# genus

“Genus” has number of distinct but compatible definitions.

In topology^{}, if $S$ is an orientable surface, its genus $g(S)$ is the number of “handles” it has.
More precisely, from the classification of surfaces^{}, we know that any orientable
surface is a sphere, or the connected sum^{} of $n$ tori. We say the sphere
has genus 0, and that the connected sum of $n$ tori has genus $n$
(alternatively, genus is additive with respect to connected sum, and the genus of a torus is 1).
Also, $g(S)=1-\chi (S)/2$ where $\chi (S)$ is the Euler characteristic^{} of $S$.

In algebraic geometry^{}, the genus of a smooth projective curve $X$ over a field $k$ is the
dimension^{} over $k$ of the vector space ${\mathrm{\Omega}}^{1}(X)$ of global regular^{}
differentials on $X$. Recall that a smooth complex curve is also a Riemann surface,
and hence topologically a surface. In this case, the two definitions of genus coincide.

Title | genus |
---|---|

Canonical name | Genus |

Date of creation | 2013-03-22 12:03:45 |

Last modified on | 2013-03-22 12:03:45 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 10 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 14H99 |