space of functions associated to a divisor

Let $C/K$ be a curve defined over the field $K$, and $D$ a divisor for that curve. We define the space of functions associated to a divisor by

$ℒ(D)=\{f\in \overline{K}{(C)}^{*}:\text{div}(f)\ge -D\}\cup \{0\},$

where $\overline{K}{(C)}^{*}$ denotes the dual to the function field of $C$.

For any $D$, $ℒ(D)$ is a finite-dimensional vector space over $\overline{K}$, the algebraic closure of $K$, and we denote its dimension by $\mathrm{\ell }(D)$, a somewhat ubiquitous number that, for example, appears in the Riemann-Roch theorem for curves.

Title	space of functions associated to a divisor
Canonical name	SpaceOfFunctionsAssociatedToADivisor
Date of creation	2013-03-22 14:12:25
Last modified on	2013-03-22 14:12:25
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	4
Author	mathcam (2727)
Entry type	Definition
Classification	msc 14H99