Let $k$ be a field with separable algebraic closure $k^{\rm s}$ and corresponding absolute Galois group $\varGamma={\rm Gal}(k^{\rm s}/\,k)$ and let $X$ and $Y$ be geometrically connected and geometrically reduced $k$-schemes locally of finite type.

Question.Are there examples where the canonical map $(\,{\rm Pic}\,X^{\rm s})^{\varGamma}\oplus(\,{\rm Pic}\,Y^{\rm s})^{\varGamma}\to{\rm Pic}\,(X^{s}\times_{\,k^{\rm s}}Y^{\rm s})^{\varGamma}$ is an isomorphism but the map ${\rm Pic}\,X^{\rm s}\oplus{\rm Pic}\,Y^{\rm s}\hookrightarrow{\rm Pic}\,(X^{s}\times_{k^{\rm s}}Y^{\rm s})$ is not?

It is known that, if $X$ and $Y$ are smooth, projective, geometrically irreducible and of finite type over $k$, then the cokernel of the latter map is naturally isomorphic to ${\rm Hom}({\rm Pic}_{ Y^{\rm s}}^{\vee},{\rm Pic}_{X^{\rm s}})$, where ${\rm Pic}_{X^{\rm s}}$ is the Picard variety of $X^{\rm s}$ (i.e., the largest reduced subscheme of the identity component of the Picard scheme of $X^{\rm s}$ over $k^{\rm s}$) and ${\rm Pic}_{Y^{\rm s}}^{\vee}$ is the dual of ${\rm Pic}_{Y^{\rm s}}$. Thus, while looking for such examples, one should assume that ${\rm Pic}_{Z}\neq 0$ for $Z=X^{s}$ and $Y^{s}$, in particular that $H^{1}(Z,\mathcal O_{Z})\neq 0$ for such $Z$ (a condition which excludes the rational varieties).

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