Riemann-Roch 02: Divisors and Degree

Riemann-Roch notes / II

Divisors before the theorem

Before the theorem can count anything, zeros and poles have to be turned into a finite object that can be added, compared, and moved.

A divisor is a finite formal integer sum of points,

\[D=\sum_P n_P P.\]

It is deliberately simple. A point receives an integer coefficient; all but finitely many coefficients are zero. This is enough to record the local orders of zeros and poles of meromorphic functions.

The degree is

\[\deg(D)=\sum_P n_P.\]

The support is the set of points where \(n_P\ne 0\). A divisor is effective, written \(D\ge 0\), if every coefficient is nonnegative.

Principal divisors

If \(f\) is a nonzero meromorphic function, define

\[(f)=\sum_P \operatorname{ord}_P(f)P.\]

Positive order means a zero, negative order means a pole. Such divisors are called principal.

Basic fact

On a compact Riemann surface,

\(\deg((f))=0.\)

One way to see this is to use the logarithmic differential \(df/f\). Its residue at \(P\) is \(\operatorname{ord}_P(f)\). The residue theorem gives

\[0=\sum_P \operatorname{res}_P(df/f)=\sum_P \operatorname{ord}_P(f).\]

The sign convention

The condition defining \(L(D)\) is

\[(f)+D\ge 0.\]

Thus positive coefficients of \(D\) allow poles. For instance, if

\[D=3P+Q,\]

then \(f\in L(D)\) may have a pole of order at most three at \(P\) and a pole of order at most one at \(Q\). If

\[D=3P-2Q,\]

then \(f\) may still have a pole of order at most three at \(P\), but it must vanish to order at least two at \(Q\).

Example on the projective line

On \(\mathbb P^1\), the affine coordinate satisfies

\[(z)=0-\infty.\]

Consequently

\[(z^m)=m\cdot 0-m\cdot\infty.\]

This is why powers of \(z\) live naturally in spaces of the form \(L(n\infty)\).

Two principal divisors that are worth knowing

First take distinct finite points \(a,b,c\in\mathbb C\) and

\[f(z)=\frac{(z-a)^2}{(z-b)^3(z-c)}\]

on \(\mathbb P^1\). The denominator has degree two more than the numerator, so \(f\) has a zero of order two at infinity. Thus

\[(f)=2[a]+2[\infty]-3[b]-[c].\]

This example is a useful antidote to the habit of forgetting the point at infinity.

Second, on the elliptic curve

\[E: y^2=x^3-x\]

with point at infinity \(O\), the coordinate functions have

\[(x)=2(0,0)-2O,\]

and

\[(y)=(0,0)+(1,0)+(-1,0)-3O.\]

The pole orders \(2\) and \(3\) at \(O\) are the first hints that \(x\) and \(y\) will generate the spaces \(L(2O)\) and \(L(3O)\) later.

The definition is modest, but it is the bookkeeping that makes the theorem possible: Riemann-Roch counts functions by reading this divisor data globally.