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).
 A: This is to elaborate on Elencwajg's nice answer. Ischebeck's sequence holds under the following general hypotheses: $k$ is any field, $X$ and $Y$ are non-empty, normal, locally of finite type and geometrically integral $k$-schemes. The proof of exactness at the middle of the sequence is Ischebeck's, while exactness at the first nontrivial term follows from a Galois descent argument of Colliot-Thelene and Sansuc's which appears in their 1977 paper on $R$-equivalence on tori (see proof of Lemma 11 on pp.188-189). By the way, the preliminary results proved by Ischebeck's in order to derive his theorem are actually rediscoveries of (particular cases of) results of Raynaud which appear in the Errata list for EGA IV_4 (Err 53). When one of the factors, say $Y$, is rational over the separable closure of $k$ and has a $k$-rational point, Sansuc 
proved the formula $Pic\, X\oplus Pic\, Y\simeq Pic (X\times_{k}Y)$ in his 1981 Crelle paper using a method that requires smoothness and differs completely from Ischebeck's (note that, by Ischebeck and Raynaud, normality suffices). Let me note that neither Sansuc nor Colliot-Thelene-Sansuc (in the 1977 paper alluded to above) cite Ischebeck's paper. Sansuc's statement can be generalized by replacing the smoothness hypothesis by normality and the existence of a $k$-rational point by the condition that the separable indices of $X$ and $Y$ are coprime. The reliance on the existence of a $k$-rational point on $Y$ can in fact be removed without much effort directly from Sansuc's statement by using a standard restriction-corestriction argument, as pointed out to me by Mathieu Florence. Beyond the base field case, one might further ask for a relative version of the question posed by Li Yutong, i.e., let $f\colon  X\to S$ be a morphism of schemes. Then what are the kernel and cokernel of the canonical homomorphism $Pic\, X\oplus Pic\, Y\to Pic (X\times_{S}Y)$? In this rarefied setting, assuming $S$ to be normal and integral (with function field $K$) seems reasonable (because of well-known counterexamples to many sorts of questions over non-normal bases). Under suitable conditions, there should exist an exact sequence of the sort
$$
Pic\,S\to Pic\, X\oplus Pic\, Y\to Pic (X\times_{S}Y)\to Pic\,(R(X_{K})\otimes_{K}R(Y_{K})),
$$
where $R(X_{K}), R(Y_{K})$ are the rings of rational functions of the generic fibers. So, if one lets $S$ shrink to $Spec\, K$, then Ischebeck's sequence is recovered (by the way, if the reader suspects that the localization sequence for the Picard group is at play above, then he/she is right indeed...)
A: Ischebeck has proved that given an algebraically closed  field $k$ and two  normal integral algebraic $k$-schemes $X,Y$  there is an exact sequence of groups $$  0\to Pic(X)\times Pic (Y)\to Pic(X\times Y)\to Pic (k(X)\otimes _k k(Y))           $$  Note that neither variety is supposed complete, nor affine, nor...
This is quite interesting because although other users have shown that the Picard group of the product of two varieties may be  bigger than the product of that of the factors, Ischebeck gives  a bound for the discrepancy.
In particular if one of the varieties, say $Y$, is rational, then the ring $k(X)\otimes _k k(Y)$ is a fraction ring $S^{-1}A$ of the polynomial ring $A=k(X)[T_1,...,T_n]$ over the field $k(X)$  and so is a UFD and thus has zero Picard group: $$Y\operatorname {rational}\implies      Pic(X\times Y) =Pic(X)\times Pic (Y)$$
This is a vast generalization of $Pic( X \times\mathbb A^1) =Pic(X)$.
A: 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).
A: If $X$ and $Y$ are two curves, then $\mbox{Pic}(X\times Y)\simeq\mbox{Pic}(X)\times\mbox{Pic}(Y)\times\mbox{Hom}(J_X,J_Y)$, where $J_X$ and $J_Y$ denote the jacobian varieties of $X$ and $Y$, respectively. In particular, for example, if $X$ and $Y$ are two isogenous elliptic curves, then $\mbox{Hom}(J_X,J_Y)\neq0$.
