I want to solve the following exercise:
Let $k$ be an algebraically closed field and consider a smooth projective irreducible curve $C$ over $k$. Set $S$ $:= C ×C$. Show that $p_{1}^* \oplus p^*_2 : \operatorname{Pic}(C)\oplus \operatorname{Pic}(C) \to \operatorname{Pic}(S)$ is injective, where $p_i : S \to C$ is the i-th projection.
I found a more general solution in the paper of Ischebeck, but as I do not understand German, I find it very difficult to follow the demonstration. I would also like to demonstrate it, if possible, in a more direct manner.
I have tried to exploit the isomorphism that exists between $\operatorname{Pic}(C)$ and the class group of Weil's divisors $\operatorname{Cl}(C)$ and tried to reason as one does for the classical example of $\mathbb{P}^1 \times \mathbb{P}^1$, but without having a clear idea of how to continue, since even having an equivalence between $C$ minus a point and an affine space $\operatorname{Spec}(A)$, I do not know two things well:
How to show that indeed the ring $A$ on which the affine space is defined is a UFD, so that then $\operatorname{Cl}(\operatorname{Spec}(A))=0$, and;
Even having that I am not given any isomorphism between $\operatorname{Cl}(C \times \operatorname{Spec}(A))$ and $\operatorname{Cl}(C)\times \operatorname{Cl}(\operatorname{Spec}(A))$, as happens in the example.
Would you have any suggestions on how such an exercise could be carried out, without exploiting Ischebeck's result?