  A Keller map is a counterexample to the Jacobian Conjecture. In dimension two
every such map, if exists, leads to a complicated set of conditions on the map
between the Picard groups of suitable compactifications of the affine plane.
This is essentially a combinatorial problem. Some solutions to it, that we call
frameworks, are described and discussed. This second version of the paper
includes several more frameworks than the first version, including one
substantially more complicated framework.
