Separation of periods of quartic surfaces

TL;DR

This paper establishes a computable lower bound on the distance between distinct periods of quartic surfaces over algebraic numbers, aiding in the study of their Diophantine properties and Picard groups.

Contribution

It introduces a method to compute lower bounds on period distances of quartic surfaces, linking height bounds on Noether--Lefschetz loci to Diophantine analysis.

Findings

Derived explicit height bounds for Noether--Lefschetz loci components

Enabled certification of numerical computations of Picard groups

Provided a new approach for studying periods of quartic surfaces

Abstract

We give a computable lower bound on the distance between two distinct periods of a given quartic surface defined over the algebraic numbers. The main ingredient is the determination of height bounds on components of the Noether--Lefschetz loci. This makes it possible to study the Diophantine properties of periods of quartic surfaces and to certify a part of the numerical computation of their Picard groups.