Genus of a smooth curve (reviewed)

The genus of a smooth projective curve $C$ defined over a field $k$ is the dimension of the $k$-vector space of regular differentials $H^0 (C, \omega_C)$. When $k=\C$ this coincides with the topological genus of the corresponding Riemann surface.

The quantity defined above is sometimes also called the algebraic genus or the geometric genus of $C$. Because of our assumption on the smoothness of $C$, it coincides with the arithmetic genus $H^1 (C,\mathcal{O}_C) - H^0 (C,\mathcal{O}_C) + 1$.