Zariski-van Kampen method and transcendental lattices of certain singular K3 surfaces

TL;DR

This paper introduces a Zariski-van Kampen type method to compute transcendental lattices of complex projective surfaces, demonstrated on singular K3 surfaces linked to specific sextics, revealing their arithmetic and geometric properties.

Contribution

The paper develops a new computational method for transcendental lattices and applies it to singular K3 surfaces associated with particular sextics, highlighting their arithmetic conjugacy.

Findings

Calculated transcendental lattices of specific singular K3 surfaces

Identified conjugate surfaces over a quadratic field

Demonstrated the method's effectiveness on complex algebraic surfaces

Abstract

We present a method of Zariski-van Kampen type for the calculation of the transcendental lattice of a complex projective surface. As an application, we calculate the transcendental lattices of complex singular K3 surfaces associated with an arithmetic Zariski pair of maximizing sextics of type $A_{10}+A_{9}$ that are defined over a real quadratic field and are conjugate to each other over the field of rational numbers.