Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Galois Orbit Bounds for Surface Degenerations

David Urbanik

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 $η\in S$ 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 $X_{s}$ 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-Nizioł.

Galois Orbit Bounds for Surface Degenerations

Abstract

Given a smooth proper family of surfaces over a number field , with an irreducible curve and its generic point, we consider the general problem of constraining the locus in of points where the Picard rank of is larger than the generic Picard rank. Assuming that the local system admits a non-trivial monodromy logarithm at infinity, we give a general condition under which certain points of of unexpectedly large Picard rank satisfy a ``Galois-orbit'' height bound. This leads to the following result of Zilber-Pink type: Let be a one-parameter family of polarized K3 surfaces admitting a non-trivial limit mixed Hodge structure and such that contains a Hodge-generic point. Then the locus in where the Picard rank jumps by 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-Nizioł.
Paper Structure (18 sections, 33 theorems, 48 equations)

This paper contains 18 sections, 33 theorems, 48 equations.

Table of Contents

  1. Motivation
  2. Heights and Galois Orbits
  3. $G$-functions
  4. The Role of Rigid Geometry
  5. Results
  6. Techniques
  7. Conventions
  8. Fields and Analytification
  9. Adelic Geometry
  10. Polynomial Algebra Norms
  11. Adelic Tubes
  12. Parameterizing Analytic Tubes
  13. Flat Deformations from $E$
  14. Relation to Derivatives:
  15. Deduction of Norm Estimates:
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose that $k = 3$ and representation of $\pi_{1}(S,s)$ on $V$ is $\mathbb{Q}$-simple. Then there exists $a, b > 0$ such that for all $s \in S(\overline{K})$ such that

Theorems & Definitions (74)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Remark
  • proof
  • Theorem 5
  • Theorem 6
  • Lemma 1.1
  • proof
  • ...and 64 more