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 $\eta \in S$ 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 $X_{s}$ is larger than the generic Picard rank. Assuming that the local system $\mathbb{V} = R^{2} g_{*} \mathbb{Z}$ admits a non-trivial monodromy logarithm $N$ at infinity, we give a general condition under which certain points of $\textrm{NL}(S)$ of unexpectedly large Picard rank satisfy a ``Galois-orbit'' height bound. This leads to the following result of Zilber-Pink type: Let $g : X \to S$ be a one-parameter family of polarized K3 surfaces admitting a non-trivial limit mixed Hodge structure and such that $S(\mathbb{C})$ contains a Hodge-generic point. Then the locus in $S(\mathbb{C})$ where the Picard rank jumps by $3$ or more is finite. Our arguments include a new technique for ``spreading out'' formal geometry, a study of the rigid geometry of equicharacteristic zero semistable surface degenerations, and use the model-free Hyodo-Kato theory of Colmez-Nizio\l.