# Riemann-Roch theorem for curves

Let $C$ be a projective nonsingular curve over an algebraically closed field. If $D$ is a divisor on $C$, then

$$\mathrm{\ell}(D)-\mathrm{\ell}(K-D)=\mathrm{deg}(D)+1-g$$ |

where $g$ is the genus of the curve, and $K$ is the canonical divisor ($\mathrm{\ell}(K)=g$). Here $\mathrm{\ell}(D)$ denotes the dimension of the http://planetmath.org/node/SpaceOfFunctionsAssociatedToADivisorspace of functions associated to a divisor.

Title | Riemann-Roch theorem for curves |
---|---|

Canonical name | RiemannRochTheoremForCurves |

Date of creation | 2013-03-22 12:03:05 |

Last modified on | 2013-03-22 12:03:05 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 12 |

Author | mathcam (2727) |

Entry type | Theorem |

Classification | msc 19L10 |

Classification | msc 14H99 |

Related topic | HurwitzGenusFormula |