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Let $A$ be an abelian surface given by the quotient of a product of two generic elliptic curves $E_1 \times E_2$ by the product $T_1 \times T_2$ of two translations by $2$-torsion points. Then $A$ admits two natural projections to $E_i':=E_i/T_i$, and the fibers are $E_j'$ with $i \neq j$.

Is the group of divisors up to numerical equivalence generated by $E_1',E_2'$? Do they intersect in $2$ points?

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up vote 3 down vote accepted

The fiber of the projection to $E'_i$, say $F_i$, is isomorphic to $E_j$ ($j\neq i$), not to $E'_j$. We have indeed $(F_1.F_2)=2$, and $F_1,F_2$ generate the group of divisors on $A$ up to numerical equivalence, because $E_1$ and $E_2$ generate the analogous group on $E_1\times E_2$.

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