Genus 2001-12-21 03:41:33 2002-12-04 12:49:33 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} ``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 $\Omega^1(X)$ 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.