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

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

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

Title	Riemann-Roch theorem for curves
Canonical name	RiemannRochTheoremForCurves
Date of creation	2013-03-22 12:03:05
Last modified on	2013-03-22 12:03:05
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	12
Author	mathcam (2727)
Entry type	Theorem
Classification	msc 19L10
Classification	msc 14H99
Related topic	HurwitzGenusFormula