I already asked this on Math.SE, but didn't receive an answer yet.

Say $E_1, \dotsc, E_n$ are elliptic curves (everything over $\mathbb C$), and $D \subset E_1 \times \dotsc \times E_n$ is an effective divisor. How can I determine the type $(d_1, \dotsc, d_n)$ of the line bundle $\mathcal O(D)$?

If $D = \sum_i \operatorname{pr}_i^* D_i$ for divisors $D_i \subset E_i$, then $d_i = \deg D_i = \deg (D \cdot \iota_i(E_i))$, where $\iota_i: E_i \to E_1 \times \dotsc \times E_n$ embeds $E_i$ by $z \mapsto (0, \dotsc, z, \dotsc, 0)$. Is it true in general that one gets $d_i$ as the degree of the intersection of $D$ with $\iota_i(E_i)$?

Concretely, I'm interested in the following two cases:

- Let $D_0 \subset E \times E$ be the union of the diagonal and the anti-diagonal,i.e. $$D_0 = \{(z,z) : z \in E\} \cup \{(z,-z):z\in E\}.$$
- Take the quotient $A = E \times E / \langle (\frac 1 2,\frac 1 2)\rangle$, by the $2$-torsion point $(\frac 1 2, \frac 1 2)$, and divisor $D_1 = \overline{D_0}$. Using the isomorphism $$E \times E \to E \times E, (z_1, z_2) \mapsto (z_1, z_1 - z_2)$$ one sees that $A$ is isomorphic to $E / \langle \frac 1 2 \rangle \times E$, so again a product of ellpitic curves.

Any help would be appreciated :)

typeof a divisor on $\prod_{i=1}^n E_i$? In other words, what is the definition of $d_i$? It is not true that $\operatorname{Pic}(\prod_i E_i) \cong \prod_i \operatorname{Pic}(E_i)$, and the same statement for $\operatorname{NS}$ also fails. $\endgroup$