Editing Riemann–Roch theorem (section)

===Examples===
The theorem will be illustrated by picking a point <math>P</math> on the surface in question and regarding the sequence of numbers

:<math>\ell(n\cdot P),n\ge0</math>

i.e., the dimension of the space of functions that are holomorphic everywhere except at <math>P</math> where the function is allowed to have a pole of order at most <math>n</math>. For <math>n=0</math>, the functions are thus required to be [[entire function|entire]], i.e., holomorphic on the whole surface <math>X</math>. By [[Liouville's theorem (complex analysis)#On compact Riemann surfaces|Liouville's theorem]], such a function is necessarily constant. Therefore, <math>\ell(0)=1</math>. In general, the sequence <math>\ell(n\cdot P)</math> is an increasing sequence.

====Genus zero====
The [[Riemann sphere]] (also called [[complex projective line]]) is [[simply connected]] and hence its first singular homology is zero. In particular its genus is zero. The sphere can be covered by two copies of <math>\Complex</math>, with [[transition map]] being given by

:<math>\Complex\setminus\{0\}\ni z\mapsto\frac{1}{z}\in\Complex\setminus\{0\}</math>.

Therefore, the form <math>\omega = dz</math> on one copy of <math>\mathbb C</math> extends to a meromorphic form on the Riemann sphere: it has a double pole at infinity, since

:<math>d\left(\frac1z\right)=-\frac1{z^2}\,dz</math>

Thus, its canonical divisor is <math>K:=\operatorname{div}(\omega)=-2P</math> (where <math>P</math> is the point at infinity).

Therefore, the theorem says that the sequence <math>\ell(n\cdot P)</math> reads

: 1, 2, 3, ... .

This sequence can also be read off from the theory of [[partial fraction]]s. Conversely if this sequence starts this way, then <math>g</math> must be zero.

====Genus one====
[[File:Torus_cycles2.svg|right|thumb|A torus]]
The next case is a Riemann surface of genus <math>g=1</math>, such as a [[torus]] <math>\Complex/\Lambda</math>, where <math>\Lambda</math> is a two-dimensional [[lattice (group)|lattice]] (a group isomorphic to <math>\Z^2</math>). Its genus is one: its first singular homology group is freely generated by two loops, as shown in the illustration at the right. The standard complex coordinate <math>z</math> on <math>C</math> yields a one-form <math>\omega=dz</math> on <math>X</math> that is everywhere holomorphic, i.e., has no poles at all. Therefore, <math>K</math>, the divisor of <math>\omega</math> is zero.

On this surface, this sequence is

:1, 1, 2, 3, 4, 5 ... ;

and this characterises the case <math>g=1</math>. Indeed, for <math>D=0</math>, <math>\ell(K-D)=\ell(0)=1</math>, as was mentioned above. For <math>D=n\cdot P</math> with <math>n>0</math>, the degree of <math>K-D</math> is strictly negative, so that the correction term is 0. The sequence of dimensions can also be derived from the theory of [[elliptic function]]s.

====Genus two and beyond====
For <math>g=2</math>, the sequence mentioned above is

:1, 1, ?, 2, 3, ... .

It is shown from this that the ? term of degree 2 is either 1 or 2, depending on the point. It can be proven that in any genus 2 curve there are exactly six points whose sequences are 1, 1, 2, 2, ... and the rest of the points have the generic sequence 1, 1, 1, 2, ... In particular, a genus 2 curve is a [[hyperelliptic curve]]. For <math>g>2</math> it is always true that at most points the sequence starts with <math>g+1</math> ones and there are finitely many points with other sequences (see [[Weierstrass point]]s).