On non Fundamental Group Equivalent Surfaces

TL;DR

This paper constructs examples of polarized K3 surfaces that are not fundamental group equivalent but have isomorphic Galois cover fundamental groups, using braid monodromy and degenerations to distinguish their topological properties.

Contribution

It provides explicit examples of K3 surfaces with non-isomorphic complement fundamental groups but isomorphic Galois cover groups, and introduces methods to distinguish their braid monodromy types.

Findings

The two K3 surfaces are not FGE equivalent.

Their Galois cover fundamental groups are isomorphic.

They are not projectively deformable into each other.

Abstract

In this paper we present an example of two polarized K3 surfaces which are not Fundamental Group Equivalent (their fundamental groups of the complement of the branch curves are not isomorphic; denoted by FGE) but the fundamental groups of their related Galois covers are isomorphic. For each surface, we consider a generic projection to CP^2 and a degenerations of the surface into a union of planes - the "pillow" degeneration for the non-prime surface and the "magician" degeneration for the prime surface. We compute the Braid Monodromy Factorization (BMF) of the branch curve of each projected surface, using the related degenerations. By these factorizations, we compute the above fundamental groups. It is known that the two surfaces are not in the same component of the Hilbert scheme of linearly embedded K3 surfaces. Here we prove that furthermore they are not FGE equivalent, and thus they are not of the same Braid Monodromy Type (BMT) (which implies that they are not a projective deformation of each other