Vipul: New page: ==Statement== Let $S$ be a fact about::compact Riemann surface. Let $K$ be a fact about::canonical divisor for $S$ , and $D$ be any [[fa...

2008-09-12T20:53:56Z

New page: ==Statement== Let <math>S</math> be a fact about::compact Riemann surface. Let <math>K</math> be a fact about::canonical divisor for <math>S</math>, and <math>D</math> be any [[fa...

New page

==Statement==

Let <math>S</math> be a [[fact about::compact Riemann surface]]. Let <math>K</math> be a [[fact about::canonical divisor]] for <math>S</math>, and <math>D</math> be any [[fact about::divisor]]. Let <math>L(D)</math> denotes the [[fact about::L-space of a divisor|L-space]] of <math>D</math>, i.e., the vector space of [[fact about::meromorphic function]]s on <math>S</math> that are either zero or have divisor greater than or equal to <math>D</math>. Then:

<math>\operatorname{dim} L(-D) = \operatorname{deg}(D) - g + 1 + \operatorname{dim} L(D - K)</math>.

==Related facts==

===Applications===

* [[Riemann's inequality]]
* [[Genus zero Riemann surface is conformally equivalent to Riemann sphere]]
* [[Every compact Riemann surface is a branched cover of the Riemann sphere]]

Riemann-Roch theorem - Revision history

Vipul: New page: ==Statement== Let S be a fact about::compact Riemann surface. Let K be a fact about::canonical divisor for S, and D be any [[fa...

Vipul: New page: ==Statement== Let $S$ be a fact about::compact Riemann surface. Let $K$ be a fact about::canonical divisor for $S$ , and $D$ be any [[fa...