Editing Riemann–Roch theorem (section)

====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.