Riemann-Roch theorem for curvesRiemannRochTheorem2001-12-12 17:31:452006-02-21 12:52:24Theorem\usepackage{amssymb}
\usepackage{amsmath}
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\usepackage{xypic}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 \PMlinkid{space of functions associated to a divisor}{SpaceOfFunctionsAssociatedToADivisor}.