Galois Orbit Bounds for Surface Degenerations

Given a smooth proper family $g : X \to S$ of surfaces over a number field $K \subset \mathbb{C}$, with $S$ an irreducible curve and $η\in S$ its generic point, we consider the general problem of constraining the locus $\textrm{NL}(S)$ in $S(\overline{K})$ of points $s$ where the Picard rank of $X_{