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?



Divisors on an abelian surface
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 $E\_1\times E\_2$.


