# 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

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

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

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

Title space of functions associated to a divisor SpaceOfFunctionsAssociatedToADivisor 2013-03-22 14:12:25 2013-03-22 14:12:25 mathcam (2727) mathcam (2727) 4 mathcam (2727) Definition msc 14H99