= Solution
The scheme $\mathbb P_k^3$ is Noetherian, integral, separated, and regular. Every open subscheme inherits these properties, so $U=\mathbb P_k^3\setminus C$ satisfies $(\star)$. Because $C$ has dimension one, it has codimension two in $\mathbb P_k^3$ and contains no <prime Weil divisor>. The <localization sequence for the divisor class group> therefore makes restriction an isomorphism
$$
\operatorname{Cl}(\mathbb P_k^3)\xrightarrow{\sim}\operatorname{Cl}(U).
$$
The <hyperplane divisor> generates the class group of projective space, so
$$
\boxed{\operatorname{Cl}(U)\cong\mathbb Z.}
$$
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