 Divisor class group of a plane-curve complement

If $C$ is an irreducible plane curve of degree $d$, the localization sequence for the divisor class group quotients $\mathrm{Cl}({\mathbb{P}}^2)=\mathbb{Z}[H]$ by the class $[C]=d[H]$. Nonsingularity of the removed curve is not required; irreducibility makes it a single removed prime Weil divisor.