Picard group of product of spaces Suppose $X,Y$ are varieties over an algebraically closed field $k$. Can we compute $\operatorname{Pic}(X \times_k Y) $ in terms of $\operatorname{Pic}(X),\operatorname{Pic}(Y)$? It seems that $\operatorname{Pic}(X \times_k Y) \cong \operatorname{Pic}(X) \times \operatorname{Pic}(Y)$ is not quite right, but I cannot figure out a counterexample. (I thought one might construct UFDs $A,B$, but their tensor product is not UFD).

In some cases it is true:

If $X$ is a projective variety over an algebraically closed field $k$ such that $H^1(X,\mathcal{O}_X)=0$, and $T$ is a connected scheme of finite type over $k$, then $\mathrm{Pic}(X \times T) \cong \mathrm{Pic}(X) \times \mathrm{Pic}(T)$. This is exercise III.12.6. in Hartshorne's book.

Since $\mathrm{Pic}(\mathbb{A}^1)=0$, a special case of the question is the "homotopy invariance" $\mathrm{Pic}(X \times \mathbb{A}^1) \cong \mathbb{Pic}(X)$. This holds when $X$ is normal, but not in general (SE/432217).