Divisors of self-intersection 2 on an Abelian surface

Question

I've came across a statement that says: If $A$ is an Abelian surface and $D$ is a positive divisor on $A$ with self-intersection number $2$, then either $D$ is a curve of genus $2$, or the sum of two positive divisors.

(1) Could anyone give a reference of this result that does not use analytic techniques? A sketch of a proof would be even better!

(2) Is there an analog of this result in higher dimension?

Thanks!

Answer 1

If $D$ is irreducible, then by adjunction formula you get $2=D^2=(K_A+D)D=2(p_a(D)-2)$, since $K_A$ is trivial. Therefore, $D$ is a curve of (arithmetic) genus $p_a(D)=2$.

If $D$ is not irreducible, well it is sum of 2 (or more) irreducible curves.

For a reference you can check the book "Compact Complex Surfaces" of Barth, Hulek, Peters and Van De Ven.

I have no idea about (2).