Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 113 / 2 / b / 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 isomorphismThe hyperplane divisor generates the class group of projective space, so