Most people will have see a geometric "explanation" of the addition law on elliptic curves given by embedding it as a cubic in the projective plane and cutting it with lines.

Is there a similar explicit, geometric definition of the addition law on (a family of?) abelian surfaces?

So the question is really: Give a nice embedding of abelian surfaces into projective space and then define the addition law using this embedding - if not for all abelian surfaces, at least for some non trivial family. In fact, it would be really nice if we could do this for the embedding that realizes the surface as a degree 10 variety using the Horrocks-Mumford bundle.