 Solution

Let $U={\mathbb{P}}^2\setminus C$. The localization sequence for the divisor class group givesEvery prime Weil divisor of $U$ extends by closure to one of ${\mathbb{P}}^2$, and the only removed prime Weil divisor is $C$. Rational functions have the same function field on the two spaces, so the kernel on divisor class groups consists exactly of multiples of $[C]$.

The divisor class group of ${\mathbb{P}}^2$ is $\mathbb{Z}$, generated by the class $[H]$ of a line. To see the degree identification, if a plane curve $D$ has degree $e$ and homogeneous equation $F_D$, then $F_D/{\ell }^e$, for a line equation $\ell$, is a rational function with principal Weil divisor $D-eH$. Degrees of principal Weil divisors are zero, so $[H]$ has infinite order. In particular, $[C]=d[H]$. The localization sequence therefore yieldsThis is the divisor class group of a plane-curve complement. It includes $d=1$, when the group is zero, and does not require the removed curve to be nonsingular.