Let $S$ be a surface defined over a field $K$, when is the albanese map $S\longrightarrow \text{Alb}(S)$ an embedding?

For curves, for example, with genus at least 2, there is a morphism between the symmetric square to the Jacobian, and it is an embedding whenever the curve is not hyperelliptic. In particular, this means that the morphism into the albanese is an embedding. I am curious if there is an intrinsic criterion.