# Riemann-Roch theorem for curves

Let $C$ be a projective nonsingular curve over an algebraically closed field. If $D$ is a divisor on $C$, then

 $\ell(D)-\ell(K-D)={\rm deg}(D)+1-g$

where $g$ is the genus of the curve, and $K$ is the canonical divisor ($\ell(K)=g$). Here $\ell(D)$ denotes the dimension of the http://planetmath.org/node/SpaceOfFunctionsAssociatedToADivisorspace of functions associated to a divisor.

Title Riemann-Roch theorem for curves RiemannRochTheoremForCurves 2013-03-22 12:03:05 2013-03-22 12:03:05 mathcam (2727) mathcam (2727) 12 mathcam (2727) Theorem msc 19L10 msc 14H99 HurwitzGenusFormula