Surface constructed using curves

Question

Suppose that $E$ and $F$ are two complex compact Riemann surfaces with genus greater or equal than $2$.
Set $$S=E \times F$$ the surface composed by the cartesian product of thees curves. What can i say about the intersection number $E.F$ and the self intersection $E^2$ or $F^2$?

Answer 1

You can calculate E.F by taking curves representing the classes [E] and [F] which intersect transversely and counting their points of intersection.

In particular, pick points $e \in E$ and $f \in F$ and take $E \times \{f\}$ and $\{e\} \times F$. Clearly these only meet at the single point $(e, f)$, and moreover it's easy to calculate the tangent spaces and see that the intersection is transverse. So $E.F = 1$.

For $E.E$, take points $f, f' \in F$ and consider the curves $E \times \{f\}$ and $E \times \{f'\}$. These don't meet at all, so they surely meet transversely. Therefore $E^2 = 0$.