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
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).
Keywords
fundamental group, generic projection, curves and
singularities, branch curve
