MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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 $E_1\times E_2$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.