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 orientablesurface is a sphere, or the connected sum of $n$ tori. We say the spherehas 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 thedimension over $k$ of the vector space ${\mathrm{\Omega }}^1(X)$ of global regulardifferentials 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.