

Cor: 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.

Cor: 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

\begin{aligned} g = 1 - \chi(\mathcal O_X) = 1 - h^0(\mathcal O_X) + h^1(\mathcal O_X) = h^1(\mathcal O_X). \end{aligned}

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{aligned} D = \sum_{P\in X} D_P\cdot P \end{aligned}

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{aligned} f^*Y = \sum_{x\in X} \text{val}_{\mathcal O_{X,x}}(\pi)\cdot x \in \operatorname{Div}(X) \end{aligned}

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{aligned} f^*:\operatorname{Div}(Y)\to \operatorname{Div}(X) \end{aligned}

Remark: Analytically,

f

is given around

x

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{aligned} \deg(f^*D) = \deg(\pi)\cdot \deg(D). \end{aligned}

Principal divisors

Let $f\in K(X)$ [i.e. $f:X\to \mathbb P^1$ ] then define

\begin{aligned} \operatorname{div}(f) = f^*(0 - \infty). \end{aligned}

Cor: For

f

above,

\deg(\operatorname{div}(f)) = 0

Any divisor $D = \operatorname{div}(f)$ 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 $D + \operatorname{div}(f)$ is effective. This is invertible