Given a family g : X -> S of smooth projective algebraic varieties over a number field K, one often wants to constrain the points s in S where the fibre X_s acquires "extra" algebraic structure. A basic sort of constraint which is important in unlikely intersection theory is that of a Galois-orbit lower bound: an inequality h(s) <= poly([K(s) : K]), where h is some logarithmic Weil height and K(s) is the field of definition of s.

Recent work has focused on how to use G-functions constructed from degenerations of g to produce such inequalities.

We describe some new results in the case where g is a one-parameter degeneration of surfaces, and the central role played by rigid and "adelic" geometry.