Genus of a curve

Remarks

For a smooth projective connected curve, Serre duality gives $g(C) = \dim_k H^1(C, \mathcal{O}_C)$, so the genus equals the arithmetic genus. Example: on $\mathbb{P}^1$ one has $\Omega = \mathcal{O}(-2)$, which has no nonzero global sections, so $g(\mathbb{P}^1) = 0$. Example: the smooth projective curve $E$ with affine equation $y^2 = x^3 - x$, in characteristic not $2$, has the form $\omega = dx/y$, regular and nowhere zero on $E$, and $g(E) = 1$. Non-example of the equality with the arithmetic genus: the nodal cubic $y^2 = x^2(x + 1)$ is singular, its normalization is $\mathbb{P}^1$, so its geometric genus is $0$, while its arithmetic genus is $1$. Over $\C$, the genus equals the topological genus of $C(\C)$ as a closed orientable surface.