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Scholl's expository paper "Classical Motives" cites Weil's "Sur les courbes algebriques et les varieties qui s'en deduisent," which I have no access to, for the following result:

If $X$ and $X'$ are smooth projective curves with Jacobian varieties $J,J'$, then $A^1(X\times X') = A^1(X)\bigoplus A^1(X')\bigoplus \operatorname{Hom}(J,J')\bigotimes \mathbb{Q}.$

$A^1(X)$ and $A^1(X')$ arise from the pullbacks of the projection maps, but I have no clue where the $\operatorname{Hom}(J,J')$ comes from. My first guess would be that it comes from some universal property of the Jacobian, but the only one I'm familiar with is that the Jacobian is a coarse moduli space for degree 0 line bundles, which I can't see how to use here.

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If $C$ is an irreducible curve in $X\times X'$. COnsider its projections $p_1, p_2$ to $X$ and $X'$ respectively. If one of $p_i(C)$ is one point $x_i$, then $C$ is "vertical" and is determined by $x_i$. Otherwise, $p_1, p_2$ are both surjective, and you can consider the effect of ${p_2}_{*}p_1^*$ on the divisors of $X_1$. –  user18119 Nov 22 '12 at 21:54

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