Divisor

Definition

Given a complex manifold $M$, a divisor on $M$ is a finite integral linear combination of the codimension 1 submanifolds in $M$. In the simple case where the manifold is a complex curve (and hence a real surface) the divisor is simply a finite integral linear combination of points. The coefficient of each point is termed its multiplicity or sometimes order.

Types of divisors

Principal divisor

Further information: Principal divisor

A divisor is said to be principal if there is a meromorphic function such that the points with nonzero coefficients in the divisor are precisely the zeroes and poles, and the coefficient of each codimension one submanifold is the multiplicity of the corresponding zero (with positive sign) or of the corresponding pole (with negative sign).

For a compact manifold, it turns out that every principal divisor must have degree zero.

Note that for a principal divisor, any two functions which have that as a principal divisor must have, as their ratio, a function with no zeroes and no poles. For a compact manifold, the only such functions are constant functions, and hence the principal divisor determines the meromorphic function upto multiplication by a scalar.

Effective divisor

Further information: Effective divisor

An effective divisor is a divisor for which there are no negative coefficients. Note that for a compact manifold, the only effective principal divisor is the zero divisor.

Constructs

Degree of a divisor

The degree of a divisor is defined as the sum of the coefficients of all the points.

Divisor class

Two divisors are said to be in the same divisor class if they differ by a principal divisor. The quotient of the Abelian group of all divisors by the Abelian group of principal divisors, is termed the divisor class group. The divisor class group of a compact manifold has the integers as a quotient because the group of principal divisors is contained in the gorup of degree zero divisors, which has quotient as the integers.