  The Coble cubics were discovered more than a century ago in connection with
genus two Riemann surfaces and theta functions. They have attracted renewed
interest ever since. Recently, they were reinterpreted in terms of alternating
trivectors in nine variables. Exploring this relation further, we show how the
Hilbert scheme of pairs of points on an abelian surface, and also its Kummer
fourfold, a very remarkable hyper-K\"ahler manifold, can very naturally be
constructed in this context. Moreover, we explain how this perspective allows
us to describe the group law of an abelian surface, in a strikingly similar way
to how the group structure of a plane cubic can be defined in terms of its
intersection with lines.
