# 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

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

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

For any $D$, $\mathcal{L}(D)$ 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 |