Hi!

Let me mention the following, which you may find helpful.

- Recall the following general facts: On a variety \( X \), say, over \( \mathbb{C} \), it holds that \( \text{CaCl}(X) \cong \text{Pic}(X) \). Moreover, if \( X \) is normal, then Cartier divisors on \( X \) correspond to (are identified with) locally principal Weil divisors on \( X \). Finally, if \( X \) is locally factorial (in particular if \( X \) is smooth), then we have isomorphisms \( \text{Cl}(X) \cong \text{CaCl}(X) \cong \text{Pic}(X) \).
- Consider also the following example: On the one hand, since \( \mathbb{C}[x_{1}, \dots, x_{n}] \) is a UFD, the (Weil) divisor class group of the affine \( n \)-space \( \mathbb{A}^{n}_{\mathbb{C}} \) over \( \mathbb{C} \) is trivial, i.e.

\[ \text{Cl} \big[ \text{Spec} \big( \mathbb{C}[x_{1}, \dots, x_{n}] \big) \big] = 0. \] On the other hand, the (Weil) divisor class group of the projective \( n \)-space \( \mathbb{P}^{n}_{\mathbb{C}} \) over \( \mathbb{C} \) is isomorphic to \( \mathbb{Z} \), i.e.

\[ \text{Cl} \big[ \text{Proj} \big( \mathbb{C}[x_{0}, x_{1}, \dots, x_{n}] \big) \big] = \mathbb{Z}. \] Finally, by the previous comment, we have also determined the corresponding Picard groups.