Genus of a smooth projective curve
= Genus of a smooth projective curve
{title2=$g=\dim_kH^1(C,\mathcal O_C)$}
For a <smooth projective curve> with $H^0(C,\mathcal O_C)=k$, its genus is $\dim_kH^1(C,\mathcal O_C)$, equivalently $\dim_kH^0(C,\Omega^1_C)$ by <Serre duality>. It equals the <arithmetic genus> $1-\chi(\mathcal O_C)$. Without the condition on constants, the general <Riemann-Roch theorem> instead uses $\chi(\mathcal O_C)=h^0(\mathcal O_C)-h^1(\mathcal O_C)$.