Galois Orbit Bounds for Surface Degenerations

TL;DR

This paper establishes bounds on Galois orbits for points where surface degenerations cause Picard rank jumps, leading to finiteness results for special loci in families of K3 surfaces.

Contribution

It introduces a new technique for spreading out formal geometry and applies Hyodo-Kato theory to study Picard rank jumps in surface degenerations.

Findings

Picard rank jumps of 3 or more are finite in certain K3 surface families.

Galois-orbit height bounds apply to points with large Picard rank.

New methods for analyzing surface degenerations and monodromy at infinity.

Abstract

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.