space of functions associated to a divisor
SpaceOfFunctionsAssociatedToADivisor
2004-02-27 12:07:38
2004-02-27 12:07:38
Definition
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Let $C/K$ be a curve defined over the field $K$, and $D$ a divisor for that curve. We define the \emph{space of functions associated to a divisor} by
\begin{align*}
\mathcal{L}(D)=\{f\in \ol{K}(C)^*:\text{div}(f)\geq -D\}\cup\{0\},
\end{align*}
where $\ol{K}(C)^*$ denotes the dual to the function field of $C$.
For any $D$, $\mc{L}(D)$ is a finite-dimensional vector space over $\ol{K}$, 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.