Divisors in A Riemann Surface
A Riemann surface is a 1-dimensional complex manifold, so its codimension 1 submanifolds are 0-dimensional. The divisors of a Riemann surface are the elements of the free abelian group of points on the surface.
Equivalently, a divisor is a finite linear combination of points of the surface with integer coefficients. The degree of a divisor is the sum of its coefficients.
We define the divisor of a meromorphic function f as
where R(f) is the set of all zeroes and poles of f, and sν is given by
A divisor that is the divisor of a meromorphic function is called principal. It follows from the fact that a meromorphic function has as many poles as zeroes, that the degree of a principal divisor is 0. Since the divisor of a product is the sum of the divisors, the set of principal divisors is a subgroup of the group of divisors. Two divisors that differ by a principal divisor are called linearly equivalent.
We define the divisor of a meromorphic 1-form similarly. Since the space of meromorphic 1-forms is a 1-dimensional vector space over the field of meromorphic functions, any two meromorphic 1-forms yield linearly equivalent divisors. The class of equivalence of these divisors is called the canonical divisor (usually denoted K).
The Riemann–Roch theorem is an important relation between the divisors of a Riemann surface and its topology.
Read more about this topic: Divisor (algebraic Geometry)
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