PlanetMath: space of functions associated to a divisor

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space of functions associated to a divisor

(Definition)

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

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

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

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

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14H99 (Algebraic geometry :: Curves :: Miscellaneous)

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