# Riemann-Roch theorem

The Riemann-Roch theorem (after the mathematician Bernhard Riemann and his student Gustav Roch ) is a central statement of the theory of compact Riemann surfaces . It indicates how many linearly independent meromorphic functions with given zeros and poles exist on a compact Riemann surface. The theorem was later extended to algebraic curves , generalized even further, and is also being further developed in current research.

## divisor

In order to be able to prescribe zero and pole positions of a function at certain points, the term divisor is introduced. Let be a Riemann surface. A function is called a divisor if it only differs from zero at isolated points . ${\ displaystyle X}$${\ displaystyle D \ colon X \ rightarrow \ mathbb {Z}}$

The divisor of a meromorphic function is denoted by and is defined in such a way that the zero or pole order of in is assigned to each point : ${\ displaystyle f: X \ rightarrow \ mathbb {P} ^ {1}}$${\ displaystyle (f)}$${\ displaystyle x \ in X}$${\ displaystyle f}$${\ displaystyle x}$

${\ displaystyle \ left (f \ right) (x): = {\ begin {cases} 0, & f {\ mbox {holomorphic and not equal to zero in}} x \\ k, & f {\ mbox {has a zero of order }} k {\ mbox {in}} x \\ - k, & f {\ mbox {has a pole of order}} k {\ mbox {in}} x \\\ infty, & f {\ mbox {disappears in one Neighborhood of}} x \ end {cases}}}$

Thus, the divisor of a function is actually a divisor according to the first definition, if the function is different from the null function on each connected component . For a meromorphic 1-form on the divisor is defined as for a function. A divisor is called a canonical divisor if it can be written as a divisor of a meromorphic 1-form , i.e. if . ${\ displaystyle X}$ ${\ displaystyle \ omega}$${\ displaystyle X}$${\ displaystyle (\ omega)}$${\ displaystyle D}$${\ displaystyle (\ omega)}$${\ displaystyle D = (\ omega)}$

For a compact Riemann area, the degree of a divisor is defined by . The sum is finite, because due to the compactness of the carrier from isolated points it must be a finite set. ${\ displaystyle D}$${\ displaystyle \ textstyle \ deg D: = \ sum _ {x \ in X} D (x)}$

Let be a compact Riemann surface of topological gender and a divisor . Then: ${\ displaystyle X}$ ${\ displaystyle g \ in \ mathbb {N} _ {0}}$${\ displaystyle D}$${\ displaystyle X}$

${\ displaystyle \ ell (D) - \ ell (KD) = \ deg D + 1-g}$

${\ displaystyle K}$stands for any canonical divisor . denotes for a divisor the dimension of the vector space of the meromorphic functions to whose zero and pole positions are restricted by the divisor as follows: ${\ displaystyle X}$${\ displaystyle \ ell (E)}$${\ displaystyle E}$${\ displaystyle \ mathbb {C}}$${\ displaystyle L (E)}$${\ displaystyle X}$

${\ displaystyle L (E): = \ left \ {f: X \ rightarrow \ mathbb {P} ^ {1} {\ mbox {meromorph}} \, | \, \ left (f \ right) (x) \ geq -E (x) \; \ forall \, x \ in X \ right \}}$

For non-singular projective algebraic curves over an algebraically closed field , the Riemann-Roch theorem is usually formulated with the help of the cohomology theory. ${\ displaystyle X}$ ${\ displaystyle K}$

${\ displaystyle \ dim _ {K} H ^ {0} \ left (X, {\ mathcal {O}} _ {X} \ right) - \ dim _ {K} H ^ {1} \ left (X, {\ mathcal {O}} _ {X} \ right) = 1-g}$

${\ displaystyle {\ mathcal {O}} _ {X}}$is the sheaf of regular features on . Instead of the topological gender, the arithmetic gender of the curve occurs, which in the case coincides with the topological gender . The duality theorem of Serre states that the formulation in the case with that matches the section on Riemannian surfaces. ${\ displaystyle X}$${\ displaystyle K = \ mathbb {C}}$${\ displaystyle K = \ mathbb {C}}$

## Consequences

• As a first classification result, it immediately follows that every Riemann surface is isomorphic by gender to the Riemann sphere , so in particular only one holomorphic structure can be defined on the sphere . For non-singular projective curves of gender it applies accordingly that they are birationally equivalent .${\ displaystyle 0}$${\ displaystyle \ mathbb {P} ^ {1}}$${\ displaystyle {S} ^ {2}}$${\ displaystyle 0}$${\ displaystyle \ mathbb {P} ^ {1}}$
• The Riemann-Hurwitz formula about the mapping behavior of holomorphic functions between two compact Riemann surfaces or about the mapping behavior of morphisms between two non-singular projective curves.
• An embedding theorem: Every compact Riemann surface or every non-singular projective curve can be embedded in the projective space .${\ displaystyle \ mathbb {P} ^ {3}}$